lyapunov bases of a vector space and maximal λ-subspaces

ISSN 0012-2661, Differential Equations, 2007, Vol. 43, No. 8, pp. 1041–1050. c© Pleiades Publishing, Ltd., 2007.Original Russian Text c© V.T. Borukhov, 2007, published in Differentsial’nye Uravneniya, 2007, Vol. 43, No. 8, pp. 1019–1028.

ORDINARY DIFFERENTIAL EQUATIONS

Lyapunov Bases of a Vector Spaceand Maximal λ-Subspaces

V. T. BorukhovInstitute of Mathematics, National Academy of Sciences, Minsk, Belarus

Received October 20, 2006

DOI: 10.1134/S0012266107080022

INTRODUCTION

The exponents (characteristic numbers) and related special bases in the solution space of a linearsystem of differential equations were defined by Lyapunov [1, p. 26]. Bogdanov [2, 3] constructedan abstract version of the theory of exponents for finite-dimensional vector spaces. He also definedλ-bases (Lyapunov bases) for infinite-dimensional vector spaces and noted some of its properties [4].Further development of the general theory of exponents in finite-dimensional vector spaces was givenin the monograph [5, Chap. 1].

Abstract exponents were defined in [2] with the use of the axiomatic approach: independentproperties of characteristic numbers are considered to be determining conditions of a class of non-linear functionals defined on a vector space (and ranging in a linearly ordered set). These functionalswere referred in [2, 3] to as Lyapunov norms; below, following [6], we call them Lyapunov–Bogdanovnorms (or functionals). In the present paper, we consider Lyapunov–Bogdanov functionals and thecorresponding Lyapunov sets, which are Lyapunov bases of subspaces of an infinite-dimensionalvector space. It turns out that such bases cannot necessarily be continued to Lyapunov bases ofthe ambient space. In the present paper, we define maximal λ-subspaces characterizing the im-possibility extension of Lyapunov sets and obtain existence conditions for Lyapunov bases. Suchconditions are important, since there exist Lyapunov–Bogdanov functionals for which there areno Lyapunov bases. The corresponding examples will be presented below. They are also coun-terexamples to Bogdanov’s assertion [4] that there exist Lyapunov bases for an arbitrary Lyapunovnorm.

The present paper consists of three sections. In the first section, we describe the standardstratification induced on a vector space by a Lyapunov–Bogdanov functional. In the second section,we define classes of founded and nonfounded Lyapunov–Bogdanov functionals, describe the localstructure of the set of maximal λ-subspaces, and obtain necessary and sufficient existence conditionsfor Lyapunov bases. The third section contains examples of Lyapunov–Bogdanov functionals: minusthe logarithmic norm of a formal power series and multidegrees of polynomials in finitely manyvariables, the lexicographic representation of Lyapunov–Bogdanov functionals, and interpretationof eigenvalues of the Laplace operator as a Lyapunov–Bogdanov functional on a vector space.

This series of Lyapunov–Bogdanov functionals can readily be continued. For example, theorders of piecewise smooth functions at a point [7, p. 175], taken with the minus sign, as well astime optimization functionals [8] of linear control discrete systems specifying [9] invariants of theaction of a feedback group on the system, belong to the class of Lyapunov–Bogdanov functionals.Lyapunov–Bogdanov functionals are closely related (see Section 1) to the notion of ultrametric fornon-Archimedean metric spaces.

Let us give some notation used throughout the following: ω = {0, 1, . . .} is the well-orderedset of positive integers; x < ω ⇔ x ∈ ω; dim X is the dimension of a vector space X (which isequal to the cardinality of an algebraic basis of the space X). The symbols + and ⊕ stand forthe algebraic and direct sum, respectively, of subspaces of a vector space. A family {Hd | d ∈ I} ofsubspaces is said to be independent if Hd∩

(+i�=d Hi

)= 0 for all d ∈ I. Note that some subspaces of

an independent family may well be zero. The algebraic sum of subspaces forming an independentfamily is a direct sum and is denoted by

⊕d∈I Hd.

1041

1042 BORUKHOV

1. DEFINING CONDITIONSAND STRATIFICATION OF A VECTOR SPACE

Let X be a given vector space over the field K, and let Λ be a linearly ordered set. A mappingλ : X → Λ satisfying the determining conditions

λ(cx) ≤ λ(x), λ(x + y) ≤ max{λ(x), λ(y)} ∀c ∈ K, ∀x, y ∈ X (1)

is called a Lyapunov–Bogdanov functional (norm). It is convenient to denote the range of a func-tional λ by the symbol |λ|.

The union of all level sets of a functional λ forms a stratification of the space X, X =⋃d∈|λ| λ

−1(d). Actually, this stratification completely characterizes the functional λ. To provide aclearer description of this stratification, we define the sets

Xd = {x | λ(x) ≤ d}, Xd = {x | λ(x) < d} ∀d ∈ |λ|.

The least element of the set |λ|, which obviously exists and is equal to λ(0), will be denotedby −∞. Obviously, X−∞ = ∅, where ∅ is the empty set. If it is necessary to emphasize thedependence of the sets Xd and Xd on the functional λ, then we write Xλd and Xλd.

