ERGODIC PROPERTIES OF ERD�OS MEASURE, THE ENTROPYOF THE GOLDENSHIFT, AND RELATED PROBLEMSNikita SIDOROV and Anatoly VERSHIKDecember 2, 1997To the memory of Paul Erd�osAbstract. We de�ne a two-sided analog of Erd�os measure on the space of two-sided expan-sions with respect to the powers of the golden ratio, or, equivalently, the Erd�os measure on the2-torus. We construct the transformation (goldenshift) preserving both Erd�os and Lebesguemeasures on T2 which is the induced automorphism with respect to the ordinary shift (or thecorresponding Fibonacci toral automorphism) and proves to be Bernoulli with respect to bothmeasures in question. This provides a direct way to obtain formulas for the entropy dimensionof the Erd�os measure on the interval, its entropy in the sense of Garsia-Alexander-Zagier andsome other results. Besides, we study central measures on the Fibonacci graph, the dynamicsof expansions and related questions.0. IntroductionAmong numerous connections between ergodic theory and the metric theory of numbers,the questions related to algebraic irrationals, expansions associated with them and ergodicproperties of related dynamical systems, are of special interest. The simplest case, i.e. thegolden ratio, the Fibonacci automorphism etc., has served as a deep source of problemsand conjectures.In 1939 P. Erd�os [Er] proved in particular the singularity of the measure on the seg-ment which is de�ned as the one corresponding to the distribution of the random variableP11 "k��k, with � being the larger golden ratio and "k independently taking the values0 and 1 (or �1) with probabilities 12 each. We think it is natural to call this measure theErd�os measure. This work gave rise to many publications and numerous generalizations(see, e.g., [AlZa] and references therein). Nevertheless, little attention was paid to theSupported in part by the INTAS grant 93-0570 and RFBR grant 96-0100676. The �rst author wassupported by the French foundation PRO MATHEMATICA. The �rst author expresses his gratitude tol'Institut de Math�ematiques de Luminy for support during his stay in Marseille in 1996-97. The secondauthor is grateful to the University of Stony Brook for support during his visit in February-March 1996and to the Institute for Advanced studies of Hebrew University for support during his being there in 1997.The previous version of this paper appeared as the Stony Brook preprint IMS 96-14.1

2 NIKITA SIDOROV AND ANATOLY VERSHIKdynamical properties of this natural measure. The aim of this paper is to begin studyingdynamical properties of Erd�os measure and its two-sided extension. We(1) de�ne the two-sided generalization of Erd�os measure (Section 1);(2) introduce a special automorphism (\goldenshift") which preserves Erd�os measureand which is a Bernoulli automorphism with a natural generator with respect toErd�os and Lebesgue measures (Section 2);(3) compute the entropy of this automorphism and prove that it is related to thepointwise dimension of the Erd�os measure de�ned in [Y] and Garsia's entropyconsidered in [AlZa] (Section 3);(4) discuss the connections with some properties of the Fibonacci graph, its centralmeasures and the adic transformation on it (see Appendix A);(5) de�ne a new kind of expansions corresponding to the goldenshift (see Appendix B).We will describe all this in more detail below.Several years ago certain connections between symbolic dynamics of toral automor-phisms and arithmetic expansions associated with their eigenvalues were established. The�rst step in this direction was also related to the golden ratio (see [Ver5]) and led to anatural description of the Markov partition in terms of the arithmetic of the 2-torus andhomoclinic points of the Fibonacci automorphism. The main idea was to consider thenatural extension of the shift in the sense of ergodic theory and the adic transformation onthe space of one-sided arithmetic expansions and to identify the set of two-sided expansionswith the 2-torus (see also [Ver6], [Ver3], [KenVer]).In the present paper we use the same idea for a detailed study of the Erd�os measure.Namely, we de�ne the two-sided Erd�os measure as a measure on the space of expansionsin�nite to both sides and identify it with a measure on the 2-torus; in the same way,Lebesgue measure on the 2-torus can be considered as a two-sided version of the Markovinvariant measure on the corresponding Markov compactum { see below and [Ver5], [Ver6].We study the properties of the ordinary shift and the goldenshift as a transformation on thespace of expansions introduced by means of the notion of block. The goldenshift turns outto preserve both Lebesgue and Erd�os measures, both being Bernoulli in the natural sensewith respect to the goldenshift; this is one of the main results of the paper (Theorem 2.7).By the way, this immediately yields a proof of the Erd�os theorem on the singularity ofErd�os measure. Moreover, the two-sided goldenshift is an induced automorphism for theFibonacci automorphism of the torus. Other important consequences of our approachfollow from the fact that the entropy of the goldenshift is directly related to the entropyof Erd�os measure in the sense of Garsia and Alexander-Zagier, i.e. to the entropy of therandom walk with equal transition measures on the Fibonacci graph (Theorem 3.3).In [AlZa] it was attempted to compute the entropy of Erd�os measure as the in�niteconvolution of discrete measures, which was in fact introduced by A. Garsia [Ga] in a moregeneral situation. Note that it proved to be the entropy of a random walk on the Fibonaccigraph. In [LePo] the authors compute the dimension of the Erd�os measure on the intervalin the sense of L.-S. Young [Y] and relate a certain two-dimensional dynamics to it.Finally, making use of a version of Shannon's theorem for random walks (see [KaVer])yields the value of the dimension of the Erd�os measure in the sense of Young (Theorem 3.7).

ERD�OS MEASURE AND THE GOLDENSHIFT 3Thus, the dynamical viewpoint for arithmetic expansions and for measures related tothem provides new information and an essential simpli�cation of computations of theinvariants invloved. One may expect that the methods of this paper will apply to moregeneral algebraic irrationals and also to some nonstationary problems.The contents of the present paper are as follows. In Section 1 we present auxiliarynotions (canonical expansions and others) and give the main de�nitions (Erd�os measureon the interval and the 2-torus, normalization, the Markov measure corresponding toLebesgue measure etc.). In addition, we deduce some preliminary facts about the one-sided and two-sided Erd�os measures. In Section 2 we study the combinatorics of blocks interms of canonical expansions, introduce the notion of the goldenshift (both one-sided andtwo-sided) and prove its Bernoulli property with respect to Lebesgue and Erd�os measures.Section 3 contains our main results on the entropy and dimension of the Erd�os measureand their relationships to the random walk on the Fibonacci graph.In the appendices we consider some related problems. Namely, in Appendix A thecombinatorial and algebraic theory of the Fibonacci graph is presented. In particular, wedescribe the ergodic central measures on this graph and the action of the adic transfor-mation, which is de�ned as the transfer to the immediate successor in the sense of thenatural lexicographic order (in our case it is just the next expansion of a given real in thesense of the natural ordering of the expansions). We study the metric type of the adictransformation with respect to the ergodic central measures. In Appendix B we considerarithmetic block expansions of almost all points of the interval. The interest in them is dueto the fact that the \digits" of the block expansions are independent with respect bothto Erd�os and Lebesgue measures. Note that there are some pecularities caused by thedi�erence between the one-sided and two-sided shifts. For instance, the one-sided Erd�osmeasure is only quasi-invariant under the one-sided shift, while the two-sided measure isshift-invariant. In Appendix C the densities of the Erd�os measure with respect to theshift and to the rotation by the golden ratio are computed by means of blocks. Finally,in Appendix D another proof of Alexander-Zagier's formula for the entropy is given. It isworthwhile because of its connection with the geometry of the Fibonacci graph.The authors express their gratitude to B. Solomyak for the fruiful discussions and tothe referees for their helpful remarks and suggestions.1. Erd�os measure on the interval and on the 2-torus.1.1. Canonical expansions. Let � = Q11 f0; 1g endowed with the one-sided Bernoullishift � and let X � � be the stationary Markov compactum with the transition matrix� 1 11 0�, i.e. the set X = f("1"2 : : : ) 2 � : "i"i+1 = 0; i � 1g endowed with the topology ofpointwise convergence. Let next � = p5+12 and L : X ! [0; 1] be the mapping acting bythe formula(1.1) L("1"2 : : : ) := 1Xk=1 "k��k:

4 NIKITA SIDOROV AND ANATOLY VERSHIKIt is well known that L is one-to-one, except for a countable number of sequences whosetail is of the form 01 or (01)1. The inverse mapping L�1 is speci�ed with the help of thegreedy algorithm. Namely, let Tx = f�xg and"k = [�T k�1x]; k � 1:We call the constructed sequence ("1(x)"2(x) : : : ) the canonical expansion of x. Notethat usually the canonical expansions are called �-expansions (for � = �). They wereintroduced in [Re] and [Ge] and thoroughly studied in [Pa].Note that the mapping L can be naturally extended to �, and let L0 stand for thisextension. However, L0(�) = [0; �], that is why we de�ne the projection � : � ! [0; 1]by the formula �(x) := ��1L0(x). It will be frequently used below. Note that � is notone-to-one.1.2. The Markov measure on X. The transformation T : (0; 1) ! (0; 1) is transferredby L to the Markov compactum X and acts as the one-sided shift � :� ("1"2"3 : : : ) := "2"3 : : :Thus, we have � = L�1TL. The transformation T has been thoroughly studied, and itwas shown that there exists the T -invariant measure m0 equivalent to Lebesgue measurem1. Its density is given by the formula�(x) = dm0dm1 = � �2=p5; 0 < x � ��1�=p5; ��1 < x � 1(see, e.g, [Ge], [Pa]). The corresponding Markov measure L�1m0 on X is the one with thestationary initial distribution � �=p5��1=p5� and the transition probability matrix � ��1 ��21 0 �.The L-preimage of the Lebesgue measure m on X di�ers from this stationary Markovmeasure only by its initial distribution � ��1��2 �. Note that for the adic transformation onX (for the de�nition see [Ver2] or Appendix A) with the alternating ordering on the pathsthe latter measure is unique invariant, as this adic transformation turns into the rotationby the angle ��1 under the mapping L (for more details see [VerSi]).1.3. Erd�os measure and normalization. Let us de�ne the Erd�os measure. By def-inition, the continuous Erd�os measure � on the unit interval is the in�nite convolution#1 � #2 � : : : , where supp#n = f0; ��n�1g, and #n(0) = #n(��n�1) = 12 (see [Er]).We are going to specify this measure more explicitly. Let p denote the product measurewith the equal multipliers �12 ; 12� on the compactum �. Then it is easy to see that � = �(p).We are also interested in the speci�cation of the Erd�os measure on the Markov com-pactum X. Of course, it is just L�1�; however, it is worthwhile to introduce a directmapping.

ERD�OS MEASURE AND THE GOLDENSHIFT 5De�nition. Let x 2 �; x = fxkg1k=1; we de�ne [0; 1] 3 c(x) = P1k=1 xk��k�1 =P1k=1 "k��k, where f"kg is the canonical expansion of c(x). We de�nen(x) := " = f"kg1k=1:The mapping n : �! X is called normalization.We will also describe the mapping n directly avoiding the expansion of a number from[0; 1]. Namely, let x = (x1; x2; : : : ) 2 �; we put x0 = 0 and look for the �rst occurrence ofthe triple 011, after which we replace it by 100. The next step is the same, i.e. we returnto the zero coordinate and start from there until we meet again 011, etc. It is easy to seethat the process leads to stabilization at a normalized sequence. Note that this algorithmis rather rough, as it is known that there exists a �nite automaton carrying out the processof normalization faster (see, e.g., [Fr]). It is obvious that this de�nition is equivalent tothe one given above.Now we can de�ne Erd�os measure also on X as the image of the product measurep =Q11 f 12 ; 12g: � = n(p)(we preserve the same notation for X as for [0; 1]).Below we will see that this de�nition of the Erd�os measure is not very suitable fordeducing its dynamical properties (the quasi-invariance under T and the rotation by thegolden ratio, etc.); we will give another de�nition related to the two-sided theory.We begin with its self-similar property which is typical for this measure and completelycharacterizes it.Lemma 1.1. The Erd�os measure � on the interval [0; 1] satis�es the following self-similarrelation: �E =8><>: 12�(�E); E � [0; ��2)12 (�(�E) + �(�E � ��1)); E � [��2; ��1)12�(�E � ��1); E � [��1; 1]for any Borel set E.Proof. Let F1 = 1; F2 = 2; : : : be the sequence of Fibonacci numbers. Let fn(k) denotethe number of representations of a nonnegative integer k as a sum of not more than n �rstFibonacci numbers. We �rst show that for n � 3,(1.2) fn(k) = 8><>: fn�1(k); 0 � k � Fn � 1fn�1(k) + fn�1(k � Fn); Fn � k � Fn+1 � 2fn�1(k � Fn); Fn+1 � 1 � k � Fn+2 � 2:To prove this, we represent fn(k) as fn(k) = f 0n(k) + f 00n (k) for each k < Fn+2 � 1, wheref 0n(k) is the number of representations with "n = 0, and f 00n (k) is the number with "n = 1.Obviously, if k � Fn � 1, then k = Pn1 "jFj = Pn�11 "jFj , whence fn(k) = f 0n(k). IfFn+1 � 1 � k � Fn+2 � 2, then fn(k) = f 00n (k). In the case Fn � k � Fn+1 � 2,

6 NIKITA SIDOROV AND ANATOLY VERSHIKobviously, f 0n(k) > 0; f 00n (k) > 0. It remains to note only that f 0n(k) = fn�1(k), andf 00n (k) = fn�1(k � Fn).Now from (1.2), and from the de�nition of the Erd�os measure it follows that if, say, aninterval E � [0; ��2), then�E = limn!1 Xk: kFn+22E fn(k)2n = 12 limn!1 Xk: kFn+12�E fn�1(k)2n�1 = 12�(�E):The other cases are studied in the same way. �Remark. The Erd�os measure � (as a Borel measure) is completely determined by the aboveself-similar relation. Indeed, by induction one can determine its values for any interval (a; b)with a; b 2 (Z+ �Z)\ [0; 1].Corollary 1.2. �(0; ��2) = �(��2; ��1) = �(��1; 1) = 13 .The next step consists in introducing a two-sided analog of the Erd�os measure, whichwill lead to the one-sided shift-invariant measure equivalent to �.1.4. Two-sided theory. Consider the two-sided space e� = Q1�1f0; 1g and its subsetthe two-sided Markov compactum eX = ff"kg1�1 : "k 2 f0; 1g; "k"k+1 = 0; k 2 Zg.1 Tode�ne the two-sided Erd�os measure on eX , we are going to construct a two-sided analogof normalization. Furthermore, we will use an arithmetic mapping from eX onto T2 whichsemiconjugates the two-sided Markov shift and the Fibonacci automorphism of the torusin order to specify Erd�os measure on the 2-torus and to study its properties (see item 1.6).Let e� denote the two-sided shift on e�, i.e. (e�x)k = xk+1, and let e� stand for thetwo-sided shift on the Markov compactum eX . We denote by em the stationary two-sidedMarkov measure on the compactum eX with the invariant initial distribution � �=p5��1=p5 � andthe transition probability matrix � ��1 ��21 0 �. As is well known, em is the unique measureof maximal entropy for the shift e� .There is an important action ofZ2 =Z+Zon eX. Namely, let w0; w1 be the generators,i.e. Z2 = fnw0 +mw1 j n;m 2 Zg. Let us describe the action Ag : eX ! eX; g 2 Z2.First, Aw1 = e��1Aw0e� , and Aw0 is addition by 1 in the sense of the arithmetic of eX. Moreprecisely, considering eX as the set of formal series fP1�1 "nwn j f"ng 2 eXg, we de�newn + wn+1 = wn�1, and 2wn = wn�1 + wn+2 (implying the representation wn $ ��n),whence the operation f"ng 7! f"ng + w0 is well de�ned for em-a.e. f"ng 2 eX, as well asthe sum of a.e. pair of sequences (see [Ver5], [Ver3]).1Henceforth the sign tilde will always stand for the two-sided objects.

