![Page 1: Multipletilings forsymmetric -expansionsfan/uploads/Main/Hejda_Fan15.pdfBasicdefinitions—multipletilings ThecollectionfR(x)g x2X S\Z[ ] isamultipletilingofdegreem m itisaunionofmtilings:](https://reader033.vdocuments.pub/reader033/viewer/2022060306/5f098b447e708231d427559b/html5/thumbnails/1.jpg)
Multiple tilingsfor symmetric β-expansions
Tom Hejda
LIAFA, Univ. Paris DiderotCzech Tech. Univ. in Prague
FAN Admont 2015
Supported by CTU in Prague SGS11/162/OHK4/3T/14, Czech Science Foundation GAČR 13-03538Sand ANR/FWF project “FAN – Fractals and Numeration” (ANR-12-IS01-0002, FWF grant I1136.)
Tom Hejda (LIAFA, Paris & CTU in Prague) Multiple tilings for symmetricβ-expansions 1
![Page 2: Multipletilings forsymmetric -expansionsfan/uploads/Main/Hejda_Fan15.pdfBasicdefinitions—multipletilings ThecollectionfR(x)g x2X S\Z[ ] isamultipletilingofdegreem m itisaunionofmtilings:](https://reader033.vdocuments.pub/reader033/viewer/2022060306/5f098b447e708231d427559b/html5/thumbnails/2.jpg)
Outline
1 Definitions
2 Examples
3 The main results
4 Methods
5 Open problems
Tom Hejda (LIAFA, Paris & CTU in Prague) Multiple tilings for symmetricβ-expansions 2
![Page 3: Multipletilings forsymmetric -expansionsfan/uploads/Main/Hejda_Fan15.pdfBasicdefinitions—multipletilings ThecollectionfR(x)g x2X S\Z[ ] isamultipletilingofdegreem m itisaunionofmtilings:](https://reader033.vdocuments.pub/reader033/viewer/2022060306/5f098b447e708231d427559b/html5/thumbnails/3.jpg)
Basic definitions — β-numerationI Base β ∈ (1, 2), Pisot unitI d-Bonacci number: Pisot root of
Xd = Xd−1 + Xd−2 + · · ·+ X+ 1
d = 2: β = 1.618 · · · (Golden ratio)d = 3: β = 1.839 · · · (Tribonacci constant)
I Symmetric β-expansion [Akiyama, Scheicher, 2007] of x ∈ [−12, 12):
the coding of its orbit by
βx−1
βx−0
βx−1
−12
β2− 1
1− β2
12
TS : XS 7→ XS
Tom Hejda (LIAFA, Paris & CTU in Prague) Multiple tilings for symmetricβ-expansions 3
![Page 4: Multipletilings forsymmetric -expansionsfan/uploads/Main/Hejda_Fan15.pdfBasicdefinitions—multipletilings ThecollectionfR(x)g x2X S\Z[ ] isamultipletilingofdegreem m itisaunionofmtilings:](https://reader033.vdocuments.pub/reader033/viewer/2022060306/5f098b447e708231d427559b/html5/thumbnails/4.jpg)
Basic definitions — β-numeration
ββ(1)
β(2)
β(3)
I Galois conjugates β(1), . . . , β(e), ignoring those with Imβ(j) < 0;
Φ(∑finite
xjβj):=
(∑xjβ
j(1), . . . ,
∑xjβ
j(e)
)∈ “Rd−1 ”
I Rauzy fractal for x ∈ Z[β] ∩ XS:
R(x) := limn→∞Φ
(βnT−nS x
)
Tom Hejda (LIAFA, Paris & CTU in Prague) Multiple tilings for symmetricβ-expansions 4
![Page 5: Multipletilings forsymmetric -expansionsfan/uploads/Main/Hejda_Fan15.pdfBasicdefinitions—multipletilings ThecollectionfR(x)g x2X S\Z[ ] isamultipletilingofdegreem m itisaunionofmtilings:](https://reader033.vdocuments.pub/reader033/viewer/2022060306/5f098b447e708231d427559b/html5/thumbnails/5.jpg)
Basic definitions — multiple tilings
The collection {R(x)}x∈XS∩Z[β] is a multiple tiling of degree mm
it is a union of m tilings:
I tiles are non-empty, compact, closures of their interiorI each tile has a zero measure boundaryI the collection has finite local complexityI the collection is locally finite (only finitely many tiles meet any
bounded set)I a.e. point of Rd−1 lies in exactly m tiles
Theorem (Kalle, Steiner, 2012)Suppose β is a Pisot unit. Then for any “well-behaving” β-transformationT : X 7→ X, the collection {R(x)}x∈X∩Z[β] is a multiple tiling.