The determining conditions (1) readily imply the following assertion.

Proposition 1. The sets Xd, d ∈ |λ|, and Xd, d ∈ |λ|\{−∞}, are subspaces of X satisfying thefollowing conditions.

1. Xd ⊂ Xd for all d ∈ |λ|\{−∞} (here and throughout the following, the symbol ⊂ stands forstrict inclusion).

2. If d1, d2 ∈ |λ| and d1 < d2, then Xd1 ⊂ Xd2 and Xd1 ⊂ Xd2 . Moreover, λ−1(d) = Xd\Xd forall d ∈ |λ|.

Therefore, the sets

X :={

Xd | d ∈ |λ|}

, X :={Xd | d ∈ |λ|\{−∞}

}, X = X ∪ X

are chains of inclusion-ordered subspaces, and the stratification of the space X induced by theLyapunov–Bogdanov functional λ has the form

X =⋃

d∈|λ|

(Xd\Xd

). (2)

By using the notation introduced in the theory of stratification manifolds, we call the set λ−1(d)a stratum and the chain X, a filtration of the space X. If |λ| is finite, then X = X. In particular,this relation holds if dimX = n < ∞, since card |λ| ≤ n + 1 in this case.

Note that in [5, Chap. I] chains of subspaces in a finite-dimensional vector space were calledpyramids and strata were called pyramid steps. The same terminology was used in [10] in theinvestigation of complete chains of subspaces in a finite-dimensional vector space. One can showthat step relations of suborder defined in [10] are naturally related to classes of Lyapunov–Bogdanovfunctionals.

Lyapunov–Bogdanov functionals can be useful in the description of classes of non-Archimedeanmetric spaces, also known as ultrametric spaces [11, p. 602 of the Russian translation]. Indeed, leta Lyapunov–Bogdanov functional λ : X → Λ satisfy the following conditions: Λ = [0,∞), λ(0) = 0,and λ(x) �= 0 if x �= 0. Then the functional � : X ×X → Λ, �(x, y) = λ(x− y), defines a metric onthe space X, since, obviously, the following determining conditions are satisfied: 1) �(x, y) ≥ 0 forall x, y ∈ X; 2) �(x, y) = 0 if and only if x = y; 3) �(x, y) = �(y, x) for all x, y ∈ X; 4) the triangleinequality �(x, y) ≤ �(x, z)+�(z, y) is valid for all x, y, z ∈ X. In addition, we have the strengthenedtriangle inequality �(x, y) ≤ max{�(x, z), �(z, y)} for all x, y, z ∈ X, which implies that � is an

DIFFERENTIAL EQUATIONS Vol. 43 No. 8 2007

LYAPUNOV BASES OF A VECTOR SPACE AND MAXIMAL λ-SUBSPACES 1043

ultrametric and the pair (X, �) is an ultrametric space. The main objects of the ultrametric space,that is, the balls Ud(a) = {x ∈ X | �(x, a) ≤ d} and U−

d (a) = {x ∈ X | �(x, a) < d} and the sphereSd(a) = {x ∈ X | �(x, a) = d}, can be explicitly described in terms of the subspaces Xd and Xd.For example Ud(a) = a + Xd, in particular, Ud(a) = Ud(0) = Xd if λ(a) ≤ d.

The following simple assertion is useful when verifying whether given functionals belong to theset of Lyapunov–Bogdanov functionals.

Proposition 2. Let λ : X → Λ be a functional defined on the vector space X. If

Xλ :={Xλd | d ∈ |λ|

}

is a filtration of the space X, then λ satisfies the determining condition (1).

Proof. Let a vector x be an arbitrary solution of the equation λ(x) = d. Since Xλd is a subspace,we have cx ∈ Xλd for all c ∈ K. Consequently, λ(cx) ≤ d for all c ∈ K. Further, if λ (x1) = d1,λ (x2) = d2, and d2 ≤ d1, then λ (x1 + x2) ≤ d1 = max {λ (x1) , λ (x2)} since x1 + x2 ∈ Xλd1 .Therefore, condition (1) is satisfied, and the proof of the proposition is complete.

2. LYAPUNOV SETS AND MAXIMAL λ-SUBSPACES

2.1. Lyapunov Sets

Recall that a nonempty set g ⊂ X is said to be linearly independent if an arbitrary linearcombination of finitely many vectors in g with nonzero coefficients is not zero. If, in addition, anarbitrary vector in X can be represented in the form of a finite linear combination of vectors in g,then g is called an algebraic basis (or a Hamel basis) of the space X.

Definition 1. A linearly independent set g is called a Lyapunov set (with respect to theLyapunov–Bogdanov functional λ) if the value of λ on any finite linear combination of vectors ing with nonzero coefficients is equal to the maximal value of λ on vectors occurring in the linearcombination. If, in addition, g is an algebraic basis, then g is called a Lyapunov algebraic basis [2–4](or a normal basis [1, 5]) of the space X.