ERD�OS MEASURE AND THE GOLDENSHIFT 7Proposition. (see [Ver5], [Ver3]). The measure em is the unique Borel measure invariantunder the action of Z2 decribed above.Remark. We de�ne the following identi�cation of some pairs of points in eX whose measureem� em is 0 in order to turn eX into the additive group. Let the equivalence relation � bede�ned as follows: (�1000000 : : : ) � (�010101 : : : );(: : : 01010100�) � (: : : 10101001�);where � denotes an arbitrary (but the same for both sequences) tail starting at the sameterm. Besides, extending the equivalence relation � by continuity, we get (0:1)1 �(1:0)1 � 01 (henceforth the point will mark the border between the negative and non-negative coordinates of a sequence). Now one can easily check that the set eX 0 = eX= � isa compact connected group in addition (see [Ver5], [Ver3]).Example. Here is an example of subtraction in eX: �� = �2 + �4 + �6 + : : : = 1 + �3 +�5 + �7 + : : : , both sequences representing one and the same element of eX 0. Similarly, forany sequence " 2 eX �nite to both sides, �" is a pair of sequences �nite to the right andco�nite to the left, i.e. with the left tail (01)1.The purpose for the de�nition of the operation of addition will be explained in item 1.6,where the automorphism ( eX; em; e� ) will be related to the 2-torus.Following the one-sided framework, we are going to de�ne the two-sided generalizationof the operation of one-sided normalization. Namely, we de�ne the two-sided normalizationen as the mapping from e� to eX. Consider the set e�0 � e� de�ned as follows:e�0 = nx 2 e� : #fk < 0 : xk = 0g =1o :On the set e�0 we will de�ne two-sided normalization. Let x 2 e�0 and 0 � k1 > k2 >: : : ; xki = 0; xk 6= 0; k 6= ki for all i. We set x(0) = x. Let n�fx(0)i g1k1+1� = f"(1)i g1i=k1 :We de�ne x(1)i = ( "(1)i i � k1xi i < k1:By induction, let n�fx(n�1)i g1kn+1� = f"(n)i g1i=kn : Then, by de�nition,x(n)i = ( "(n)i i � knxi i < kn:Obviously, the process leads to the stabilization of x(n)i in n.

8 NIKITA SIDOROV AND ANATOLY VERSHIKDe�nition. The two-sided normalization en at x 2 e�0 is de�ned as follows:en(fxig1�1) = limn!1fx(n)i g:De�nition. The two-sided Erd�os measure e� on the Markov compactum eX is the imageunder the mapping en of the measure ep, which is the product of in�nite factors �12 ; 12� onthe full compactum e�.Since the set e�0 has full measure ep, we have the homomorphism of the measure spaces:en : (e�; ep)! ( eX; e�):Let now � : e�! �; �(fxng) = (x1; x2; : : : );�0 : eX ! X; �(f"ng) = ("0; "1; : : : )be the projections.Let us write the following diagram:� � ���� � � ���� e� e� ���� e�not commutes not commutes commutesn??y n??y en??y en??yX � ���� X �0 ����� eX e� ���� eXNote that �0en 6= n�; �n 6= n�:This is the reason why the one-sided theory has some di�culties. The two-sided theory ismore natural for this purpose (the right part of the diagram does commute).Propostion 1.3. The two-sided Erd�os measure e� is invariant under the two-sided shift,i.e. e�e� = e� (cf. the one-sided case, where it does not take place).Proof. From the above speci�cation of the mapping en it follows that(1.3) ene� = e�en;hence e�(e��1E) = ep(en�1e��1E) = ep(e��1en�1E) = (e�ep)(en�1E) = ep(en�1E) = e�(E) for anyBorel set E � eX.Proposition 1.4. For any cylinder eC = ("1 = i1; : : : ; "r = ir) � eX, its measure e� isstrictly positive.Proof. It follows from the direct speci�cation of the two-sided normalization describedabove that for the cylinder C 0 = ("0 = 0; "1 = i1; : : : ; "r = ir ; "r+1 = 0; "r+2 = 0) � e�, wehave en�1( eC) � C 0, whence, by de�nition of the Erd�os measure, e�( eC) � 2�r�3.

ERD�OS MEASURE AND THE GOLDENSHIFT 9Proposition 1.5. The two-sided shift e� on the Markov compactum eX with the two-sidedErd�os measure is a Bernoulli automorphism.Proof. We observe that this dynamical system is a factor of the Bernoulli shift e� : e�! e� with the product measure � 12 ; 12� (see relation (1.3)) and apply the theorem due toD. Ornstein [Or] on the Bernoullicity of all Bernoulli factors.Remark. Note that the measure e� is not Markov on the compactum eX. It would beinteresting to prove that the two-sided Erd�os measure is a Gibbs measure for a certainnatural potential.1.5. Dynamical properties of Erd�os measure. Let the measure � on the one-sidedcompactum X be de�ned as the projection of the two-sided Erd�os measure e�. In otherwords, the dynamical system ( eX; e�; e� ) is the natural extension of (X; �; � ). We recall thatit means by de�nition that for any cylinder C = ("1 = i1; : : : ; "k = ik) � X its measure �equals e�( eC), where eC = ("1 = i1; : : : ; "k = ik) � eX. Below we will denote the L-image of� on the interval [0; 1] by the same letter.Proposition 1.6. The measure � is � -invariant and ergodic.Proof. The � -invariance of � is a consequence of the e� -invariance of e�. Furthermore, sincethe automorphism ( eX; e�; e� ) is Bernoulli, it is ergodic, thus, the endomorphism (X; �; � ) isalso ergodic.We are going to establish a relation between the mappings �0en and n� in order to provethe equivalence of the measures � and � on the interval. Note �rst that in these terms� = (n�)(ep), and � = (�0en)(ep). We also note that the fact that � � � is shown easily, whilein the opposite direction it is not straightforward. The following claims yield the sameproofs for both sides.Lemma 1.7. There exists a subset of e� of full measure ep and its countable partition intothe sets fEkg1k=1 and the corresponding set ffkg1k=1 of �nite sequences in e� such that(1.4) (�0en)(x) = (n�)(x + fk); x 2 Ekwith group addition in the set e� = Q1�1Z=2. Similarly, there exists a ep-a.e. partition ofe� into the sets fDkg1k=1 and the corresponding set fgkg1k=1 of �nite sequences in e� suchthat(1.5) (n�)(x) = (�0en)(x + gk); x 2 Dk:Remark. The actions x 7! x + fk; x 7! x+ gk change �nitely many coordinates of x. Weneed only this property.Proof. Both assertions are proved in the same way. Let us prove the �rst one. The ideaof the proof is based on the fact that a ep-typical sequence from e� can be splitted into�nite pieces so that its normalization splits into the concatenation of the correspondingnormalizations.

10 NIKITA SIDOROV AND ANATOLY VERSHIKWe assume that x has two successive zero coordinates with negative indices and foursuccessive zero coordinates with positive indices. So, let x = (A000A10000A2), where A0and A2 are in�nite and A1 is a �nite fragment of x containing x0 and x1. Then en(x) is theconcatenation of the normalizations of its pieces A00; 0A1000 and 0A2, whence(1.6) (�0en)(x) = (C0n(0A2))with a certain �nite admissible word C ending with two zeroes. We have a countablenumber of possibilities for C. Let Ek := EC be de�ned as the set of x 2 e�0 such thatrelation (1.6) holds with some A2.To construct fk, we consider two cases. If C begins with 0, i.e. if C = 0C 0, then weset x0 := x + fk = (A00 : : : 0 j C 000A2), where \j" denotes the border between positiveand nonpositive coordinates. For such an x0 relation (1.4) is satis�ed. If C begins with1, i.e. if C = (10)j0C 0 for some j � 1 and admissible C 0, then we set x0 := (A00 : : : 0 j10(11)j�1C 000A2). The proof is complete.Relations (1.4) and (1.5) together yield the main assertion.Proposition 1.8. The measures � and � are equivalent.Proof. Consider the group which acts on e� by adding the �nite sequences (= by changinga �nite number of coordinates). Since the action of this group preserves the measure ep,we obtain from relation (1.4) (�0en)(ep) � (n�)(ep), and from relation (1.5) we get (n�)(ep) �(�0en)(ep), whence � � �, and � � �. �Remark. It is possible to show that there exist two positive constants C1 and C2 such thatC1�(E) � �(E) � �C2�(E) log�(E)for any Borel E. Note that the right estimate is attainable, for instance, at the sequenceof sets E = En = (0; ��n), as by Lemma 1.1 and Corollary 1.2, �(0; ��n) = 43 � 2�n, while�(0; ��n) � n2�n (see the proof of Proposition 1.10 below). However, 0 is the only pointof the interval [0; 1], where the density d�d� is unbounded (see Appendix C).Let R denote the rotation of the circle R=Zby the angle ��1.Corollary 1.9. The Erd�os measure � is quasi-invariant ergodic with respect to T and R.Proof. The �rst claim follows directly from the equivalence of the measures � and �. Toprove the second one, we note that by Lemma 1.1,(1.7) �(T�1E) = 12��E + �(E + ��2 mod 1)�;whence follows the required assertion.We recall the following well-known claim which is a corollary of the individual ergodictheorem. Namely, two Borel measures invariant and ergodic with respect to one and thesame transformation of a metric space, either coincide or are mutually singular.Now we can present a new (dynamical) proof of the Erd�os theorem on the singularityof the measure �.

ERD�OS MEASURE AND THE GOLDENSHIFT 11Proposition 1.10. (Erd�os theorem, see [Er])The Erd�os measure � is singular with respectto Lebesgue measure m.Proof. We proved that T� = �, and above it was noted that Tm0 = m0 (see item 1.2),hence by the corollary of the ergodic theorem, either � ? m0 or � = m0. Indeed, we canapply it, because � is ergodic by Proposition 1.6, and and the ergodicity of m0 is a classicalfact (see, e.g., [Ge], [Re]). To show that � 6= m0, we observe that m0(0; ��n) � ��n,while �(0; ��n) = e�("1 = "2 = � � � = "n = 0) = O(n2�n), because if for a sequencex = fxng 2 e�; en(x) 2 ("1 = "2 = � � � = "n = 0), then either xi = 0; 1 � i � n, orxi = 0; k + 1 � i � n, and xk = 1; xk�1 = 1; xk�2 = 0; xk�3 = 1; xk�4 = 0, etc. for somek. So, � ? m0, hence, � ?m1, as m0 �m1; � � �. �Remark 1. Note that the initial proof of Erd�os followed the traditions of those times andwas based on the study of the Fourier transform of �.Remark 2. The present proof �lls a gap in the proof of this statement in the previousjoint paper by the authors [VerSi]. Another dynamical proof of Erd�os theorem is given inCorollary 2.8 (see below).Remark 3. The problem of computing the densities d�d� ; d(R�)d� and d(T�)d� will be solved inAppendix C. Note that all these densities prove to be piecewise constant and unbounded.We recall that em is the Markov measure on eX with maximal entropy (see item 1.4)Proposition 1.11. The two-sided Erd�os measure e� is singular with respect to the Markovmeasure em.Proof. We again apply the corollary of the ergodic theorem to the transformation e� andthe measures em and e� on the two-sided Markov compactum. The distinction of the twomeasures is a consequence of the noncoincidence of their one-sided restrictions (see Propo-sition 1.10), therefore, em ? e�.At the end of the item we prove a claim which we will need in the next item. Recallthat eX has an additive structure (item 1.4).Proposition 1.12. The Erd�os measure e� is invariant under the transformation i : f"ng 7!�f"ng.Proof. Note that i(f"ng) = en(f"0ng), where f"0ng 2 e�, and "0n = 1� "n. Therefore, for anyBorel E � eX and any sequence fxng from the set en�1(�E) there exists a unique sequencefx0ng 2 en�1(E) such that x0n = 1� xn. Now the claim of the proposition follows from thede�nition of e� and the symmetricity of the measure ep on e�.

12 NIKITA SIDOROV AND ANATOLY VERSHIK1.6. Erd�os measure on the 2-torus. There exists an important smooth interpretationof the two-sided theory. It is related to a general arithmetic approach to the coding ofthe hyperbolic automorphisms of the torus. Here we will give only some de�nitions andprimary claims whose aim is to describe a two-sided analog of the Erd�os measure. Somenecessary bibliographic references will be given at the end of the item.Consider the Fibonacci automorphism eT of the 2-torus, i.e. the automorphism with thematrix � 1 11 0�. Later it will be clear that this automorphism can be considered as a naturalextension of the endomorphism Tx = f�xg of the interval.There exists a natural way to de�ne a mapping semiconjugating the shift e� on theMarkov compactum eX and the Fibonacci automorphism, namely the mapping which nat-urally generalizes the �-expansions with � = � to the two-sided case. It is de�ned by theformula(1.8) el(f"kg1�1) = 1Xk=�1 "k��k mod 1; 1Xk=�1 "k��k�1 mod 1! :The convergence of the series involved follows from � being a PV number.2 Indeed, as ���1is the Galois conjugate of �, we have k�nk � ��n for any n � 1, where as usual, kxk :=min (fxg; 1 � fxg). Let us explain the background of formula (1.8). Consider �rst x � 0and its expansion x =P1k=�1 "k��k which is the canonical expansion natural extended toall nonnegative reals with "k � 0 for k � K(x). So, we identify the set of sequences �niteto the left with R+. Consider now the inclusion R+$R = f(fxg; f��1xg) j x � 0g � T2.Since the set R is the half-leaf of the unstable foliation for the Fibonacci automorphism(corresponding to its eigenvalue �), we make sure that (ele�)(f"ng) = eTel(f"ng), where f"ngis �nite to the left.As the set R is dense in the 2-torus, as well as the set of sequences �nite to the left isdense in eX, we can extend the relation above to the whole compactum eX, i.e.ele� = eTeleverywhere.Besides, el is surjective and from the proposition cited in item 1.4 and the fact thatLebesgue measure m2 is the only measure invariant under the translations by a dense setof points of the 2-torus, it follows that m2 = el(em).The important property of the mapping el is that it is not bijective a.e. Nevertheless,as is well known, the automorphisms (T2;m2; eT ) and ( eX; em; e� ) are metrically isomorphic,see, e.g., [AdWe]. Below we will introduce a conjugating mapping for these dynamicalsystems which will be bijective a.e.We recall that after a certain identi�cation of pairs of sequences of measure zero, theMarkov compactum eX becomes a group in addition which we denoted by eX 0 (see item 1.4).Note that the mapping el is well de�ned on eX 0, i.e. it is constant on the equivalence classes.2I.e. an algebraic integer greater than 1 whose Galois conjugates have the moduli less than 1, see, e.g.,[Cas].