Pisot conjecture for β-numeration: It is a tiling for greedy expansionsProved 3 weeks ago [Barge, arXiv:1505.04408]
Tom Hejda (LIAFA, Paris & CTU in Prague) Multiple tilings for symmetricβ-expansions 5
![Page 6: Multipletilings forsymmetric -expansionsfan/uploads/Main/Hejda_Fan15.pdfBasicdefinitions—multipletilings ThecollectionfR(x)g x2X S\Z[ ] isamultipletilingofdegreem m itisaunionofmtilings:](https://reader033.vdocuments.pub/reader033/viewer/2022060306/5f098b447e708231d427559b/html5/thumbnails/6.jpg)
0(01-1)ω
1-1001(-110)ω
(1-10)ω
0-110(-101)ω
0(0-11)ω
01-11-10(10-1)ω
1-10(01-1)ω
01-110(-101)ω
-11-110(-101)ω
-1010(-101)ω
001(-110)ω
0-10(10-1)ω
00(-101)ω
-101(-110)ω
00-10(10-1)ω
10(0-11)ω
(-101)ω
10(-101)ω
100-1(1-10)ω
0(10-1)ω
1(-110)ω
-100(10-1)ω
0-10(01-1)ω
(0-11)ω
01-1(1-10)ω
-10(10-1)ω
01(-110)ω
-110(-101)ω
0-11-10(10-1)ω
-1001(-110)ω
-101-110(-101)ω
010(0-11)ω
-11(-110)ω
(01-1)ω
1-10(10-1)ω
-1(1-10)ω
0-101(-110)ω
010-1(1-10)ω
100(-101)ω
10-11-10(10-1)ω
010(-101)ω
00(10-1)ω
0(-101)ω
0010(-101)ω
0-11(-110)ω
00-1(1-10)ω
-10(01-1)ω
0-11-110(-101)ω
-110(0-11)ω
1-11-10(10-1)ω
10-1(1-10)ω
-1100-1(1-10)ω
1-1(1-10)ω
10-10(10-1)ω
(-110)ω
0-1(1-10)ω-11-10(10-1)ω
(10-1)ω
01-10(10-1)ω
1-110(-101)ω
![Page 7: Multipletilings forsymmetric -expansionsfan/uploads/Main/Hejda_Fan15.pdfBasicdefinitions—multipletilings ThecollectionfR(x)g x2X S\Z[ ] isamultipletilingofdegreem m itisaunionofmtilings:](https://reader033.vdocuments.pub/reader033/viewer/2022060306/5f098b447e708231d427559b/html5/thumbnails/7.jpg)
Tom Hejda (LIAFA, Paris & CTU in Prague) Multiple tilings for symmetricβ-expansions 7
![Page 8: Multipletilings forsymmetric -expansionsfan/uploads/Main/Hejda_Fan15.pdfBasicdefinitions—multipletilings ThecollectionfR(x)g x2X S\Z[ ] isamultipletilingofdegreem m itisaunionofmtilings:](https://reader033.vdocuments.pub/reader033/viewer/2022060306/5f098b447e708231d427559b/html5/thumbnails/8.jpg)
10-11-1(1-100)ω
0010-1(01-10)ω
00-11-1(1-100)ω
01000-1(1-100)ω
-11000-1(1-100)ω
0(-1001)ω1-11-1(1-100)ω
(01-10)ω(1-100)ω