Since below we consider only algebraic Lyapunov bases, as a rule, we omit the term “algebraic.”We denote the set of all Lyapunov sets partially ordered by inclusion by the symbol L(λ,X) and theset of all linearly independent sets by B(X). Following the terminology of the theory of partiallyordered sets, we say that an element g ∈ L(λ,X) is maximal if the conditions f ∈ L(λ,X) andg ⊆ f imply that f = g.

It is well known that the maximal elements of the set B(X) (and only these elements) arealgebraic bases of the space X. In addition, any linearly independent set either is an algebraicbasis or can be complemented to an algebraic basis. For maximal elements of the set L(λ,X), thesituation can be quite different. Lyapunov bases are simultaneously maximal elements in L(λ,X),but there are pairs (λ,X) for which some maximal elements are not Lyapunov bases, which isequivalent to the fact that they cannot be continued to Lyapunov bases. There are also pairs(λ,X) for which Lyapunov bases do not exist. To investigate the above-mentioned properties ofthe set L(λ,X), we introduce the following notion.

Definition 2. A subspace H ⊆ X represented as the direct sum H =⊕

d∈|λ| Hd of subspacesHd satisfying the conditions Xd ⊕ Hd ⊆ Xd for all d ∈ |λ| is called a λ-subspace; if Xd ⊕ Hd = Xd

for all d ∈ |λ|, then it is called a maximal λ-subspace.

Therefore, the definition of a λ-subspace is equivalent to the definition of an independent set{Hd | d ∈ |λ|} of subspaces satisfying the conditions Xd ⊕ Hd ⊆ Xd for all d ∈ |λ|.

Now let g ∈ B(X). It follows from the stratification (2) that the set g admits the disjointpartition

g =⋃

d∈|λ|

gd, gd ⊂ Xd\Xd. (3)

By {gd}Kwe denote the linear span of the set gd. Set Gd = {gd}K

and G = {g}K.


1044 BORUKHOV

Theorem 1. A linearly independent set g belongs to L(λ,X) if and only if the direct sum

G =⊕

d∈|λ|

Gd (4)

is a λ-subspace.

Proof. Suppose that the direct sum occurring in (4) is a λ-subspace. By virtue of (3), an arbi-trary linear combination of vectors in the set g can be represented in the form

z = c0l0 + · · · + cmlm + · · · + cklk, ci ∈ K\0 ∀i ≤ k, (5)

where l0, . . . , lk are vectors satisfying the conditions

li ∈ gd ∀i ≤ m, d = λ (l0) = · · · = λ (lm) > λ (lm+1) ≥ · · · ≥ λ (lk) . (6)

It follows from (1) and (5) that λ(z) ≤ λ (c0l0 + · · · + cmlm). Actually, by virtue of (6) and theassumption Xd ⊕ Gd ⊆ Xd, this inequality is an equality. Therefore,

λ(z) = λ (c0l0 + · · · + cmlm) = λ (l0) ,

which, together with the definition of a Lyapunov set, implies that g ∈ L(λ,X).Conversely, let g ∈ L(λ,X). Let us show that G is a λ-subspace. To this end, we note that

the algebraic sum Xd + Gd is a direct sum and Xd + Gd ⊆ Xd for all d ∈ |λ|\{−∞}. Indeed, thelast inclusion is obvious. Suppose that Xd + Gd is not a direct sum. Then there exists a vector zsatisfying the conditions z �= 0 and z ∈ Gd ∩ Xd. Since z ∈ Xd, we have λ(z) < d. On the otherhand, z ∈ Gd ⊂ Xd\Xd; therefore, λ(z) = d. The resulting contradiction completes the proof.

Theorem 2. A Lyapunov set g is a maximal element of the set L(λ,X) if and only if the directsum (4) is a maximal λ-subspace.

Proof. Suppose that G is a maximal λ-subspace but g is not a maximal element of the setL(λ,X). Then there exists an f ∈ L(λ,X) such that g ⊂ f . Consequently, there exists a vectorz that belongs to the set f but does not belong to g. Since Xd ⊕ Gd = Xd, where d = λ(z),it follows that z can be represented in the form (5), (6). By (6) and the condition g ⊂ f , theset {z, l0, . . . , lm} is a subset of the set f ; moreover, λ(z) = λ (l0) = · · · = λ (lm) = d. Sincef is a Lyapunov set, it follows from Theorem 1 that Xd ⊕ {z, l0, . . . , lm}

K⊆ Xd. In particu-

lar, λ (z − c0l0 − · · · − cmlm) = d. On the other hand, relations (5) and (6) imply the inequalityλ (z − c0l0 − · · · − cmlm) = λ (cm+1lm+1 + · · · + cklk) < d. From the resulting contradiction, we findthat f = g; i.e., g is a maximal element of the set L(λ,X).