ERD�OS MEASURE AND THE GOLDENSHIFT 13Lemma 1.13. The mapping el : eX 0 ! T2 is a group homomorphism.Proof. It su�ces to check that el(en�1) = el(en) + el(en+1), where ek is a sequence having 1at the k'th place and 0 at the other places. This follows directly from formula (1.8).Now we are going to prove the following assertion.Proposition 1.14. The mapping el is 5-to-1 a.e.We are going to give two di�erent proofs of this proposition.First (geometric) proof. Consider an arbitrary sequence " = f"ng1�1 2 eX. We split it intotwo pieces f"ng0�1 and f"ng11 and de�ne x1(") :=P1k=1 "k��k; x2 =P1k=0 "�k(��)�k.It is a direct inspection that x1 2 [0; 1]; x2 2 [�1; �]. Using the relation f�ng =f(�1)n+1��ng; n � 0, we make sure that P1�1 "n��n = x1 � x2 mod 1 and similarly,P1�1 "n��n�1 = ��1x1 + �x2 mod 1. Thus, we have the sequence of mappingseX '�! R2 b�! R2;where '(") = (x1; x2), and b(x1; x2) = (x1 � x2; ��1x1 + �x2). Nowel(") = (b')(") mod Z2:Note that since ("0; "1) 6= (1; 1), the '-image of eX is in fact the di�erence of the rectangles� = ([0; 1]�[�1; �])n([��1; 1]�[��1; �]). As the area of � is �2���2 = p5, and the lineartransformation b = � 1 �1��1 � � from R2 to R2 has determinant p5, the image (b')( eX) hasarea 5. Since this mentioned image is also the di�erence of some rectangles whose verticesare (1;��); (2;�1); (��2; �); (0; ��1p5); (�1; 3); (��; �2), one can immediately checkthat this set is a 5-tuple fundamental domain with respect to the lattice Z2. �Remark. On the other hand, the �nal statement of the proof follows from Lemma 1.13, asfrom its claim it follows that the el-preimage of a.e. point of the 2-torus has one and thesame capacity.Second (algebraic) proof. From Lemma 1.13 it follows that one needs only to describethe kernel of the homomorphism el. From the general considerations we conclude that themapping el is bounded-to-one, whence #Kerel < +1. Furthermore, this kernel is obviouslyinvariant under the shift e� , hence it consists of purely periodic sequences only.Let " = ("1; : : : ; "r)1 be such a sequence. From formula (1.8) it follows thatk��nk ! 0; n! +1;where � = �(") = (Pr1 "k��k)=(�r � 1) 2 Q(�). Let G := f� : k��nk ! 0; n ! 1g.Obviously, G is a group in addition. The following lemma answers the question on thestructure of the group G which is important itself and will be used below. Let Z[�] denotethe additive group of the ring of all Laurent polynomials in powers of �, i.e. fm + n� jm;n 2Zg.

14 NIKITA SIDOROV AND ANATOLY VERSHIKLemma 1.15. The group G is isomorphic to Z2. Its elements are described as follows:G = �m+ n�5 : m;n 2Z; 2n�m � 0 (mod 5)� :The factor group H = G=Z[�] is the cyclic group fa�+25 j a 2Z=5Zg.Proof. By the theorem of Pisot-Vijayaraghavan on the structure of the group G (see [Cas]),� is necessarily algebraic and belongs to the �eld Q(�), and the necessary and su�cientcondition for � to belong to G is Tr(�) 2 Z;Tr(��) 2 Z, where Tr denotes the trace ofan algebraic number. Solving these equations for � = (m + n�)=q with m;n 2 Z; q 2 Nand at least one of the numbers m; n being coprime with q, we come to the system ofcongruences � 2m+ n � 0 (mod q)m+ 3n � 0 (mod q);whence 5m � 0 (mod q); 5n � 0 (mod q), and thus, q = 1 or 5. Besides, the systemabove with q = 5 is equivalent to one congruence 2n�m � 0 (mod 5). The second claimof the lemma is a direct computation.Return to the second proof of the proposition. From the formula for � = �(") above itfollows that � belongs to Z[�] if and only if its period r � 2, i.e. when the sequence " isequivalent to 01 in eX 0 (see the de�nition of eX 0 in item 1.4). Besides, if for two periodicsequences " and "0, �(") and �("0) belong to one and the same element of the factor groupH, we see that el(") = el("0).Thus, #Kerel � 5. Let the sequence "(0) = (0:100)1; consider the setH0 = f0; "(j); 0 �j � 3g, where "(j) = e� j("(0)).It is veri�ed directly that H0 is a subgroup of eX 0. We claim that H0 = Kerel. By theabove, it su�ces to prove the inclusionH0 � Kerel. From formula (1.8), el("(j)) = (a; a); 0 �j � 3, where(1.9) a = limn!1 �n�4 � 1 = limn!1 �n�2 + 1 = limn!1 �np5 = 0;as �n = Fn�1�+Fn�2; n � 3, and we have �n � Fn�1p5, whence k 1p5�nk ! 0 as n!1.Remark. There exists a simple explanation of the origin of the sequences "(k). Note �rstthat if we identify a positive integer n with the sequence equal to ne0, we obtain a naturalinclusion N � eX. For example, 2 = �+ ��2; 3 = �2 +��2, etc. It is a direct computationthat the Fibonacci numbers have the following representations in the compactum eX:Fk = �k�1 + �k�5 + � � �+ ��k+3 + ��k; k even;Fk = �k�1 + �k�5 + � � �+ ��k+5 + ��k+1; k odd:Thus, "(j) = limk(F4k+je0); 0 � j � 3 in the compactum eX. Since limk k�Fkk = 0, wehave el("(j)) = (0; 0).Now we are going to present a simple modi�cation of el which proves to be a bijection.

ERD�OS MEASURE AND THE GOLDENSHIFT 15De�nition. Let the mapping eL : eX ! T2 be de�ned by the formula(1.10) eL(f"kg1�1) = 1Xk=�1 "k ��kp5 mod 1; 1Xk=�1 "k ��k�1p5 mod 1! :By formula (1.9), a series Pk2Z "k ��kp5 converges modulo 1 for any f"kg 2 eX.Remark. In fact, we can treat formula (1.10) as formula (1.8) with the set of digitsf0; 1=p5g instead of f0; 1g (see the remark about the references at the end of the item).Furthermore, eL semiconjugates the shift and the Fibonacci automorphism, and by thesame purposes as for el, we have m2 = eL(em).Proposition 1.16. The mapping eL : eX ! T2 is bijective a.e.Proof. Let the projection P : eX ! eX be de�ned by the formula P (en) = en�1 + en+1and extended to the whole compactum eX by linearity (here en is, as above, the sequencehaving the only 1 at the n'th place). Then by the relation �k = �k�1=p5 + �k+1=p5 andformulas (1.8) and (1.10), we have el = eLP . Now, considering A = eLP (eL)�1 : T2! T2, wesee that A = eT + eT�1 = � 1 22 �1�. Since jdet(A)j = 5, el is 5-to-1 and el = eLA, we completethe proof.So, we proved the following theorem.Theorem 1.17. The mapping eL is a metric isomorphism of the two-sided Markov shift( eX; em; e� ) and the Fibonacci automorphism (T2;m2; eT ).Remark 1. Furthermore, eL is a group isomorpism of the groups eX 0 and T2.Remark 2. Actually, the mappings el and eL lead to di�erent interpretations of the torusas the image of eX. Namely, T2 �= (Q(�) R)=(Z+ �Z) for el, while for eL we haveT2 �= (Q(�) R).Z+�Zp5 .Now we are ready to de�ne the two-dimensional Erd�os measure.De�nition. Let the measure e� on the 2-torus be de�ned as e� = eL(e�). We call it thetwo-dimensional Erd�os measure.Let us show that in a sense the two-dimensional Erd�os measure is de�ned canonically.Note that any mapping from eX onto T2 such that(1) e� = eT,(2) (" + "0) = (") + ("0) for a.e. "; "0 2 eX,(3) is one-to-one a.e.

16 NIKITA SIDOROV AND ANATOLY VERSHIKis of the form(1.11) (f"kg1�1) = el�(f"kg1�1) = 1Xk=�1 "k���k mod 1; 1Xk=�1 "k���k�1 mod 1! ;where � is some real number.Remark. Dealing with the leaf of the stable foliation for eT (see the explanation afterformula (1.8) above), we come to a similar family of mappings:e��(f"kg1�1) = 1Xk=�1 "k�(��)k mod 1; 1Xk=�1 "k�(��)k+1 mod 1! :However, it does not yields anything new, since below we will show that � should be aquadratic irrational, and it is easy to see that for such a �, e�� = el��, where � denotes thealgebraic conjugate of �.We formulate the claim which answers the question on the possible values for � and thebijectivity of el�.Proposition 1.18.(1) For the series in formula (1.11) to converge, � must belong to the group G.(2) The mapping el� is one-to-one a.e. if and only if � = � �kp5 for k 2Z.Proof. The �rst claim is a consequence of the condition k��nk ! 0; n!1, which de�nesthe group G. Let us prove the second one. Using the same geometric arguments and thesame notation as in the proof of Proposition 1.14 and also the fact that ��n = �(��)�n,we obtain the following sequence of mappingseX '�! � b��! R2;where b� = � � ����1� �� � ;and (el�)(") = (b�')(") mod Z2. The area of the set (b�')( eX) is 5j��j = 5jN(�)j (thealgebraic norm of �). So, the condition of the bijectivity of el a.e. isN(�) = �15 ;where � 2 G. We are going to use Lemma 1.15; let � = m+n�5 with the extra condition(*) 2n�m � 0 (mod 5):

ERD�OS MEASURE AND THE GOLDENSHIFT 17Then N(�) = 125(m2 +mn� n2), and we come to the Diophantine equationm2 +mn� n2 = �5together with (*). Putting u = n; v = 2m+n5 2Z, we get a classical Pell's equationu2 � 5v2 = �4:Using the usual method (see, e.g., the monograph [Lev, vol. 1, Th. 8{7]), we obtain itsgeneral solution in the form (u; v) = (uk; vk), whereuk +p5vk = �2�k; k 2Z:Returning to the variables m = 5v�u2 and n = u, we have2n�m+ (2m + n)�5 = ��k; k 2Z;whence � = � �kp5 ; k 2Z: �Remark 1. Note that the condition (*) proved to be satis�ed automatically.Remark 2. Thus, any value of a parameter � yielding the bijectivity of el� is of the form 1p5times an arbitrary unit of the �eld Q(�).Corollary 1.19. For any bijection el� from eX onto T2 having the form (1.11), the imageof the Erd�os measure e� under it will coincide with e�.Proof. It su�ces to use the form of any mapping of such a kind deduced in the previousproposition and the invariance of e� under the shift e� (Proposition 1.3) and the operation" 7! �" (Proposition 1.12).So, the two-dimensional Erd�os measure is de�ned canonically, and we show that theresult analogous to the Erd�os theorem holds for the two-dimensional Erd�os measure.Theorem 1.20. The two-dimensional Erd�os measure e� on T2 is singular with respect toLebesgue measure.Proof. It follows from the Proposition 1.11 that em is singular with respect to e�, and bythe bijectivity of eL a.e. with respect to both measures, we deduce the mutual singularityof their images.Remark. Let us formulate a number of open questions about the properties of the two-dimensional Erd�os measure.(1) Is the measure e� Gibbs with respect to the Fibonacci automorphism for somenatural potential?(2) Is e� invariant under the endomorphismA = � 1 22 �1� (if it was so, the mapping el(e�)would coincide with e�)?(3) Both questions can be reformulated in terms of the compactum eX and remain validfor it.

18 NIKITA SIDOROV AND ANATOLY VERSHIKWe conclude this item by mentioning some necessary references. The �rst precise sym-bolic coding of the hyperbolic automorphisms of the 2-torus had been proposed in [AdWe]and was developing then by a number of authors (see the references in [Ver6], [KenVer]).In [Ber] in connection with the arithmetic of PV numbers were considered two-sided ex-pansions and the corresponding mapping semicongugating the two-sided (in general, so�c)shift and the endomorphism of the torus with a companion matrix. Note that for the caseof the golden ratio it coincides with the mapping el.In the works [Ver3], [Ver5], [Ver6], [KenVer] an arithmetic approach to the coding ofhyperboilc automorphisms of the torus has been developing. In one of the versions ofsuch an approach which generalizes adic transformation to the two-sided case, it leads tothe two-sided �-expansions, and another one being applicable to a more general algebraicnumbers (not only PV) leads to a scheme of eL; it uses the digits from the �eld of anirrationality being not always integers (see [Ver6], [KenVer]). Later this approach wasdeveloped and detailed in the dissertation [Leb].The group H0 and its action on eX (see above) were considered in the recent work [FrSa]in connection with the study of certain �nite automata.More detailed analysis of bijective arithmetic codings for hyperbolic automorphisms ofthe 2-torus was recently given in [SiVer].1.7. Numerical properties of the measure �. We are going �rst to give now anexplicit formula for �.Proposition 1.21. The following relation holds:(1.12) �E = 8><>: 23�E + 13�(E + ��2) + 16�(E + ��1); E � [0; ��2)23�E + 13�(E + ��2); E � [��2; ��1)12�E + 13�(E � ��1); E � [��1; 1]:Proof. Let the measure �0 be de�ned by formula (1.12). We need to show that �0 = �. Note�rst that the T -invariance of �0 is checked directly using Lemma 1.1. Let, say, E � (0; ��2).Then T�1E = ��1E [ (��1E + ��1). Hence�0(T�1E) = �0(��1E) + �0(��1E + ��1)= 23�(��1E) + 13�(��1E + ��2) + 16�(��1E + ��1) + 12�(��1E + ��1)+ 13�(��1E)= 12�E + 13�(E + ��2) + 16�(E + ��1) + 16�E (by Lemma 1.1)= 23�E + 13�(E + ��2) + 16�(E + ��1)= �0(E):The cases E � (��2; ��1) and E � (��1; 1) are studied in the same way.