100(-1001)ω
1-110-1(01-10)ω
0100(-1001)ω
0-11-1(1-100)ω
-1100(-1001)ω
(001-1)ω
(-1001)ω
0-1(1-100)ω
01-11-1(1-100)ω
0(001-1)ω
-1(01-10)ω
-1(1-100)ω
1-1(1-100)ω
00-1(1-100)ω
Tom Hejda (LIAFA, Paris & CTU in Prague) Multiple tilings for symmetricβ-expansions 7
![Page 9: Multipletilings forsymmetric -expansionsfan/uploads/Main/Hejda_Fan15.pdfBasicdefinitions—multipletilings ThecollectionfR(x)g x2X S\Z[ ] isamultipletilingofdegreem m itisaunionofmtilings:](https://reader033.vdocuments.pub/reader033/viewer/2022060306/5f098b447e708231d427559b/html5/thumbnails/9.jpg)
1000-1(10-10)ω
10-110(0-101)ω
-10001(-1010)ω
-101-10(010-1)ω
00-10(010-1)ω
0010(0-101)ω (1-1)ω
(-11)ω
1(-1010)ω
0(0-101)ω
1-110(0-101)ω (0-101)ω
(10-10)ω
-1(10-10)ω
0-110(0-101)ω
0100-1(10-10)ω
(-1010)ω
-11-10(010-1)ω
0-1001(-1010)ω
01-10(010-1)ω
-1010(0-101)ω
0(010-1)ω
10-10(010-1)ω
(010-1)ω
Tom Hejda (LIAFA, Paris & CTU in Prague) Multiple tilings for symmetricβ-expansions 7
![Page 10: Multipletilings forsymmetric -expansionsfan/uploads/Main/Hejda_Fan15.pdfBasicdefinitions—multipletilings ThecollectionfR(x)g x2X S\Z[ ] isamultipletilingofdegreem m itisaunionofmtilings:](https://reader033.vdocuments.pub/reader033/viewer/2022060306/5f098b447e708231d427559b/html5/thumbnails/10.jpg)
Tom Hejda (LIAFA, Paris & CTU in Prague) Multiple tilings for symmetricβ-expansions 7
![Page 11: Multipletilings forsymmetric -expansionsfan/uploads/Main/Hejda_Fan15.pdfBasicdefinitions—multipletilings ThecollectionfR(x)g x2X S\Z[ ] isamultipletilingofdegreem m itisaunionofmtilings:](https://reader033.vdocuments.pub/reader033/viewer/2022060306/5f098b447e708231d427559b/html5/thumbnails/11.jpg)
Two main results
Theorem ASuppose β ∈ (1, 2) is a Pisot unit. Then {R(x)}x∈XS∩Z[β] forms a (single) tilingif and only if:
1 β− 1 is a unit; and2 {RB(y)}y∈XB∩Z[β] forms a (single) tiling (RB and XB explained in the next slide).
Theorem BLet d > 2 and βd = βd−1 + · · ·+ β+ 1. Then the symmetric β-transformationinduces a multiple tiling of Rd−1 with covering degree d− 1.