Conversely, let g be a maximal element. It follows from Theorem 1 that the algebraic sumXd + Gd is a direct sum. Let us show that Xd ⊕Gd = Xd for all d ∈ |λ|. We suppose the contrary,complement Xd ⊕ Gd to the space Xd, Xd = Xd ⊕ Gd ⊕ Kd, and choose an algebraic basis in Kd.As a result, we obtain a maximal λ-space and a Lyapunov set f such that g ⊂ f , which contradictsthe fact that g is a maximal element. Therefore, Xd⊕Gd = Xd for all d ∈ |λ|; i.e., G is the maximalλ-subspace. The proof of the theorem is complete.

2.2. Maximal λ-Subspaces

We denote the set of λ-subspaces and the set of maximal λ-subspaces by S(λ) and Sm(λ), respec-tively. Theorems 1 and 2 imply that it is useful to investigate various characteristics of the sets S(λ)and Sm(λ), for example, for the description of nonextendible Lyapunov sets. In particular, thesecharacteristics depend on whether the range |λ| of the functional λ is nonfounded, in other words,whether the linearly ordered set |λ| is well ordered. In this connection, a Lyapunov–Bogdanovfunctional λ is said to be founded if |λ| is a well-ordered set and nonfounded otherwise.

It is known (e.g., see [12, p. 233]) that a linearly ordered set is not well ordered if and onlyif it contains a subset of the ordinal type ω∗, that is, a subset ordinally isomorphic to the set{. . . ,−2,−1, 0}. This readily implies the characteristic property of the class of nonfounded func-tionals.



Proposition 3. An Lyapunov–Bogdanov functional λ belongs to the class of nonfounded func-tionals if and only if |λ| contains a subset of the form

d∗ = {di | di+1 < di, i < ω} .

The description of the structure of the set Sm(λ) for a nonfounded functional λ is a very difficultproblem in the general case. Here we consider the local structure of the set Sm(λ). Throughout thefollowing, we index individual vectors in the set Xd\Xd by the symbol d; for example, the symbolxdi

means that λ (xdi) = di.

Let H ∈ Sm(λ), let λ be a nonfounded functional, and let d∗ = {d0, d1, . . .} be the subsetindicated in Proposition 3. It follows from Definition 2 that there exist vectors xdi

∈ Hdifor

all di ∈ d∗.

Lemma 1. Let Hdibe a subspace represented by a direct sum

Hdi= {xdi

}K⊕ Jdi

∀di ∈ d∗.

Then the subspace K =⊕

d∈|λ| Kd, where

Kd ={

Hd if d �∈ d∗{xdi

+ xdi+1

}K⊕ Jdi

if d = di ∈ d∗ (7)

satisfies the conditions K ∈ Sm(λ), K ⊂ H, and dimH/K = 1.

Proof. Since di+1 < di, it follows that the algebraic sum Xdi+

{xdi

+ xdi+1

}K

+ Jdiis a direct

sum; in addition, xdi+1 ∈ Xdi. This, together with the condition H ∈ Sm(λ), implies the chain of

equalities

Xdi⊕ Kdi

= Xdi⊕

{xdi

+ xdi+1

}K⊕ Jdi

= Xdi⊕ {xdi

}K⊕ Jdi

= Xdi⊕ Hdi

= Xdi. (8)

The relation Xd ⊕Kd = Xd, which is valid for any d ∈ |λ| by virtue of (7) and (8), implies thatK ∈ Sm(λ).

Obviously, K ⊆ H. Therefore, it remains to show that dimH/K = 1. To this end, we notethat the vector xd0 does not belong to K. Indeed, suppose the contrary. Then the relationxd0 = c0 (xd0 + xd1) + · · · + cl

(xdl

+ xdl+1

)should be valid for some set ci ∈ K, i = 0, . . . , l.

Since the vectors xd0 , . . . , xdl+1 are linearly independent, it follows that the vector c = (c0, . . . , cl)satisfies the system of equations

c0 − 1 = 0, c0 + c1 = 0, . . . , cl−1 + cl = 0, cl = 0,

which has no solution. Consequently, xd0 �∈ K. Consider the direct sum {xd0}K⊕ K. We have

{xd0}K⊕ K = {xd0}K

⊕(

⊕

i<ω

({xdi

+ xdi+1

}K⊕ Jdi

))

⊕(

⊕

d �∈d∗

Hd

)

=⊕

i∈ω

({xdi}

K⊕ Jdi

) ⊕(

⊕

d �∈d∗

Hd

)

=⊕

d∈|λ|

Hd = H.

This readily implies the relation dimH/K = 1 and completes the proof of the lemma.

Corollary 1. Let λ be a nonfounded Lyapunov–Bogdanov functional. Then for any H ∈ Sm(λ),there exist chains of the form {Mi | i < ω} satisfying the conditions Mi ∈ Sm(λ), dimMi/Mi+1 = 1for all i < ω, and M0 = H.


1046 BORUKHOV

Lemma 2. Let H ∈ Sm(λ) and H �= X. Then the subspace G = {a}K ⊕ H belongs to the setSm(λ) for any vector a �∈ H.