ERD�OS MEASURE AND THE GOLDENSHIFT 19To prove now that �0 = �, we observe that since � is quasi-invariant with respect to Tand the rotations by ��1 and ��2, the measure �0 is also ergodic with respect to T (see(1.7) and (1.12)). Since � � � and � � �0, we have � � �0, and by the corollary of theergodic theorem, �0 = �. �Corollary 1.22. �(0; ��2) = 49 ; �(��2; ��1) = �(��1; 1) = 518 .Proposition 1.23. The following relation holds:� = limn Tn�:Proof. The sketch of the proof is as follows. Let, for example, E � (0; ��2). Consideringsuccessively the sets T�nE; n � 1 and using Lemma 1.1, we deduce similarly to rela-tion (1.7) that �(T�2E) = 34�E+ 14�(E+��2)+ 14�(E+��1); �(T�3E) = 58�E+ 38�(E+��2)+ 18�(E+��1), etc., whence by induction, �(T�nE) = �12 + 14 � 18 + � � �+ (�1)n2n ��E+� 12 � 14 + � � �+ (�1)n+12n ��(E+��2)+� 14 � 18 + � � �+ (�1)n2n ��(E+��1) = 23�E+ 13�(E+��2)+ 16�(E+��1)+O(2�n) = �E+O(2�n) by formula (1.12). The cases E � (��2; ��1)and E � (��1; 1) are considered in the same way.Remark. It is appropriate, following the well-known framework of the baker's transforma-tion which serves as a model for the full two-sided shift on e�, to represent the two-sidedshift on eX as the Fibonacci-baker's transformation.Namely, we split a sequence f"kg 2 eX into the two one-sided sequences, i.e. into("1"2 : : : ) 2 X and ("0"�1 : : : ) 2 X with regard to the fact that "0"�1 = 0. This lastcondition leads to the space Y = ([0; 1] � [0; 1]) n ([��1; 1] � [��1; 1]) similar to the set �described in the proof of Proposition 1.14.Fig. 1. The natural domain for the Fibonacci-baker's transformationThus, the shift e� on the two-sided Markov compactum eX is isomorphic to the transfor-mation F on the space Y withF (x; y) = � ��x; ��1y�; x 2 [0; ��1]��x� 1; ��1y + ��1�; x 2 (��1; 1]:We call F the Fibonacci-baker's transformation on the set Y (see Fig. 1). For more generalmodels this transformation was considered in [DaKrSo].

20 NIKITA SIDOROV AND ANATOLY VERSHIK2. Symbolic dynamics of expansionsIn this section we will study in detail the combinatorics of all possible representationsof a real x of the form (1.1) with "k 2 f0; 1g for all k.2.1. Blocks. Let us give an important technical de�nition.De�nition. A �nite 0-1 sequence without pairs of adjacent 1's starting from 1 and endingby an even number of zeroes will be called a block if it does not contain any piece \1(00)l1"with l � 1.Let us make some remarks. Note �rst that each block has odd length; the simplestexample of a block is \100". Next, there are exactly 2n�1 blocks of length 2n + 1.This assertion follows from the fact that a block B can be represented in the form1(00)a1(01)a2(00)a3 : : : (01)at�1(00)at for t odd or 1(01)a1(00)a2 : : : (01)at�1(00)at for t even.Thus, any block B is naturally parametrized by means of a �nite sequence of positive in-tegers a1; : : : ; at, and we will write B = B(a1; : : : ; at).Let I0 := [��1; 1), i.e. the interval corresponding to the cylinder ("1 = 1) � X.De�nition. Let x lie in the interval I0, and let the canonical expansion of x have in�nitelymany pieces \1(00)l1" with l � 1. We call such a point x regular.Almost every point x in I0 with respect to Lebesgue measure is regular. Now we splitthe canonical expansion of a regular x into blocks as follows. Since x 2 I0, its canonicalexpansion starts with 1. It is just the beginning of the �rst block B1 = B1(x). The �rstblock ends, when an even number of zeroes followed by 1 appears for the �rst time. This1 begins the second block B2 = B2(x) of the canonical expansion of x, etc. We de�ne thusa one-to-one mapping acting from the set of all regular points of (��1; 1) into the spaceof block sequences.De�nition. The sequence (B1(x); B2(x); : : : ) = (x) will be called the block expansionof a regular x.2.2. The cardinality of a 0-1 sequence and its properties. We are going to de�nean equivalence relation on the set of all �nite 0-1 sequnces.De�nition. Two 0-1 sequences (�nite or not) (x1x2 : : : ) and (x01x02 : : : ) are called equiv-alent if Pk xk��k = Pk x0k��k (or, equivalently, if their normalizations coincide | seeSection 1). Let for a �nite 0-1 sequence x, E(x) denote the set of all 0-1 sequences equiva-lent to x; this set is always �nite. Let f(x) := #E(x). We call f the cardinality of a �nitesequence (or the cardinality of an equivalence class).Note that this function (of positive integers) was considered in [Car], [AlZa] and recentlyin [DuSiTh] and [Pu].The assertions below answer the question about the cardinality of a block and explainthe purpose of the introduction of blocks as natural structural units in this theory.

ERD�OS MEASURE AND THE GOLDENSHIFT 21Lemma 2.1. Let a �nite word x be a block, i.e. x = 1(00)a1(01)a2(00)a3 : : : (01)at�1(00)ator 1(01)a1(00)a2 : : : (01)at�1(00)at . Let p=q = [a1; : : : ; at] be a �nite continued fraction.Then f(x) = p+ q:Proof. Note �rst that f(100) = 2 = p+ q. The desired relations for the blocks 10000 and10100 follow by direct inspection. Next, let {k = {k(a1; : : : ; ak) = f(x). We need to showthat similarly to the numerators and denominators of the convergents, {t = at{t�1+{t�2,whence the required assertion will follow.Let, say, t be odd and B = 1(00)a1 : : : (00)at�2(01)at�1(00)at . We �rst consider all 0-1sequences equivalent to B and ending by (00)at . Their number is obviously {t�2, as theyin fact should end by (01)at�1(00)at . Now we consider 0-1 sequences equivalent to B butnot ending by (00)at and will show that their number is at{t�1. Namely, letB(1) = 1(00)a1 : : : (00)at�2(01)at�1�10011(00)at�1;B(2) = 1(00)a1 : : : (00)at�2(01)at�1�10(01)21(00)at�2;: : :B(at) = 1(00)a1 : : : (00)at�2(01)at�1�10(01)at1:It is clear that any B(j) is equivalent to B; now we observe that the number of 0-1 wordsequivalent to B and ending by (00)at�j ; 1 � j � at, is exactly {t�1, as the replaceablepart of B(j) with the �xed end (00)at�j is in fact 1(00)a1 : : : (00)at�2(01)at�1�100, hence{t = at{t�1 + {t�2, as [a1; : : : ; at�1 � 1; 1] = [a1; : : : ; at�1]. The case of even t is studiedin the same way. �Remark. To any rational r 2 (0; 1) exactly two blocks correspond, namely with r =[a1; : : : ; at] = [a1; : : : ; at�1; at � 1; 1], and the unique block \100" corresponds to r = 1.IfE1 andE2 are two sets of sequences, then henceforwardE = E1E2 is the concatenationof these two sets, i.e. any sequence in E begins with a word from E1 and ends with a wordfrom E2.Lemma 2.2. For any blocks B1; : : : ; Bk,(1) E(B1 : : : Bk) = E(B1) : : : E(Bk).(2) The cardinality is blockwise multiplicative, i.e.f(B1 : : : Bk) = kY1 f(Bi):

22 NIKITA SIDOROV AND ANATOLY VERSHIKProof. It su�ces to prove item (1). Let us restrict ourselves to the case k = 2 (thegeneral case is studied in the same way). We will see that there is no sequence in E(B1B2)containing a triple 011 or 100 which crosses the \border" between the �rst jB1j digits andthe last jB2j. In other words, we will show that any sequence equivalent to B1B2 can beconstructed as the concatenation of a sequence equivalent to B1 and a sequence equivalentto B2.Let \j" below denote the border in question. First, any sequence from E(B2) mustbegin either with 10 or with 01, hence, the situation (1j00) or (0j11) is impossible. Next,a sequence in E(B1) ends in either 00 or 11 (see the proof of the previous lemma), neitherleading to (10j0) or (01j1). �This simple result shows that the space of all equivalent in�nite 0-1 sequences for agiven regular x splits into the direct in�nite product of spaces, the k'th space consistingof all �nite sequences equivalent to the block Bk(x). So, we see that the notion of block,initially arising in terms of the canonical expansion, can be naturally extended to allrepresentations. Below we will explain the geometric sense of a block in terms of theFibonacci graph (see p. 3.1).Remark. Note that this block partition appeared for the �rst time in [Pu] in other termsand for algebraic and combinatorial purposes. Namely, let the partial ordering on a spaceE(x) for some �nite word x be de�ned as follows. We set x � x0 if there exists k � 2 suchthat xk�1 = 0; xk = 1; xk+1 = 1; x0k�1 = 1; x0k = 0; x0k+1 = 0, and xj = x0j ; jk � jj � 2.Next, one extends this ordering by transitivity. It was shown in [Pu] that any equivalenceclass has the structure of a distributive lattice in the sense of this order.2.3. Goldenshift. We are going to give one of the central de�nitions of the presentpaper.De�nition. The transformation S acting from the set of regular points of the interval(��1; 1) into itself by the formulaSx = �n(x)x; x is regular,where n(x) is the length of the �rst block of the block expansion of x, is called the gold-enshift.Remark 1. The transformation S is piecewise linear. More precisely, if (��1; 1) = Sr�ris the partition of (��1; 1) mod 0 into intervals corresponding to that of B into the statesof the �rst block, then S is linear inside �r =: [�r; �r), and S(�r) = ��1; S(�r) = 1.Remark 2. The transformation S is a generalized power of � in the sense of Dye (see, e.g.,[Bel]). In other words, the goldenshift is a random power of the ordinary shift, as thenumber of shifted coordinates depends on the length of the �rst block. Note that thegoldenshift is not an induced endomorphism for � but for the two-sided case it is (seeProposition 2.3 and Theorem 2.12 below).Remark 3. Let X denote the space of block sequences. The goldenshift may be treated asthe one-sided shift in the space X, i.e. S(B1B2B3 : : : ) = (B2B3 : : : ).

ERD�OS MEASURE AND THE GOLDENSHIFT 23Now we are going to de�ne the two-sided goldenshift. Let, as above, e� denote the two-sided shift on eX. In order to de�ne the two-sided goldenshift on the Markov compactumeX, we give the following de�nition.De�nition. The e�-invariant set eXreg � eX is de�ned as the set consisting of all sequencescontaining pieces \102l1" with l � 1 in�nitely many times both to the left and to the rightwith respect to the �rst coordinate.Obviously, e�� eXreg� = 1, as by Proposition 1.4, the measure e� of any cylinder in eX ispositive, and by Proposition 1.5, the automomorphism ( eX; e�; e� ) is ergodic, so, it su�cesto apply the ergodic theorem. Let eX0 := S1k=1(x�2k = 1; x�2k+1 = � � � = x0 = 0; x1 = 1),and let eXreg0 = eXreg \ eX0.De�nition. The two-sided goldenshift eS : eXreg0 ! eXreg0 on the Markov compactum eX is,by de�nition, the shift by the length of B1.Remark. Considering the space of two-sided block sequences eX = Q1�1B and implyingthat B1 begins with the �rst coordinate of eX (i.e. with "1 = 1), we see that the two-sidedgoldenshift eS is the shift in the space eX.Proposition 2.3. The two-sided shift e� on the set eXreg is a special automorphism overthe goldenshift eS on the set eXreg0 . The number of steps over a sequence ("k) 2 eXreg0 isequal to the length of the block beginning with "1 = 1.Proof. It su�ces to present the steps of the corresponding tower. Let, by de�nition,eXreg1 = e� eXreg0 , and eXreg2j = e� eXreg2j�1; eXreg2j+1 = e� eXreg2j n ("1 = 1); j � 1. This completes theproof, as eXreg = S10 eXregj , the union being disjoint.Fig. 2. The steps of the special automorphism e�Below the corresponding result will be established for the metric case with the two-sidedErd�os measure.