Proof. By Definition 2, one should indicate a representation G =⊕

d∈|λ| Gd, Xd ⊕Gd = Xd forall d ∈ |λ|. We redenote the vector a by setting a = xd0 . Recall that, by our convention, d0 = λ(a).Since Xd0 ⊕ Hd0 = Xd0 and xd0 �∈ Hd0 , it follows that there exists an expansion xd0 = zd0 + xd1 ,where zd0 ∈ Hd0 and xd1 ∈ Xd0 . We rewrite the subspace Hd0 in the form of the direct sumHd0 = {zd0}K

⊕ Ld0 and set Gd0 = {xd0}K⊕ Ld0 . Note that the condition xd1 ∈ Xd1 implies the

inequality d0 > d1. In addition, we have the relation Xd0 ⊕ Gd0 = Xd0 , since

Xd0 ⊕ Gd0 = Xd0 ⊕ {zd0 + xd1}K⊕ Ld0 = Xd0 ⊕ {zd0}K

⊕ Ld0 = Xd0 .

We define the spaces Gdifor all i ∈ ω0 by induction on i: for the (i + 1)st induction step, we

have Gdi+1 ={xdi+1

}K⊕Ldi+1 , where the index di+1 and the vector xdi+1 are given by the expansion

xdi= zdi

+ xdi+1 (zdi∈ Hdi

, xdi+1 ∈ Xdi)

on the ith induction step and the subspace Ldi+1 is given by the expansion

xdi+1 = zdi+1 + xdi+2 (zdi+1 ∈ Hdi+1, xdi+2 ∈ Xdi+1)

and the direct sum Hdi+1 ={zdi+1

}K⊕Ldi+1 . The above-performed construction readily implies the

inequalities d0 > d1 > · · · for elements of the set d∗ = {d0, d1, . . .} and the relations Xdi⊕Gdi

= Xdi

for all di ∈ d∗. Hence it follows that the subspace G′ =⊕

d∈|λ| Gd, where Gd = Gdiif d ∈ d∗ and

Gd = Hd if d ∈ |λ|\d∗, belongs to the set Sm(λ).Let us prove the relation G′ = G. Since xd0 ∈ G and zd0 ∈ Hd0 ∈ G, we have xd1 = xd0 −zd0 ∈ G

and further, by induction, xdi∈ G for all i ∈ ω0. This, together with the inclusions Ldi

⊂ Hdi⊂ G

for all di ∈ d∗, implies that Gdi⊂ G for all di ∈ d∗. Now, from the conditions Gd ⊂ G for all

d ∈ |λ|\d∗, we obtain G′ ⊆ G. On the other hand, zdi= xdi

−xdi+1 ∈ G′ for all di ∈ d∗. In addition,xd0 ∈ Gd0 ⊆ G′ and Ldi

⊂ G′ for all di ∈ d∗; consequently, Hdi⊂ G′ for all di ∈ d∗. This, together

with the relations Hd = Gd for all d ∈ |λ|\d∗, implies that G ⊆ G′. Therefore, G′ = G. The proofof the lemma is complete.

Corollary 2. Let H ∈ Sm(λ). Then the bundle of subspaces

OH = {M | M ⊇ H, dim M/H < ω}

consists of maximal λ-subspaces.

Let us state necessary and sufficient conditions for the existence of a Lyapunov basis for thepair (λ,X). Recall that a partially ordered set Y is said to be inductive if, for each chain A ⊆ Yin Y , there exists an upper bound (an element b ∈ Y such that a ≤ b for all a ∈ A). By [13, p. 55 ofthe Russian translation], an inductive set contains at least one maximal element (the Zorn lemma),and each chain in a partially ordered set is contained in some maximal chain (the Kuratowskilemma).

Theorem 3. The following assertions are equivalent :(a) there exists a Lyapunov basis of the pair (λ,X);(b) the space X is a maximal λ-subspace;(c) the set Sm(λ) of maximal λ-subspaces partially ordered by inclusion is inductive.

Proof. The implication (a) ⇒ (b) is a consequence of Theorem 2. The implication (b) ⇒ (c)is valid, since the maximal λ-subspace X is an upper bound for an arbitrary chain A ⊂ Sm(λ).To prove the implication (c) ⇒ (a), we consider the maximal, in the sense of the Zorn lemma,



element H ∈ Sm(λ). Suppose that H �= X. Then, by Lemma 2, there exists a maximal λ-subspaceG such that G ⊃ H. But the inclusion H ⊂ G contradicts the requirement that H is maximalin Sm(λ). Consequently, H = X. This, together with Theorem 2, implies the existence of aLyapunov basis and completes the proof of the theorem.

It follows from Lemmas 1 and 2 and the Kuratowski lemma that maximal chains of maximalλ-subspaces are locally complete.

2.3. Invariant Sequence

An important characteristic of an arbitrary Lyapunov–Bogdanov functional is given by thesequence

q(λ) =(qd(λ) = dim Xλd/Xλd | d ∈ |λ|

).

Here q−∞(λ) = dim Xλ−∞. Note two properties of the sequence q(λ). First, it is invariant underthe action of the group GL(X) of linear invertible transformations on the space X. Indeed, ifT ∈ GL(X), then one can readily see that q(λ) = q (λT ), where λT (x) := λ(Tx) for all x ∈ X andλT is a Lyapunov–Bogdanov functional. Second, the sequence q(λ) is uniquely determined by anarbitrary maximal λ-subspace. Indeed, if H ∈ Sm(λ), then, obviously, the sequence q(λ) can becomputed by the formula q(λ) = (qd(λ) = dim Hd | d ∈ |λ|).

Theorem 3, together with the relations qi(λ) = dimHi for all i ∈ |λ|, provides necessary existenceconditions for a Lyapunov basis via the sequence q(λ).

Corollary 3. If there exists a Lyapunov basis for an Lyapunov–Bogdanov functional λ, then∑

i≤d

qi(λ) = dim Xλd ∀d ∈ |λ|, (9)

where summation is performed by the arithmetic rules for cardinal numbers (e.g., see [12, p. 189 ofthe Russian translation]).

Note that relation (9) does not necessarily guarantee the existence of a Lyapunov basis. A relatedexample is given below, in item 3.1.

2.4. Founded Lyapunov–Bogdanov Functionals

Consider the set Sm(λ) of maximal λ-subspaces in the case of a founded functional λ. In thiscase, the structure of Sm(λ) is very simple.

Theorem 4. The set Sm(λ) of a founded functional λ consists of the unique element, thespace X.

Proof. Let H ∈ Sm(λ), and let λ be a founded Lyapunov–Bogdanov functional. If H =⊕d∈|λ| Hd �= X, then there exists a vector zd0 �∈ H. Since Xd0 = Xd0 ⊕ Hd0 , it follows that there

exists an expansion zd0 = xd0 +zd1 , where d0 > d1, xd0 ∈ Hd0 , and zd1 ∈ Xd1 . In addition, zd1 �∈ Hd1 ,since if zd1 ∈ Hd1 , then necessarily zd0 ∈ H. By expanding the vector zd1 in the same way andby proceeding in a similar way, we obtain a subset d∗ = {di | di+1 < di, i < ω} of the set |λ|.By Proposition 3, the existence of such a subset contradicts the assumption that λ is a foundedLyapunov–Bogdanov functional. Therefore, H = X. The proof of the theorem is complete.

Corollary 4. The set S(λ) of λ-subspaces of a founded Lyapunov–Bogdanov functional coin-cides with the lattice of all subspaces of the space X.

Proof. It is known that an arbitrary subset of a well-ordered set inherits the property ofbeing well ordered; therefore, the restriction of a founded Lyapunov–Bogdanov functional λ toan arbitrary subspace of X is also founded. This, together with Theorem 4, implies that eachsubspace of X belongs to the set S(λ). The proof of the corollary is complete.

Corollary 5. If λ is a founded Lyapunov–Bogdanov functional, then an arbitrary Lyapunov seteither is a Lyapunov basis or can be continued to a Lyapunov basis.


1048 BORUKHOV

3. EXAMPLES OF LYAPUNOV–BOGDANOV FUNCTIONALS

3.1. Logarithmic Norm

Consider the vector space Xs of formal power series of the form x(µ) = x0 + x1µ + · · · (xi ∈ R

for all i < ω) over the field R of real numbers and its subspace Xp of polynomials. Note that thesets of series and polynomials treated either as vector spaces or as rings with respect to standardoperations of addition and multiplication of series (in this case, the notation R[[µ]] and R[µ] isused) are widely used in mathematical theory of systems for the investigation of cell automata,n-D–systems, etc.

An important characteristic of the ring R[[µ]] is given by the logarithmic norm (the lower degreein other notation) [14, p. 320] deg x of a vector x, which is equal to k if xk �= 0 and xj = 0 forall j < k and is infinite if x = 0. By setting λs(x) = − degx, we obtain the Lyapunov–Bogdanovfunctional λs : Xs → Λs = |λs| = {−∞, . . . ,−1, 0} = {−∞} ∪ ω∗.

By restricting λs to the space Xp, we obtain the Lyapunov–Bogdanov functional λp : Xp →|λp| = {−∞} ∪ ω∗. Both functionals λs and λp belong to the class of nonfounded functionals.

The spaces Xsd, Xsd, Xpd, and Xpd satisfy the conditions

dim Xsd/Xsd = dim Xpd/Xpd = 1 ∀d ∈ ω∗, dim Xs−∞ = dim Xp−∞ = 0. (10)

Therefore, q (λs) = q (λp) = (qd = 1, q−∞ = 0 | ∀d ∈ ω∗). This, together with the relationdim Xs = card R > card ω, implies that the necessary condition (9) for the existence of a Lyapunovbasis for the pair (λs,Xs) fails.

On the other hand, the algebraic basis {µi | i < ω} of the space Xp is a Lyapunov basis forXp. Since Xsd ⊕

{µ−d

}R

= Xsd for all d ∈ ω∗, it follows that Xp is a maximal λ-subspacein Xs. Obviously, by (10), an arbitrary maximal λ-subspace in Xs in the general case has the form{fi | i < ω}

R, where fi ∈ Xs and deg fi = i for all i < ω.