24 NIKITA SIDOROV AND ANATOLY VERSHIK2.4. Bernoullicity of the goldenshift. In this subsection we will show that the gold-enshift (one-sided or two-sided) is a Bernoulli shift in the space X (respectively eX) withrespect both to Lebesgue and Erd�os measures and compute their one-dimensional distri-butions.We recall that above we denoted the set of blocks by B; let the same letter stand forthe totality of cylinder sets fB1 = Bg � X for all blocks B. Let Bn denote all cylindersets fB1 = Bg � X such that the length of B is 2n+ 1. Thus, B = S1n=1Bn.Recall that the mapping assigns to each regular x its block expansion, i.e. a certainsequence in the space X. We denote bym the normalized Lebesgue measure on the interval(��1; 1); let mX stand for the measure (m) in the space X, and, similarly, let �X denote(�I0) (this measure is well de�ned, as �-a.e. point x 2 (��1; 1) is also regular).We recall that to any block we associated the interval �r de�ned as the image of thecylinder \B(r)1" in X by the mapping (1.1) (see Remark 1 after the de�nition of thegoldenshift).We are going to show that the measures mX and �X are Bernoulli in the space X andto compute their one-dimensional distributions.Theorem 2.4. The measure mX in the space X is a product measure with equal multi-pliers, i.e. a Bernoulli measure.Proof. By the linearity of S on each interval �r and the fact that S(�r) = (��1; 1), wehave for any Borel set E � (��1; 1),(2.1) m(S�1E \�r) =mE �m�r;whence the required assertion immediately follows by virtue of the obvious S-invariance ofm, and by setting E = �r0 in relation (2.1) for any r0, which yields the mX-independenceof the �rst and the second blocks.So, it remains to compute the one-dimensional distribution of mX.Corollary 2.5. For any cylinder set fB1 = Bg �Bn its measuremX equals ��2n�1. Themeasure mX of Bn is equal to 12� � 2�2 �n.Proposition 2.6. The measure �X is a product measure on X with equal multipliers.Proof. It su�ces to establish a relation similar to (2.1) for the measure �X and for any�nite block sequence E = B1 : : : Bk. Note �rst that by virtue of Lemma 2.2, E(B1 : : : Bk) =E(B1) : : : E(Bk) for any blocks B1; : : : ; Bk. Next,n�1(B1 : : : Bk1) = E(B1) : : : E(Bk)n�1("1 = 1):We are going to show that(2.2) �XfB1 : : : Bkg = �XfB1g : : : �XfBkg = f(B1)2jB1j � � � f(Bk)2jBkj ; k � 1:

ERD�OS MEASURE AND THE GOLDENSHIFT 25To do this, we use previous remarks and the de�nition of the Erd�os measure onX by meansof the normalization (see Section 1). We have �XfB1 : : : Bkg = �(B1 : : : Bk1)=�("1 = 1),and �(B1 : : : Bk1) = p(n�1(B1 : : : Bk1)) = p(E(B1) : : : E(Bk)n�1("1 = 1))= f(B1)2jB1j � � � f(Bk)2jBkj � �("1 = 1)(by Lemma 2.2), whence the required assertion follows.Thus, we have proved one of the main results of the present paper.Theorem 2.7. The goldenshift S is a Bernoulli automorphism with respect to Lebesgueand Erd�os measures.Now we are ready to give the second proof of Erd�os theorem (see Section 1).Corollary 2.8. (a new proof of Erd�os theorem) The Erd�os measure is singular withrespect to Lebesgue measure.Proof. In fact, by the corollary of the ergodic theorem, the measures involved are mutuallysingular on the interval (��1; 1) (recall that �(0; ��2) = 13 , whence the measure � di�ersfrom the Lebesgue measure). This yields the assertion of the corollary, as any in�niteconvolution of discrete measures, if it is not stochastically constant, is known to be eithersingular or absolutely continuous with respect to Lebesgue measure (the \Law of PureTypes", see [JeWi])).The one-dimensional distribution of �X is a bit more sophisticated than for mX. It isdescribed as follows (see formula (2.2)):Proposition 2.9. For a block B = B(a1; : : : ; at) of length 2n+ 1,�XfB1 = Bg = f(B)2jBj = p+ q22n+1 ;where, as usual, p=q = [a1; : : : ; at]. The measure �X of the set Bn equals 13 � �34�n.2.5. Concluding remarks on the Erd�os measure. We conclude the study of ergodicproperties of Erd�os measures (one-sided and two-sided) and the transformations of shiftand goldenshift.Recall that the Erd�os measure � is quasi-invariant under the one-sided shift � , and theequivalent measure � is � -invariant. It is worthwhile to know the behavior of � with respectto the goldenshift S.Let �X be de�ned in the same way as �X. We formulate the following claim (for moredetails see Appendix C).

26 NIKITA SIDOROV AND ANATOLY VERSHIKProposition 2.10. The measure �X on the space X of block sequences is quasi-invariantunder the goldenshift S. More precisely, any two cylinders fBj = B0jg and fBi = B0ig withi 6= j are �X-independent, and for B = B(a1; : : : ; at),�XfBk = Bg = �XfBk = Bg = f(B)2jBj = p + q2 � 4a1+���+at ; k � 2;�XfB1 = Bg = ( 45p+65 q2�4a1+���+at ; B = 100 : : :65p+45 q2�4a1+���+at ; B = 101 : : :Now we will prove a numerical claim useful for the next section.Corollary 2.11. e� eX0 = 19 .Proof. We have by the de�nition of the set eX0, Proposition 2.10 and the fact that �("1 =1) = 518 (see Corollary 1.22),e� eX0 = 1Xk=1 e��1(00)k1� = 518 1Xk=1 45 + 65k1 + k � k + 122k+1 = 19 :Finally, we prove a metric version of Proposition 2.3.Theorem 2.12. The two-sided shift e� with the measure e� is a special automorphism overthe goldenshift eS with the measure e� on the space eX0. The step function is de�ned as thelength of the block beginning with the �rst coordinate.Proof. It su�ces to show that the lifting measure for e� coincides with e�. This in turnis implied again by the corollary of the ergodic theorem applied to e� that preserves bothmeasures which are ergodic. Since they are clearly equivalent, we are done.3. The entropy of the goldenshift and applicationsIn this section we will establish a relationship between the entropy of the Erd�os measurein the sense of A. Garsia and the entropy of the goldenshift with respect to �, i.e. betweentwo di�erent entropies. As an application, we will reprove the formula for Garsia's entropyproved in [AlZa]. Further, we use random walk theory to compute the dimension of theErd�os measure on the interval.3.1. Fibonacci graph, randomwalk on it and Garsia's entropy. The combinatoricsof equivalent 0-1 sequences may be expressed graphically, namely by means of the Fibonaccigraph introduced in [AlZa]. Let, as in Section 1, � = Q11 f0; 1g, and let the mapping� : �! [0; 1] be de�ned as(3.1) �("1; "2; : : : ) = 1Xk=1 "k��k�1:Since the "n assume the values 0 and 1 without any restrictions, a typical x will have acontinuum number of representations, and they all may be illustrated with the help of theFibonacci graph depicted in Fig. 3. This �gure appeared for the �rst time in the work dueto J. C. Alexander and D. Zagier [AlZa]. Let us give the precise de�nition.

ERD�OS MEASURE AND THE GOLDENSHIFT 27De�nition. The Fibonacci graph � is a binary graph with the edges labeled with 0 ifto the left and 1 if to the right. Any vertex at the n'th level corresponds to a certainx, for which some representation (3.1) is �nite with the length n (obviously, in this casex = fN�g for some N 2Z). The paths are 0-1 sequences treated as representations of theform (3.1).3Fig. 3. The Fibonacci graph �Remark. The vertices of the n'th level of the graph � can be treated as the nonnegativeintegers from 0 to Fn+2 � 2. Namely, if a path ("1"2 : : : "n) goes to a vertex k, then, byde�nition, k =Pn1 "jFn�j (obviously, this sum does not depend on the choice of a path).Let Y (�) denote the set of paths in the graph �. Obviously, Y (�) is naturally isomor-phic to �, and sometimes we will not make a distinction between them. Let ("1"2 : : : ) be apath, and let the projection from Y (�) onto [0; 1] be also denoted by � (see formula (3.1)).Let, as above, fn(k) denote the number of representations of a nonnegative integer kas a sum of not more than n �rst Fibonacci numbers. It is easy to see that fn(k) is alsothe frequency of the vertex k on the n'th level of the graph �.4 Let Dn = fk : k =Pnk=1 "kFn�k; "k 2 f0; 1gg (or, equivalently, the n'th level of the Fibonacci graph), andD0n = fw : w = Pnk=1 "k��k�1; "k 2 f0; 1gg. These sets are clearly isomorphic (w $ k),and #Dn = #D0n = Fn+2�1. The use ofDn instead of D0n is due only to technical reasons.We recall that the sequence of distributions (2�nfn(w))1n=1 tends to the distribution ofErd�os measure (see Section 1).We have proved in the previous section that block is an object de�ned on the space ofalmost all 0-1 sequences (not only admissible), i.e. on the Fibonacci graph. Let us nowexplain the geometric treatment of the block expansion.3The term \Fibonacci graph" is overloaded, as the authors know several di�erent graphs also called\Fibonacci". Nevertheless, we hope that there will be no confusion with any of them.4Thus, relation (1.2) completely determines the whole graph �.

28 NIKITA SIDOROV AND ANATOLY VERSHIKNote �rst that an odd level 2n + 1 contains 2n�1 speci�c vertices which we will call,following [AlZa], the Euclidean vertices. They are de�ned recursively. The central vertexon the third level is Euclidean; then, any Euclidean vertex generates exactly two newEuclidean vertices on the next odd level by means of the arcs 00 and 11. We call the setof all Euclidean vertices the Euclidean tree.It is evident that for a given path in the Fibonacci graph its �rst block can end only atsome Euclidean vertex. To �nd out, if it does end at a vertex v, we consider the inducedFibonacci graph with v as the top and also the induced Euclidean tree. The criterion inquestion is that a given path should go through some induced Euclidean vertex earlierthan through any initial one. Next, considering the induced Fibonacci graph, we can �ndthe second block of the block expansion, etc.We denote the entropy of the discrete distribution on D0n decribed above, by H(n).Thus, H(n) = � Fn+2�2Xk=0 fn(k)2n log fn(k)2n :Then, by de�nition, H� := limn!1 H(n)n log �(this limit is known to exist and is obviously independent of the choice of the base oflogarithms, see [Ga]). Now we choose once and for all � as the base of logarithms.The quantity H� can be considered as the entropy of the random walk on the Fibonaccigraph with the probabilities (12 ; 12 ). In the next item it will be shown that in fact H� isproportional to the entropy of the goldenshift. The Erd�os measure is the projection of theMarkov measure (12 ; 12) on the graph � under the mapping �. We consider the randomwalk on the Fibonacci graph with the equal transition measures.De�nition. The Fibonacci semigroup (resp. group) is by de�nition, the semigroup (resp.group) with the generators a; b and the relation ab2 = ba2.The following claims are straightforward.Proposition 3.1. The Fibonacci graph is the Cayley graph of the Fibonacci semigroup.In [Av] (see also [KaVer]) was introduced the notion of the entropy of a random walkon a �nitely generated group (or semigroup). The following claim establishes a relationbetween the two notions of entropy. Note that the Fibonacci semigroup can be naturallyembedded into the Fibonacci group, that is why we can use the theory of random walkson groups.Proposition 3.2. The entropy H� is equal to the entropy of the random walk on theFibonacci semigroup with the probabilities � 12 ; 12�.Proof. Follows from the de�nitions and Proposition 3.1.

ERD�OS MEASURE AND THE GOLDENSHIFT 293.2. Main theorem. We prove an assertion that is one of the central points of thepresent paper. Note that in [AlZa] Garsia's entropy was computed by means of generatingfunctions. We will see that H� is closely connected with the entropy of the goldenshift,which gives a new and simpli�ed proof of their relation and relates it to the dynamics ofthe Erd�os measure.Theorem 3.3. The following relation holds:h�(S) = 9H�:Proof. We are going to apply Abramov's formula for the entropy of the special automor-phism (see [Ab]) to the dynamical systems ( eX; e�; e� ) and ( eX0; e�; eS). By Abramov's formulaand Theorem 2.12, he�( eS) = 1e� eX0he�(e�):From this relation we will deduce the required one.1. Since the dynamical system ( eX; e�; eS) is the natural extension of (X;�;S), we havehe�( eS) = h�(S) and by the same reason, he�(e� ) = h�(� ).2. By Corollary 2.11, e� eX0 = 19 .3. The rest of the proof is devoted to establishing the validity of the relationh�(� ) = H�:Let � denote the partition of X into the cylinders ("1 = 0) and ("1 = 1). Since � isa generating partition for e� , we have h�(� ) = h�(�; �) by Kolmogorov's theorem. Byde�nition, h�(�; �) = limn!1 1nH�(�(n));where �(n) is the partition of X into Fn+1 admissible cylinders of the form ("1 = i1; : : : ;"n = in). We need to prove that H�(�(n)) � H(n):By virtue of the equivalence of the measures � and � it su�ces to show this for H�(�(n))instead of H�(�(n)). Let �n(k) = 2�nfn(k). Then, by relation (1.2), for n � 3,�n(k) =8><>: 12�n�1(k); 0 � k � Fn � 112 (�n�1(k) + �n�1(k � Fn)); Fn � k � Fn+1 � 212�n�1(k � Fn); Fn+1 � 1 � k � Fn+2 � 2:We are going to obtain almost the same recurrence relation for the distribution �(n). Todo this, we return to the interval [0; 1] and denote by �(n) the partiton into Fn+1 intervalswhich is the image of the corresponding partition of X with the help of the canonical

30 NIKITA SIDOROV AND ANATOLY VERSHIKexpansion. So, let �(n+1) =: (Jn(k))Fn+2�1k=0 with the ordered intervals Jn(k). Finally, let�n(k) := �Jn(k). Then by Lemma 1.1, for n � 3,�n(k) = 8><>: 12�n�1(k); 0 � k � Fn � 112 (�n�1(k) + �n�1(k � Fn)); Fn � k � Fn+1 � 112�n�1(k � Fn); Fn+1 � k � Fn+2 � 3:Also, �n(Fn+2 � 2) = O(2�n); �n(Fn+2 � 1) = O(2�n). Thus, by induction on n, thereexists C > 0 such that 1C � �n(k)�n(k) � C; 0 � k � Fn+2 � 2;Since the measure � is � -invariant and ergodic, we can apply the Shannon-McMillan-Breiman theorem and deduce that the entropies of the distributions �n(k) and �n(k) areequivalent. �3.3. Alexander-Zagier's theorem. We �rst compute the entropy of the goldenshiftwith respect to the Erd�os and Lebesgue measures.Proposition 3.4. The metric entropies of the goldenshift S with respect to the twomeasures in question are computed as follows:hm(S) = �XB2BmX(B) log�mX(B) = 4�+ 3 = 9:4721356 : : : ;h�(S) = �XB2B�X(B) log� �X(B) = � 1Xn=1 XB:jBj=2n+1 p+ q22n+1 log� p+ q22n+1 = 8:961417 : : :Proof. This is a direct computation using the Bernoullicity of S with respect to bothmeasures and Corollaries 2.5 and 2.9.Corollary 3.5. Let � := log� 2, andkn = Xt�1(a1;:::;at)2Nta1+���+at=np=q=[a1;:::;at](p+ q) log�(p + q):Then(3.2) h�(S) = 9 �� 118 1Xn=1 kn4n! :

ERD�OS MEASURE AND THE GOLDENSHIFT 31Proof. We haveh�(S) = � 1Xn=1 XB:jBj=2n+1 p+ q22n+1 log� p+ q22n+1= �12 1Xn=1 4�n kn ��(2n+ 1) Xa1+���+at=n(p+ q)! :It su�ces now to recall that Xt�1(a1;:::;at)2Nta1+���+at=np=q=[a1;:::;at](p + q) = 2 � 3n�1and to compute the value of the corresponding series. �Note that the quantity kn appeared for the �rst time in [AlZa] in somewhat di�erentnotation. Namely, let k and i be positive integers, and let e(k; i) denote the length of thesimple Euclidean algorithm for k and i (formally: e(i; i) = 0; e(i + k; i) = e(i + k; k) =e(i; k) + 1). Then obviously kn = Xk>i>0gcd(k;i)=1; e(k;i)=nk log� k:In the cited work J. C. Alexander and D. Zagier used this de�nition of kn to deduce aformula for H� in terms of kn. We will prove their assertion in two di�erent ways. The �rstis an immediate consequence of Theorem 3.3 and relation (3.2), while the second is ratherlong but reveals a more essential relationship between certain structures on the Fibonaccigraph (see Appendix D).Proposition 3.6. (Alexander-Zagier, 1991). The following relation holds:(3.3) H� = �� 118 1Xn=1 kn4n = 0:995713 : : :Proof. An application of Theorem 3.3 and of relation (3.2).3.4. The dimension of the Erd�os measure. As an application of the treatment ofthe Erd�os measure as the projection of the measure of the uniform random walk on theFibonacci graph, we will compute the dimension of � in the sense of L.-S. Young.We �rst give a number of necessary de�nitions (see [Y]).