If fi ∈ Xp for all i ∈ ω, then the subspace {fi | ∀i < ω}R

is also a maximal λ-subspace in Xp.Therefore, sets of the form {fi ∈ Xs | ∀i < ω} and {fi ∈ Xp | ∀i < ω} describe the entire set ofnonextendible Lyapunov sets in the spaces Xs and Xp, respectively. Among maximal λ-subspacesin Xp, we consider the class of subspaces of the form {ϕ(µ)µi | i < ω}

R, where λp(ϕ) = 0. It

coincides with the part of the set of ideals of the ring R[µ] generated by polynomials with zerologarithmic norm. Note also that, by using Lemmas 1 and 2, one can readily construct examplesof chains of maximal λ-subspaces in Xp and Xs as well as examples of maximal chains of maximalλ-subspaces in Xp.

The functional λs corresponds to an ultrametric space known as the Baire space [11, p. 388 of theRussian translation]. Indeed, by setting λB(x) = (1 − λs(x))−1, we obtain the Lyapunov–Bogdanovfunctional λB : Xs → [0, 1]. Then �(x, y) = λB(x − y) is an ultrametric in Xs making Xs a Bairespace.

As was mentioned above, for the functional λs, there is no Lyapunov basis, since condition (9)fails. In the following example, condition (9) holds, but there is no Lyapunov basis.

Consider the linearly ordered set

Λe := {−∞,−ω, . . . ,−1, 0} = {−∞} ∪ {−ω} ∪ ω∗ (−∞ < −ω < −i < −(i − 1) ∀i < ω)

and the functional λe : Xe → Λe, where Xe = Xs × Xs, λe(x) = λs (x2) if x2 �= 0 (here andthroughout the following, x = (x1, x2), x1, x2 ∈ Xs); λe(x) = −ω if x2 = 0 and x1 �= 0; andλe(x) = −∞ if x1 = x2 = 0. The sets Xd, d ∈ |λe|, corresponding to λe have the form Xd = Xs ×{x2 | λs (x2) ≤ d} for all d ∈ ω∗, X−ω = Xs×0, X−∞ = 0×0, where 0 is the zero subspace in Xs. Onecan readily see that the family {Xd | d ∈ |λe|} of subspaces is a filtration of the space Xe; therefore,by using Proposition 2, one can find that λe is a Lyapunov–Bogdanov functional. The invariantsequence q (λe) has the form q (λe) = (qd = 1 ∀d ∈ ω∗, q−ω = dim Xs, q−∞ = 0). This implies thevalidity of relation (9) for λe. On the other hand, an arbitrary maximal λ-subspace for the functionalλe has the form H = H0 ⊕ · · · ⊕ H−ω ⊕ H−∞, where H−∞ = 0, H−ω = Xs × 0, and Hd = 0 × Ld

(dim Ld = 1) for all d ∈ ω∗. Since⊕

d∈ω∗ Ld ⊂ Xs, we have H ⊂ Xe for all H ∈ Sm (λe). This,together with Theorem 3, implies that there is no Lyapunov basis for the functional λe.



3.2. Multidegrees

Along with a nonfounded functional λp, the founded Lyapunov–Bogdanov functional λp1 : Xp →{−∞} ∪ ω, λp1(x) = deg x if x �= 0, λp1(0) = −∞, is defined on the space Xp. Here deg x is theordinal (upper) degree of the polynomial x(µ). The set of multidegrees of a polynomial in finitelymany variables [15, p. 83 of the Russian translation] is a natural generalization of the notion of thedegree of a polynomial in a single variable.

Let x (µ1, . . . , µk) =∑

α xαµα be a nonzero polynomial in the ring R [µ1, . . . , µk]. Here α :=(α1, . . . , αk) is a multi-index (α ∈ Z

k+), and µα := µα1

1 . . . µαk

k is a monomial in the variablesµ1, . . . , µk. The multidegree multideg x of the polynomial x is equal to max

{α ∈ Z

k+ | xα �= 0

},

where the maximum is taken with respect to a fixed complete order on Zk+ satisfying the following

condition: if α ≥ β and γ ∈ Zk+, then α + γ ≥ β + γ. In particular, the so-called pure lexicographic

order, given by the condition α ≥lex β if there exists an l ≤ k such that α1 = β1, . . . , αl−1 = βl−1,αl > βl, defines the multidegree of the polynomial x (µ1, . . . , µk).

By [15, p. 83 of the Russian translation], an arbitrary multidegree satisfies the conditions

multideg(x + y) ≤ max{multideg x,multideg y}, multideg(cx) = multideg x

for x, y �= 0, x + y �= 0, and c �= 0. Consequently, the relations λpk(x) = multideg x for x �= 0 andλpk(0) = −∞ define the set of founded Lyapunov–Bogdanov functionals

λpk : Xpk → {−∞} ∪ Zk+,

where Xpk is the vector space of real polynomials in the variables µ1, . . . , µk. Among Lyapunovbases of the pairs (λpk,Xpk), we note the monomial algebraic basis

{µα1

1 . . . µαk

k | (α1, . . . , αk) ∈ Zk+

}.

It is a Lyapunov basis for an arbitrary pair (λpk,Xpk) regardless of the choice of a complete orderon Z

k+.

3.3. Lexicographic Representation of Lyapunov–Bogdanov Functionals

The definition of functionals of the form λpk can be represented by the following general con-struction. Consider an algebraic basis θ = {zα | α ∈ K} of the space X, where K is an index setof the basis. Let Λ0 be a given linearly ordered set (an alphabet of the basis θ) coordinated withK so as to ensure that K admits the disjunctive partition K =

⋃d∈Λ0

Kd. In this connection,we represent an arbitrary index α ∈ Kd in the form αd. Then an arbitrary nonzero vector z ∈ Xadmits the unique representation

z = cα0d0zα0d0 + · · · + cαmdmzαmdm

, (11)

where cαidi∈ K\0, αi ∈ Kdi

for all i < m, m < ω, and dm ≤ dm−1 ≤ · · · ≤ d0.We combine the set Λ0 with the formal symbol −∞ and set −∞ = inf Λ0 if inf Λ0 exists; −∞ < d

for all d ∈ Λ0 if inf Λ0 does not exist. By setting λlex(z) = d0 for z of the form (11), λlex(0) = −∞,we obtain the Lyapunov–Bogdanov functional λlex : X → Λ0 ∪ {−∞}. It follows from Definition 1that an arbitrary Lyapunov–Bogdanov functional for which there exists a Lyapunov basis admitsa lexicographic representation λlex. An example of a lexicographic representation of a nonfoundedLyapunov–Bogdanov functional related to the problem of Poincare asymptotic expansions wasconsidered in [16]. It was also noted there that chains of maximal λ-subspaces can be used for theinvestigation of properties of equivalent asymptotic sequences appearing in the theory of asymptoticexpansions.

3.4. Eigenvalues of the Laplace Operator

As a rule, the choice of a basis θ and an alphabet Λ0 for a lexicographic representation ofan Lyapunov–Bogdanov functional is specified by a particular problem. For example, the eigen-functions of the Laplace operator ∆ on a Riemannian connected compact manifold are naturally


1050 BORUKHOV

ordered in decreasing order of eigenvalues λ0, λ1, . . . , λi+1 < λi for all i < ω. This order defines alexicographic representation of the Lyapunov–Bogdanov functional λ0

L : Z0 → Λλ ∪ {−∞}, whereZ0 is the linear span of the set of eigenfunctions, Λλ := {λi | λi+1 < λi, i < ω}. The closure of Z0

in an appropriate norm ‖ · ‖ provides a separable Hilbert space Z = Z0. In this case, the Lyapunovcharacteristic number

λL(z) = limt→∞

t−1 ln ‖(exp ∆t)z‖ (12)

is defined for an arbitrary vector z ∈ Z\0. It [together with the condition λL(0) = −∞] can betreated as an extension of the functional λ0

L to the space Z.The algebraic structure of the functional λL : Z → Λλ ∪{−∞} given by (12) is similar to that of

the functional λs (see Subsection 3.1). In particular, there is no algebraic Lyapunov basis for thepair (λL, Z), and, in the case of a simple spectrum ΛL, maximal λ-subspaces and the correspondingnoncontinuable Lyapunov sets have the form {G}R, G = {zd | d ∈ Λλ}, where the index d impliesthat λL (zd) = d. The multiplicity of the spectrum is determined by the invariant sequence q (λL):the spectrum of the operator ∆ is simple if and only if qλi

= 1 for all i < ω.In conclusion, note that the absence of an algebraic basis does not imply that an appropriately

defined topological Lyapunov basis cannot exist. In the case of the functional λL, such a basis canbe chosen in the form of the set of orthonormal eigenfunctions of the Laplace operator.

ACKNOWLEDGMENTS

The work was financially supported by the Foundation for Basic Research of Republic Belarus(project no. F05-023).

REFERENCES

1. Lyapunov, A.M., in Sobr. soch. (Collected Papers), Moscow: Akad. Nauk SSSR, 1956, vol. 2, pp. 7–263.2. Bogdanov, Yu.S., Dokl. Akad. Nauk , 1957, vol. 113, no. 2, pp. 255–257.3. Bogdanov, Yu.S., Mat. Sb., 1959, vol. 49, no. 2, pp. 225–231.4. Bogdanov, Yu.S., in Funkts. analiz i ego primenenie: Tr. V Vsesoyuz. konf. po funkts. analizu i ego

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14. Kurosh, A.G., Lektsii po obshchei algebre (Lectures on General Algebra), Moscow: Nauka, 1973.15. Cox, D., Little, J., and O’Shea, D., Ideals, Varieties, and Algorithms. An Introduction to Computational

Algebraic Geometry and Commutive Algebra, New York: Springer-Verlag, 1997. Translated under thetitle Idealy mnogoobraziya i algoritmy. Vvedenie v vychislitel’nye aspekty algebraicheskoi geometrii ikommutativnoi algebry, Moscow: Mir, 2000.

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lyapunov bases of a vector space and maximal λ-subspaces

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