32 NIKITA SIDOROV AND ANATOLY VERSHIKDe�nition. Let � be a Borel probability measure on a compact space Y . The quantitiesdimH � = inffdimH A : A � Y; �A = 1g;C(�) = lim sup�!0 inffC(A) : A � Y; �A � 1� �g;C(�) = lim inf�!0 inffC(A) : A � Y; �A � 1� �g(where C(A) and C(A) are respectively the upper and lower capacities of A) are called theHausdor� dimension of a measure � and the upper and lower capacities of �, respectively.Let next N("; �) denote the minimal number of balls of radius " > 0 which are necessaryto cover a set of �-measure � 1� �.De�nition. The quantitiesCL(�) = lim sup�!0 lim inf"!0 logN("; �)log(1=") ;CL(�) = lim sup�!0 lim sup"!0 logN("; �)log(1=")are called the lower and upper Ledrappier capacities of �.De�nition. Let H�(") = inffH�(�) : diam � � "g, where H�(�) is the entropy of a �nitepartition �. The quantities R(�) = lim sup"!0 H�(")log(1=") ;R(�) = lim inf"!0 H�(")log(1=")are called respectively the upper and lower informational dimensions of � (= R�enyi di-mensions).Theorem. (L.-S. Young [Y], 1982). Let � be a Borel probability measure on a metricspace Y , and let B(x; r) denote the ball with the center at x of radius r. If�(x) := limr!0 log �B(x; r)log r � �for �-a.e. point x 2 Y , thendimH � = C(�) = C(�) = CL(�) = CL(�) = R(�) = R(�) = �:De�nition. If the condition of Young's theorem is satis�ed, then this � is called thepointwise dimension of a measure � and is denoted by dim(�). This notion was alsoproposed in [Y].

ERD�OS MEASURE AND THE GOLDENSHIFT 33Theorem 3.7. For the Erd�os measure �,dim(�) = H�:Proof. Fix a path " = ("1"2 : : : ) 2 Y (�), and let x = �(") and Yn = Yn(") be theinterval whose every point x0 has a path "0 2 Y (�) such that "i � "0i; 1 � i � n.Clearly, Yn(") = �Pn1 "k��k�1;Pn1 "k��k�1 + ��n�, We recall that � = #1 � #2 � : : : (seeSection 1). Let �(n) := #1 � � � � � #n. Then by Shannon's theorem for the random walks(see Theorem 2.1 in [KaVer] and also [De]) and because H� is the entropy of the randomwalk on the Fibonacci semigroup (see Proposition 3.2),limn!1 log� �(n)(Yn("))n = �H�for �-a.e. x 2 [0; 1]. We set Y 0n(") = �Pn1 "k��k�1;Pn1 "k��k�1 + ��n+2� � Yn(").Obviously, �(n)(Y 0n(")) � �(n)(Yn(")). Given h > 0, we choose n = n(h) such thatY 0n+1(") � (x; x + h) � Y 0n(") for any " 2 ��1fxg. Hence it follows that for �-a.e. x,limh!0 log� �(x; x + h)log� h = � limn!1 log� �Yn(")n = H�;as h � ��n. �Remark. When the present paper was in preparation, the authors were told that theclaim of Theorem 3.7 can be obtained as a corollary of several results including the newone due to F. Ledrappier and A. Porzio. More precisely, it was shown in [AlYo] thatH� = R(�) = R(�), and in [LePo] it was proved that the limit in Young's theorem doesexist for the Erd�os measure. This proves Theorem 3.7.Our proof is straightforward and, what is more important, is a direct corollary of aShannon-like theorem, so far it leads to new connections between geometric and dynamicalproperties of the Erd�os measure.Corollary 3.8.dimH � = C(�) = C(�) = CL(�) = CL(�) = R(�) = R(�) = H� = 0:995713 : : :Remark 1. Another proof of Theorem 3.7 can be obtained by using the Bernoulli structureof the measure �. More precisely, for a regular x 2 (��1; 1) having a normal blockexpansion with respect to the measure �X, as is easy to compute, the limit in the de�nitionof the dimension equals 19h�(S). This yields also another proof of Theorem 3.3. The detailsare left to the interested reader.Remark 2. In fact, we have computed the �-typical Lipschitz exponent of the distributionfunction of the Erd�os measure. Note that in [Si] it was proved that the best possibleLipschitz exponent of this function for all x is �� 12 = 0:9404 : : : .

34 NIKITA SIDOROV AND ANATOLY VERSHIKRemark 3. As a conjecture, we claim that for the two-dimensional Erd�os measure e� (seeSection 1), dim(e�) = 2dim(�) = 1:991426 : : :The proof could apparently follow from the theorem due to L.-S. Young [Y] relating thepointwise dimension to the entropy of an automorphism and Lyapunov exponents of anergodic measure. Besides, we think that the measure e� has a local structure of directproduct, which would also explain this relation.3.5. Fibonacci exponents. We recall that in Section 2 we de�ned the function f actingon the set of �nite 0-1 words and counting the number of words equivalent to an argument.Let x 2 (0; 1) and ("1"2 : : : ) be its canonical representation. Let now the �nite wordxn = ("1 : : : "n).De�nition. The limit E(x) = limn!1 log� f(xn)n(if exists) will be called the Fibonacci exponent of a point x.We will show that for a.e. x with respect to Erd�os measure the Fibonacci exponentexists and is the same, as well as for Lebesgue measure. Besides, we reprove one theoremdue to S. Lalley (see [Lal] and references therein). The proof in [Lal] can be applied to anyPV number � but for the golden ratio our proof is more direct.Proposition 3.9. ([Lal, Th. 2]). For �-a.e. x,E(x) = E� := ��H� = 0:44469 : : :Proof. It su�ces to consider a regular x 2 (��1; 1). Let B1B2 : : : be its block expansion.Then E(x) = limk!1 log� f(B1 : : : Bk)jB1j+ � � �+ jBkj = E� log� f(B1)E�jB1j(where E� denotes mathematical expectation with respect to Erd�os measure) by the er-godic theorem applied to the goldenshift and Erd�os measure. Now it su�ces to observethat E�jB1j = 9, and E� log� f(B1) = 12Pn�1 kn4n and apply Proposition 3.6. �In the same way we obtainProposition 3.10. For a.e. x with respect to Lebesgue measure,E(x) := Em = Em log� f(B1)EmjB1j = P1n=1 `n��2n�14�+ 3 ;where `n = Xt�1(a1;:::;at)2Nta1+���+at=np=q=[a1;:::;at] log�(p + q):

ERD�OS MEASURE AND THE GOLDENSHIFT 35Remark 1. We established in Proposition 3.9 a relation between the pointwise dimensionof the Erd�os measure and its typical Fibonacci exponent. This gives us an occasion tostate without proof a similar claim for Lebesgue measure:limh!0 log�(x; x + h)logh = ��Emfor a.e. x with respect to the Lebesgue measure. The proof is the same as the onementioned in Remark 1 after Corollary 3.8. Apparently, this dimensional characteristichas not been considered yet.Remark 2. Let f(n) be the number of representations of a positive integer n as a sum of dis-tinct Fibonacci numbers. If n =Pk1 "jFj is such a representation with "j 2 f0; 1g "j"j+1 =0; 1 � j � k � 1, and "k = 1, then evidently f(n) = f("k : : : "1) in the usual sense, whichexplains the choice of the notation. It is known that f(n) = O(pn) is an attainable esti-mate (see [Pu]) and that the average behavior of f in the sense of its summation function isPn<N f(n) � N� (for more precise results see [DuSiTh]). It is worth asking the questionabout the \typical exponent" of f(n) with respect to density, i.e.Ed = limn!1n2J log f(n)logn ;where J � N is a subsequence of density 1. We conjecture that this exponent exists, andEd = Em, i.e. density 1 corresponds to full Lebesgue measure.Appendix A. The ergodic central measures andthe adic transformation on the Fibonacci graphIn this appendix we will study in detail some properties of the space of paths Y (�) ofthe Fibonacci graph introduced in Section 3. We �rst give a necessary de�nition which isclose to the de�ntion of canonical expansions but re ects the fact that 0 and 1 have thesame rights in the graph �.De�nition. The generalized canonical expansion of a point in (0; 1) is de�ned as follows.We construct the sequence ("1"2 : : : ) such that relation (3.1) holds, and either ("1"2 : : : ) 2X, or "1 = � � � = "m = 1 for some m 2 N, and the tail is in X. The algorithm is a clearmodi�cation of the greedy algorithm.Remark. In fact, the generalized canonical expansions lead to a normal form in the semi-group corresponding to the group G (see Section 3).The tail partition �(�) of Y (�) is de�ned as follows.De�nition. Paths ("n) and ("0n), by de�nition, belong to one and the same element of�(�) i�(i) �("1"2 : : : ) = �("01"02 : : : ), and(ii) there exists N 2 N such that "n � "0n; n > N .The partial lexicographic ordering on Y (�) is de�ned for paths belonging to one andthe same element of the tail partition �(�).

36 NIKITA SIDOROV AND ANATOLY VERSHIKDe�nition. Let two paths " = ("1"2 : : : ) and "0 = ("01"02 : : : ) belong to one and the sameelement of �(�). If "k�1 = 0; "k = 1; "k+1 = 1 and "0k�1 = 1; "0k = 0; "0k+1 = 0 for somek � 2, and "j � "0j for k � j � 2, then, by de�nition, " � "0. Next, by transitivity," � "0; "0 � "00 implies " � "00.Remark. This de�nition is consistent, because any element of �(�) is isomorphic to a �nitenumber of �nite paths, and they all can be transfered into one another with the help ofreplacements 011 $ 100. Note also that this linear ordering on each element of �(�)is stronger than the partial ordering introduced in [Pu] (see the end of item 2.2). Forexample, (100011�) � (011100�) in the above sense but in the sense of the partial orderthey are noncomparable.De�nition. The adic transformation T� assigns (if possible) to a path " 2 Y (�) the path"0 such that "0 belongs to the same element of the tail partition as " and is the immediatesuccessor of " in the sense of the lexicographic order.It is clear that the adic transformation T� is not everywhere well de�ned. More precisely,it is well de�ned on the paths " containing at least one triple "k = 0; "k+1 = 1; "k+2 = 1.Let us describe its action in more detail. Let ("1"2 : : : ) 2 Y (�) be as described. After�nding the �rst triple "k = 0; "k+1 = 1; "k+2 = 1, we1) replace it by "k = 1; "k+1 = 0; "k+2 = 0,2) leave the tail ("k+3"k+4; : : : ) without changes,3) �nd the minimal possible ("01 : : : "0k�1) equivalent to ("1 : : : "k�1) in the sense of Section 2.To carry out 3), we may use the algorithm of \anti-normalization", i.e. the processanalogous to the ordinary normalization but changing \100" to \011" (cf. Section 1).So, the generalized canonical expansions are just the maximal paths, i.e. the ones whereT� is not well de�ned; thus, the set of maximal paths is naturally isomorphic mod 0 tothe interval [0; 1]. Geometrically the generalized canonical expansion corresponds to theright most possible path descending to a given vertex. Similarly, the minimal paths (i.e.the ones, on which T�1� is not well de�ned) are just the so-called lazy expansions (for thede�nition see, e.g., [ErJoKo]).For more general de�nitions of adic transformations and investigation of their propertiessee [Ver1], [Ver2], [LivVer] and [VerSi].Let us formulate several well-known de�nitions related to graded graphs (see [StVo] and[VerKe] for more details). We recall that topologically the space Y (�) is a nonstationaryMarkov compactum (see [Ver2] for de�nition).De�nition. A measure � on � with the distribution �n on its n'th level is called Markovif the sequence (Dn; �n) of random variables is a (nonhomogenous) Markov chain.Now we can de�ne for a Markov measure the notion of conditional measures.De�nition. A Markov measure � on the graph � is called central if any of the followingequivalent conditions is satis�ed:(1) For any vertex in this graph the conditional measure on the set of all paths de-scending to this vertex, is uniform.(2) � is T�-invariant.(3) � is invariant with respect to the tail partition �(�).

ERD�OS MEASURE AND THE GOLDENSHIFT 37De�nition. A central measure � on � is called ergodic if either of the following twoequivalent conditions is satis�ed:(1) The adic transformation T� is ergodic with respect to it.(2) The tail partiton is �-trivial, i.e. contains only sets whose �-measure is either 0 or1.The aim of this section is to describe1) all ergodic central measures on �.2) the action of the adic transformation T� on �.In the following theorem we will describe the ergodic central measures and the corre-sponding components of the action of T�. As was noted above, T� interchanges repre-sentations of one and the same x. We will see that the regularity or irregularity of thegeneralized canonical expansion of a given x leads to three types of possible ergodic com-ponents of the action of T�, namely, to a \full" odometer, an irrational rotation of thecircle or a special automorphism over a rotation.Theorem A.1. 1. The ergodic central measures on � are naturally parametrized by thepoints of the interval [0; 1]. We denote by �x the measure corresponding to x.2. The measure �x is continuous if and only if x 6= fN�g for any N 2Z(or, equivalently,if x has in�nite canonical expansion).3. The action of the adic transformation T� is not transitive, and its trajectories aredescribed as follows. Let x be as in the previous item, and let 'x denote the space of pathsin � such that 'x = supp�x. The set 'x is invariant under T� and we have the followingalternatives.a. If the generalized canonical expansion of x contains in�nitely many pieces \1(00)l1"with l � 1 (let us call such a piece even), then T�j'x is strictly ergodic andmetrically isomorphic to the shift by 1 on the group of certain a-adic integerswith a = (a1; a2; : : : ) being a sequence of positive integers, generally speaking,nonstationary. Thus, T�j'x has a purely discrete rational spectrum.b. If the generalized canonical expansion of x does not contain even pieces at all,then T�j'x is also strictly ergodic and metrically isomorphic to a certain irrationalrotation of the circle.c. Finally, if the generalized canonical expansion of x contains a �nite number of evenpieces, then T�j'x is metrically isomorphic to some special automorphism over arotation of the circle, i.e. to a shift on the space S1 nZ=k for some k 2 N.Proof. (1) Let 'x be the set of all paths projecting to x 2 [0; 1] (a �-�ber over x). Obvi-ously, the set 'x is invariant under T� for any x. Thus, T� is not transitive, and its actionsplits into components, each acting on a certain 'x (below we will see that for all x, exceptfor some countable set, the action of T�j'x is strictly ergodic).(2) We have the following cases. If x = 0 or x = 1, then #'x = 1. If x = fN�gfor some N 2 Z, then it is easy to see that 'x is countable and that the unique in-variant measure for T� is concentrated in a �nite number of paths (see Example 1 be-low). Henceforward in this proof we assume that x has in�nite canonical expansion. Letx = P1j=1 "j��j�1 be the generalized canonical expansion of x. We �rst split it in the

38 NIKITA SIDOROV AND ANATOLY VERSHIKfollowing way: ("1"2 : : : ) = B(0)B(1), where B(0) is either 0s or 1s for some s � 0 (ifs = 0, then B(0) = ;), and B(1) begins with \10". Such a splitting is caused by the trivialreason that the action of T� does not touch at all the set B(0), as T� only interchangescertain triples \100" and \011". So, we have the following cases (they correspond to thoseenumerated in the theorem).a. If B(1) contains in�nitely many even pieces, then x is regular (see Section 2), hence,B(1) = B1B2B3 : : : ;whereBj = 1(00)a(j)1 (01)a(j)2 (00)a(j)3 : : : (00)a(j)tj or Bj = 1(01)a(j)1 (00)a(j)2 (01)a(j)3 : : : (00)a(j)tjwith a(j)i 2 N and tj <1.b. If B(1) does not contain any even piece, then obviouslyB(1) = 1(00)a1(01)a2(00)a3 : : : or B(1) = 1(01)a1(00)a2(01)a3 : : :with aj 2 N for any j � 1.c. Finally, if the number of even pieces is �nite (but nonzero), thenB(1) = B1B2 : : : Bm eB(1);where B1; : : : ; Bm have the form described in the previous item, and eB(1) is an in�niteblock of the form described in item b.(3) Consider items a, b, c from the viewpoint of the action of Tx := T�j'x .a. The idea of the study of Tx in this case is based on two assertions of the previoussection, namely on Lemmas 2.1 and 2.2. In particular, from Lemma 2.2 it follows thatblocks Bi and Bi+1 for all i 2 N are replaced by any equivalent sequences independently,and hence it is clear that for such a point x the transformation Tx is the shift by 1 in thegroup of a-adic integers with a = (p1 + q1; p2 + q2; : : : ). This transformation Tx is knownto be strictly ergodic, i.e. there is a unique (product) measure �x invariant under it.b. We recall that in this caseB(1) = 1(00)a1(01)a2(00)a3 : : : or B(1) = 1(01)a1(00)a2(01)a3 : : :Let � = [1; a1; a2; : : : ] denote a (regular) continued fraction. We claim that in this case thetransformation Tx is strictly ergodic and metrically isomorphic to the rotation through �.The idea of the proof lies in recoding the space 'x into the second model of the adicrealization of the rotation from [VerSi] (see Section 3 of the cited work and Example 3below). The unique invariant measure can be described with the help of Theorem 2.3 fromthe cited work.c. This case in a sense is a \mixture" of the previous ones. One can easily see that ifeB(1) is parametrized by the in�nite sequence (a1; a2; : : : ) in the sense of the previous item,and if Bj = Bj(a(j)1 ; : : : ; a(j)tj ), then Tx acts on 'x as the special automorphism over the

ERD�OS MEASURE AND THE GOLDENSHIFT 39rotation through � = [1; a1; a2; : : : ] with the constant step function (� 1) and the numberof upper steps equal to Qm1 (pj + qj )� 1. So, Tx is again strictly ergodic. The proof of thetheorem is complete.Remarks. 1. It is known (see, e.g., [VerKe]) that any ergodic central measure on the Pascalgraph is also parametrized by a real in [0; 1] but in a completely di�erent manner, namely bymeans of the �rst transition measure. It is appropriate to compare that situation with theFibonacci graph. We see that in the graph �, for any � 2 [0; 1] there exists a central ergodicmeasure � such that �("1 = 0) = �. If � is irrational, then this measure is unique, namely� = �x for x =Pj "j��j�1 with � = [1; a1; a2; : : : ] and ("1"2 : : : ) = 1(00)a1(01)a2 : : : for� > 12 , and 1 � � = [1; a1; a2; : : : ] and ("1"2 : : : ) = 1(01)a1(00)a2 : : : otherwise. If � isrational, then there exists a whole interval of x in [0; 1] such that �x("1 = 0) = �.2. A typical x from the viewpoint of Lebesgue measure, of course, corresponds to thecase a. of the theorem.Examples. We illustrate the possible situations in the previous theorem with four exam-ples. For a better illustration we will use the following convention:Fig. 4We will use generalized canonical expansions, writing x � ("1"2 : : : ).1. x = ��2 � (1000 : : : ). Here 'x is countable and isomorphic to the stationary Markovcompactum with the matrix � 1 10 1�. This compactum consists of the sequence 01 and thesequences 0k11 for k � 0. For a central measure �x, �x(0k11) = �x(0l11) for any k; l,hence, this measure is concerntrated on the path 01 corresponding to the path (010101 : : : )in the initial compactum (see Fig. 5 below).

40 NIKITA SIDOROV AND ANATOLY VERSHIKFig. 5. The case x = ��22. x = 12 � (100)1. Here B(0) = ;; B(1) = �1(00)1�1. We have 'x = Q11 f011; 100g,and thus Tx is isomorphic to the 2-adic shift, i.e. the shift by 1 in the group of dyadicintegers. Therefore, Tx has the binary rational purely discrete spectrum. We depict theway of recoding the paths in 'x into the full dyadic compactum by the rule \011 � 0; 100 �1"(see Fig. 6).

Fig. 6. Recoding the paths for x = 12 , case a3. x � (1(0001)1). Here � = [1; 1; 1; : : : ] = ��1, and Tx acts as the rotation by thegolden ratio. Fig. 8 shows the way of recoding the paths in 'x into the usual model forthis rotation (\Fibonacci compactum"). Note that the natural ordering of these paths isalternating (see Fig. 7, 8).

ERD�OS MEASURE AND THE GOLDENSHIFT 41Fig. 7. Transition measures for � = ��1

Fig. 8. Recoding the paths for � = ��1, case b4. x � (1001(0001)1). For this x, the transformation Tx acts as the special automor-phism over the rotation by ��1 with a single step equal to the base (see Fig. 9).

42 NIKITA SIDOROV AND ANATOLY VERSHIKFig. 9. Recoding the paths for case cAppendix B. Arithmetic expression for block expansionsWe recall that in Section 2 we have de�ned the mapping assigning to a regularx 2 (��1; 1) the sequence of blocks B1(x); B2(x); : : : . In this Appendix we are goingto specify the mapping in an arithmetic way. To this end, we gather the canonicalexpansion of a given regular x blockwise.Recall that similarly to the canonical expansion (1.1) of reals, there exists the corre-sponding representation of positive integers. Namely, each N 2 N has a unique represen-tation in the form N = kXi=1 "iFi;where "i 2 f0; 1g; "i"i+1 = 0; "k = 1 for some k 2 N. It is usually called the Zeckendorfdecomposition. We denote by F the class of positive integers whose Zeckendorf decom-position has "1 = 1 and "i � 0 for all even i. Obviously, F as a subset of N has zerodensity. Let the height of e with a �nite canonical expansion of the form e =Pj "j��j be,by de�nition, the positive integer h(e) := maxfj : "j = 1g.Proposition B.1. Each regular x 2 (��1; 1) has a unique representation of the form(B.1) x = 1Xj=1 ej(x)�� jPi=1ni(x);where(i) ej (x) =m�� n; n 2 F ; n = [m�].(ii) nj is odd, nj � 3 for j � 1.(iii) nj � 2h(ej) + 1 for all j.

ERD�OS MEASURE AND THE GOLDENSHIFT 43Proof. Let (x) = B1B2 : : : be the block expansion of x. Suppose Bj = Bj(a(j)1 ; : : : ; a(j)tj ).We set nj(x) := 2Ptji=1 a(j)i + 1; j � 1, i.e. nj is the length of the j'th block. Letm(j)0 = 0; m(j)k = Pki=1 a(j)i , and let ej be the \value" of Bj in the sense of formula (1.1)as if it were the �rst block, i.e.ej(x) = ��1 + ��1 12 (tj�1)Xk=1 a(j)2kX�=1��2�m(j)2k�1+��= ��1 + ��2 12 (tj�1)Xk=1 ���2m(j)2k�1 � ��2m(j)2k � ; tj odd;ej(x) = ��1 + ��1 12 tjXk=1 a(j)2k+1X�=1 ��2�m(j)2k�2+��= ��1 + ��2 12 tjXk=1���2m(j)2k�2 � ��2m(j)2k�1� ; tj even:The uniqueness of expansion (B.1) follows from the condition (iii) and from the uniquenessof expansion (1.1) for any �nite sequence (and, therefore, for any block). The fact thatn 2 F follows from the de�nition of block.De�nition. We call the expansion of x 2 (��1; 1) of the form (B.1) satisfying the condi-tions (i){(iii) the arithmetic block expansion.Remark 1. nj and ej depend on Bj only.Remark 2. In fact, series (B.1) is nothing but series (1.1) rewritten in a di�erent notation.However, we will see that it has its own dynamical sense (see relation (B.2) below).Remark 3. By Item (iii), the quantities ej and nj are not completely independent. Letlj := h(ej). Taking into consideration new quantities sj := nj � 2lj and representing njas the sum sj and 2lj in formula (B.1), we come to independent multipliers, but this newform of the block representation does not seem to be natural.Remark 4. In terms of arithmetic block expansions the goldenshift acts as(B.2) S(x) = 1Xj=2 ej(x)�� jPi=2ni(x):

44 NIKITA SIDOROV AND ANATOLY VERSHIKAppendix C. Computation of densities and the polymorphism �We return to the subject of the �rst section. Recall that we have already denoted thetransformation x 7! f�xg by T , and R stands for the rotation of the circle R=Zby theangle ��1.C.1. Computation of densities.Proposition C.1. The densities d(R�)d� ; d(��)d� and d�d� are unbounded and piecewise con-stant with a countable number of steps.Proof. By virtue of the results of Section 1, it su�ces to prove the proposition only ford(R�)d� . Let E be a Borel subset of (0; 1). The idea of the study lies in the fact that R doesnot change any block beginning with the second. As usual, we consider three cases.I. E � (0; ��2). If E � (��2k; ��2k+1); k � 1, then each point x of the set E has thecanonical expansion (1.1) of the form 02k�110�. Hence the canonical expansion of x+��1is 1(00)k�110�, and �(E + ��1)�E � k;as f(1(00)k�1) = k. If, on the contrary, E � (��2k�1; ��2k); k � 1, then the situationis as follows. This interval in terms of the canonical expansion is SB 02kB mod 0, wherethe union runs over all closed blocks B.5 We have two subcases.Ia. Let in terms of the canonical expansion, E � 02k1(00)a1(01)a2 : : : (00)at1. Here E +��1 � 1(00)k�1(01)(00)a1 (01)a2 : : : (00)at1, hence �(E+��1)�E = f(B0)f(B) , where B0 is the closedblock de�ned as B0 = B0(k � 1; 1; a1; a2; : : : ; at). So, we conclude from Lemma 2.1 that�(E + ��1)�E = kp+ (k + 1)qp+ q ;where, as usual, pq = [a1; a2; : : : ; at].Ib. In the same terms, suppose E � 02k1(01)a1(00)a2 : : : (00)at1. Similarly to the above,�(E + ��1)�E = (k + 1)p + kqp+ q :II. Let E � (��2; ��1). This case is analogous to Case I. If E � (��2 + ��2k�3; ��2 +��2k�2); k � 1, then �(E � ��2)�E � 1k :If E � (��2 + ��2k�2; ��2 + ��2k�1); k � 1, then�(E � ��2)�E � ( p+qkp+(k+1)q ; E � 01(00)k�101(00)a1(01)a2 : : : (00)at1p+q(k+1)p+kq ; E � 01(00)k�101(01)a1(00)a2 : : : (00)at1:5We say that a 0-1 word is a closed block if it has the form B1 for some block B.

ERD�OS MEASURE AND THE GOLDENSHIFT 45III. Let E � (��1; 1). If E � (��1; ��1 + ��4), then �(E � ��2) = �E. If E �(��1 + ��4; 1), then E � ��2 � (��2; ��1), hence E � ��2 � 010�.IIIa. Let E � ��2 � 1(00)a1(01)a2 : : : (00)at1, then�(E � ��2)�E = 1 + pq :IIIb. Let E � ��2 � 1(01)a1(00)a2 : : : (00)at1. Here�(E � ��2)�E = 1 + qp :The proof is complete.Remark 1. Let d = d(R�)d� . Then by relation (1.7), d(��)d� = 12 (d+1), and by formula (1.12),d�d� (x) =8><>: 23 + 13d(x) + 16d�1(x + ��1); x 2 [0; ��2)23 + 13d(x); x 2 [��2; ��1)12 + 13d(x); x 2 [��1; 1]:Remark 2. From this relation follows Proposition 2.10.C.2. The polymorphism �. Let as above � : �! � be the one-sided shift. Let us askthe natural question: what is the image of � on the interval [0; 1] under the mapping �de�ned by the formula (3.1)?Note �rst that the partition into �-preimages of singletons is not invariant under �.Indeed, if, say, x = 01101, then �n(x) = 01, while n�(x) = 1101. Thus, � := ����1 :[0; 1]! [0; 1] is a multivalued mapping, i.e. a polymorphism by the terminology of [Ver4].Recall that a measure-preserving polymorphism of a measure space (X;A; �) is thediagram (X;�) �1 � (Y; �) �2�! (X;�);where �1; �2 are homomorphisms of measure spaces such that �i� = �; i = 1; 2. Instead ofan arbitrary Y it su�ces to consider X �X with the coordinate projections and a certain\bistochastic" measure � (i.e. a measure on the sigma-algebra A�A with given marginalmeasures). Such a polymorphism is called reduced.If T is an automorphismof the space Y with an invariantmeasure and � is a measurablepartition, one can de�ne the polymorphism T� : (Y� ; �) into itself as follows. Considertwo partitions of Y , namely, � and T�1�. Identifying Y� and YT�1� in the natural way, weobtain the diagram (Y� ; ��) � (Y; �) �! (Y� ; ��):Let � = � _ T�1�. Then Y� � Y� � YT�1� = Y� � Y�, and the reduced automorphism(Y� ; ��) � (Y�; ��) �! (Y� ; ��)

46 NIKITA SIDOROV AND ANATOLY VERSHIKcan be easily interpreted: it is a Markov (multivalued) mapping of the factor space Y�with the invariant measure ��. If � is a T -invariant partition (i.e. T�1� � �), then thepolymorphism is the factor endomorphism T� : Y� ! Y� . That is why the polymorphismin this context is a generalization of the notion of endomorphism (see [Ver4]).We are going to make use of these notions in our case. We de�ne the polymorphism �as the subset of [0; 1]� [0; 1] de�ned as�(x) = 8><>: �x; 0 � x < ��2�x [ �x� ��1; ��2 � x < ��1�x� ��1; ��1 � x � 1and providedwith the measure � which is the image of the product measure p. By de�nitionof the polymorphism and the Erd�os measure, �� = ��1� = �, and � being the projectionof � to both axes.Fig. 10. The polymorphism �This polymorphism was considered in [VerSi]. Note that for a Borel set E, ��1E =���1��1E = ��1E [ ��1E + ��2. Let = ( 1; 2) be the corresponding partition of�. It is possible to show that there exists a countable partition of [0; 1] into the intervalsfGkg1k=1 such that for any k and any G � Gk the ratio �(��1G \ 1)=�((��1G \ 2) isconstant. In particular, if G � (��2n; ��2n+1) for n � 1, then this ratio is equal to n(speci�cally, for G � (��1; ��2), it equals 1). The method of the proof is the same as inProposition C.1.

ERD�OS MEASURE AND THE GOLDENSHIFT 47Appendix D. An independent proof of Alexander-Zagier's formulaIn this appendix we will present the second proof of formula (3.3) which reveals somenew relations between certain structures of the Fibonacci graph �.We �rst recall that the quantity fn(k) is nothing but the frequency of the k'th vertexon the n'th level of the Fibonacci graph which was denoted by Dn (see the beginning ofSection 3). We have #Dn = Fn+2 � 1.Consider level n of the Fibonacci graph for n = 2N + 1. We denote the middle part ofDn, i.e. the segment from Fn to Fn+1 � 1, by D0n. The Erd�os measure of D0n obviouslyequals 13 + O(��n), and we will introduce the partition of D0n into 2N�1 � 1 intervals ofvertices in the following way.Recall that a Euclidean vertex is one, where the �rst block can end (they are markedin Fig. 3 for D3 and D5) and that these vertices form the Euclidean binary tree (seeSection 3). There are 2N�1 such vertices at level 2N + 1, and all of them lie in D0n. LetV (N)k denote the k'th Euclidean vertex from the left on level 2N + 1.De�nition. An open interval of vertices (N)k := (V (N)k ; V (N)k+1 ) will be called a Euclideaninterval.So, we have divided the set of vertices D0n into 2N�1 Euclidean vertices �V (N)k �2N�1k=1and 2N�1 � 1 open intervals (N)k . Now we introduce the subgraph �V associated witheach Euclidean vertex V . It is de�ned as the one containing all the successors of V in thesense of the Fibonacci graph, except any other Euclidean vertices (see Fig. 11).Fig. 11. The graph �VWe state a straightforward lemma.

48 NIKITA SIDOROV AND ANATOLY VERSHIKLemmaD.1. For any Euclidean interval (N)k there is a unique Euclidean vertex V (j)i ; j <N , such that (N)k � �V (j)i .So, any Euclidean interval is determined by a certain Euclidean vertex on one of thepreceding odd levels of the Fibonacci graph. Moreover, in the notation of the above lemma,the entropy of (N)k may be computed in terms of the frequency of V (j)i and the entropy ofeD2N+1�j. Namely, let Hn := Fn+1�1Pk=Fn fn(k) log� fn(k), and let next H2N+1 =PNj=1H(j)2N+1,where H(j)2N+1 denotes the sum over the vertices V 2 �V (j)i for all i � 2j�1. So, H(j)2N+1corresponds to all Euclidean vertices of level 2j + 1.Let next '(j)i denote the frequency of V (j)i . For instance, for j = 3, '(3)1 = '(3)4 =4; '(3)2 = '(3)3 = 5. In this notation kj =P2j�1i=1 '(j)i log� '(j)i .For any v 2 (V (N)k ; V (N)k+1 ) its frequency equals f(V (j)i ) times the frequency of the cor-responding vertex of the central part of level 2N + 1 � j. So, we established an essentialrelationship between the central part of level 2N + 1 of the Fibonacci graph and all Eu-clidean vertices V (j)i ; 1 � j � N; 1 � i � 2j�1.Lemma D.2. The following recurrence relation holds:(D.1) H2N+1 = 23 N�1Xj=1 3jH2N�2j + 13 � 4N � NXj=1 kj4j +O0@ NXj=1 kj1A ; N !1:Proof. By the above considerations,H(j)2N+1 = 2j�1Xi=1 F2N�2j+1�1Xk=F2N�2j '(j)i f2N�2j(k) log� �'(j)i f2N�2j(k)�= 2j�1Xi=1 '(j)i �H2N�2j + 13 log� '(j)i � �4N�j +O(1)��= 2H2N�2j � 3j�1 + 13 � kj4j � 4N +O(kj )(we used the fact that P2j�1i=1 '(j)i = 2 � 3j�1 easily obtained from Proposition 2.9). Hencerelation (D.1) follows.Remark. Formula (D.1) shows that the entropy of the n'th level with n odd can be com-puted by means of the entropies of the previous even levels and the entropy of the Euclideantree.Now we are ready to complete the second proof of formula (3.3). We havenH� � Fn+2�2Xk=0 fn(k)2n log� 2nfn(k) ;

ERD�OS MEASURE AND THE GOLDENSHIFT 49whence nH� � 3 Fn+1�1Xk=Fn fn(k)2n log� 2nfn(k) ;and(D.2) Hn � 13���H��n2n:From relation (D.2) it follows that in the sum N�1Pj=1 3jH2N�2j the �rst terms are morevaluable than the last. Thus, from formulas (D.1) and (D.2) and from the fact thatkN =P2N�1i=1 '(N)i log� '(N)i < 2(N � 1)3N�1 it follows that13���H�� � (2N + 1)22N+1 � 23 NXj=1 3j � 13���H��(2N � 2j)4N�j+ 13 � 4N NXj=1 kj4j ;whence, after straightforward computations,18���H�� � NXj=1 kj4j ; N !1: �Remark. The Euclidean tree, being symmetric, naturally splits into two binary subtrees(left and right) being symmetric. If we label each vertex of the left subtree with thecorresponding rational p=q, then this left subtree turns out to coincide with the Farey treeintroduced and studied in detail in [Lag].References[Ab] L. M. Abramov, The entropy of an induced automorphism, Sov. Dokl. 128 (1959), 647{650.[AdWe] R. L. Adler and B. Weiss, Entropy, a complete metric invariant for automorphisms of the torus,Proc. Nat. Acad. Sci. USA 57 (67), 1573{1576.[AlYo] J. C. Alexander and J. A. Yorke, Fat baker's tranformations, Ergod. Theory Dynam. Systems4 (1984), 1{23.[AlZa] J. C. Alexander and D. Zagier, The entropy of a certain in�nitely convolved Bernoulli measure,J. London Math. Soc. 44 (1991), 121{134.[Av] A. Avez, Entropie des groupes de type �ni, C. R. Acad. Sci. Paris 275A (1972), 1363{1366.[Bel] R. M. Belinskaya, Generalized powers of an automorphism and entropy, Siberian Math. J. 11(1970), 739{749.[Ber] A. Bertrand-Mathis, Developpement en base �, r�epartition modulo un de la suite (x�n)n�0;langages cod�es et �-shift, Bull. Math. Soc. Fr. 114 (1986), 271{323.[Car] L. Carlitz, Fibonacci representations I, Fibonacci Quart. 6 (1968), 193{220.[Cas] J. Cassels, An Introduction in Diophantine Approximation, Cambridge Univ. Press, 1957.

50 NIKITA SIDOROV AND ANATOLY VERSHIK[DaKrSo] K. Dajani, C. Kraaikamp and B. Solomyak, The natural extension of the �-transformation,Acta Math. Hung. 73 (1996), 97{109.[De] Y. Derrenic, Quelques applications du th�eor�eme ergodique sous-additif, Ast�erisque 74 (1980),183{201.[DuSiTh] J. M. Dumont, N. Sidorov and A. Thomas, Number of representations related to a linear recur-rent basis, Preprint of Institut de Math�ematiques de Luminy, Marseille, Pr�etirage no96 � 26.[Er] P. Erd�os, On a family of symmetric Bernoulli convolutions, Amer. J. Math. 61 (1939), 974{976.[ErJoKo] P. Erd�os, I. Jo�o and V. Komornik, Characterization of the unique expansions 1 = 1Pi=1 q�ni andrelated problems, Bull. Soc. Math. Fr. 118 (1990), 377{390.[Fr] Ch. Frougny, Representations of numbers and �nite automata, Math. Systems Theory 25(1992), 37{60.[FrSa] Ch. Frougny and J. Sakarovitch, Automatic conversion from Fibonacci to golden mean, andgeneralization, to appear in Int. J. of Alg. and Comput.[Ga] A. Garsia, Entropy and singularity of in�nite convolutions, Pac. J. Math. 13 (1963), 1159{1169.[Ge] A. O. Gelfond, On a certain general property of number systems, Izvestiya Akad. Nauk SSSR,ser. math. 23 (1959), 809{814.[JeWi] B. Jessen and A. Wintner, Distribution functions and the Riemann zeta function, Trans. Amer.Math. Soc. 38 (1938), 48{88.[KaVer] V. Kaimanovich and A. Vershik, Random walks on discrete group: boundary and entropy, Ann.Prob. 11 (1983), 457{490.[KenVer] R. Kenyon and A. Vershik, Arithmetic construction of so�c partitions of hyperbolic toral auto-morphisms, Erg. Theory Dynam. Systems 18 (1998), 357{372.[Lag] J. C. Lagarias, Number theory and dynamical systems, Proc. Symp. Applied Math. 46 (1992),35{72.[Lal] S. Lalley, Random series in powers of algebraic integers: Hausdor� dimension of the limitdistribution, to appear in J. London Math. Soc..[Leb] S. Le Borgne, Dynamique symbolique et propri�et�es stochastiques des automorphisms du tore :cas hyperbolique et quasi-hyperbolique, Th�ese doctorale (1997).[LePo] F. Ledrappier and A. Porzio, A dimension formula for Bernoulli convolutions, J. Stat. Phys.76 (1994), 1307{1327.[Lev] W. J. LeVeque, Topics in Number Theory, Addison-Wesley, 1956.[LivVer] A. N. Livshits and A. M. Vershik, Adic models of ergodic transformations, spectral theory andrelated topics, Adv. in Soviet Math. 9 (1992), 185{204.[Or] D. Ornstein, Ergodic theory, randomness and dynamical systems, New Haven and London, YaleUniv. Press, 1974.[Pa] W. Parry, On the �-expansions of real numbers, Acta Math. Hungar. 11 (1960), 401{416.[Pu] I. Pushkarev, Multizigzag ideal lattices and the enumeration of Fibonacci partitions, Zap. Na-uchn. Sem. POMI 223 (1995), 280{312. (in Russian)[Re] A. R�enyi, Representations for real numbers and their ergodic properties, Acta Math. Hungar. 8(1957), 477{493.[Si] N. A. Sidorov, The summation function for the number of Fibonacci representations, PDMIpreprint 15/1995.[SiVer] N. Sidorov and A. Vershik, Bijective arithmetic codings of hyperbolic automorphisms of the2-torus, and binary quadratic forms, J. Dynam. Control Sys. 4 (1998), 365{400.[StVo] S. Str�atil�a and D. Voiculescu, Representations of AF algebras and of the group U(1), LectureNotes in Math. 486 (1975), Springer-Verlag, Berlin-Heidelberg-New York.[Ver1] A. M. Vershik, Uniform algebraic approximation of shift and multiplication operators, Dokl.Akad. Nauk. SSSR 259 (1981), 526{529; English transl. Soviet Math. Dokl 24 (1981), 97{100.[Ver2] A. M. Vershik, A theorem on Markov periodic approximation in ergodic theory, Zap. Nauchn.Sem. LOMI 115 (1982), 72{82 (in Russian); English transl. J. Soviet Math. 28 (1985), 667{673.[Ver3] A. M. Vershik, Locally transversal symbolic dynamics, Algebra and Analysis 6 (1994), no. 3 (inRussian); English transl. St. Petersburg Math. J. 6 (1995), 529{540.

ERD�OS MEASURE AND THE GOLDENSHIFT 51[Ver4] A. M. Vershik, Multivalued mappings with invariant measure (polymorphisms) and Markovoperators, Zap. Nauchn. Sem. LOMI 72 (1977), 26{61 (in Russian); English transl. J. SovietMath. 23 (1983), 2243{2266.[Ver5] A. M. Vershik, The �badic expansions of real numbers and adic transformation, Prep. ReportInst. Mittag{Le�er, 1991/1992, pp. 1{9.[Ver6] A. M. Vershik,Arithmetic isomorphism of the toral hyperbolic automorphisms and so�c systems,Funct. Anal. Appl. 26 (1992), 22{24.[VerKe] A. Vershik and S. Kerov, Locally semisimple algebras. Combinatorial theory and K0-functor,Current problems in Mathematics. Newest results. Itogi Nauki i Tehniki. VINITI 26 (1985),3{56; English transl. J. Soviet Math. 38 (1987), 1701{1733.[VerSi] A. Vershik and N. Sidorov, Arithmetic expansions associated with rotation of the circle andwith continued fractions, Algebra and Analysis 5 (1993), 97{115 (in Russian); English transl.St. Petersburg Math. J. 5 (1994), 1121{1136.[Y] L.-S. Young, Dimension, entropy and Lyapunov exponents, Ergod. Theory Dynam. Systems 2(1982), 109{124.St. Petersburg Department of Steklov Mathematical Institute, Fontanka 27, St. Pe-tersburg 191011, RussiaE-mail address: sidorov@pdmi.ras.ru, vershik@pdmi.ras.ru