Tom Hejda (LIAFA, Paris & CTU in Prague) Multiple tilings for symmetricβ-expansions 8
![Page 12: Multipletilings forsymmetric -expansionsfan/uploads/Main/Hejda_Fan15.pdfBasicdefinitions—multipletilings ThecollectionfR(x)g x2X S\Z[ ] isamultipletilingofdegreem m itisaunionofmtilings:](https://reader033.vdocuments.pub/reader033/viewer/2022060306/5f098b447e708231d427559b/html5/thumbnails/12.jpg)
Symmetric vs. balanced β-transformation
βx−1
βx−0
βx−0
βx−1
−12
β2− 1
1− β2
12
βx−0
βx−1
2−β2β−2
12β−2
β2β−2
Tom Hejda (LIAFA, Paris & CTU in Prague) Multiple tilings for symmetricβ-expansions 9
![Page 13: Multipletilings forsymmetric -expansionsfan/uploads/Main/Hejda_Fan15.pdfBasicdefinitions—multipletilings ThecollectionfR(x)g x2X S\Z[ ] isamultipletilingofdegreem m itisaunionofmtilings:](https://reader033.vdocuments.pub/reader033/viewer/2022060306/5f098b447e708231d427559b/html5/thumbnails/13.jpg)
Symmetric vs. balanced β-transformation
βx−1
βx−0
βx−0
βx−1
−12
β2− 1
1− β2
12
βx−0
βx−1
2−β2β−2
12β−2
β2β−2
Tom Hejda (LIAFA, Paris & CTU in Prague) Multiple tilings for symmetricβ-expansions 9
![Page 14: Multipletilings forsymmetric -expansionsfan/uploads/Main/Hejda_Fan15.pdfBasicdefinitions—multipletilings ThecollectionfR(x)g x2X S\Z[ ] isamultipletilingofdegreem m itisaunionofmtilings:](https://reader033.vdocuments.pub/reader033/viewer/2022060306/5f098b447e708231d427559b/html5/thumbnails/14.jpg)
Symmetric vs. balanced β-transformation
βx−1
βx−0
βx−0
βx−1
−12
β2− 1
1− β2
12
βx−0
βx−1
2−β2β−2
12β−2
β2β−2
symmetric balancedXS −−−−−−−−−−−−−−−−−−−−−−−−→ XB
ψ :
x 7−−−−−−−−−−−−−−−−−−−−−→
{1β−1x if x > 01β−1 (x+ 1) if x < 0
I TS(x) = ψ−1TBψ(x)
Tom Hejda (LIAFA, Paris & CTU in Prague) Multiple tilings for symmetricβ-expansions 9
![Page 15: Multipletilings forsymmetric -expansionsfan/uploads/Main/Hejda_Fan15.pdfBasicdefinitions—multipletilings ThecollectionfR(x)g x2X S\Z[ ] isamultipletilingofdegreem m itisaunionofmtilings:](https://reader033.vdocuments.pub/reader033/viewer/2022060306/5f098b447e708231d427559b/html5/thumbnails/15.jpg)
The idea — Theorem A
PropositionSuppose β ∈ (1, 2) is a Pisot unit. Let
I mB be the covering degree of {RB(y)}y∈XB∩Z[β]I mS be the covering degree of {RS(x)}x∈XS∩Z[β]
ThenmS =
∣∣N(β− 1)∣∣ ·mB
RS(x)Φ←−−−
n→∞ βnT−nS x = βnψ−1T−nB ψx = βn((β− 1)T−nB ψx− ?
)= (β− 1)βnT−nB ψx− βn? Φ−−−→
n→∞ Φ(β− 1)×RB(ψx) if y := ψx ∈ Z[β]
I ψx = 1β−1 (x+ ?)
I ψ−1y = (β− 1)y− ?
Tom Hejda (LIAFA, Paris & CTU in Prague) Multiple tilings for symmetricβ-expansions 10
![Page 16: Multipletilings forsymmetric -expansionsfan/uploads/Main/Hejda_Fan15.pdfBasicdefinitions—multipletilings ThecollectionfR(x)g x2X S\Z[ ] isamultipletilingofdegreem m itisaunionofmtilings:](https://reader033.vdocuments.pub/reader033/viewer/2022060306/5f098b447e708231d427559b/html5/thumbnails/16.jpg)
The idea — Theorem A
PropositionSuppose β ∈ (1, 2) is a Pisot unit. Let
I mB be the covering degree of {RB(y)}y∈XB∩Z[β]I mS be the covering degree of {RS(x)}x∈XS∩Z[β]
ThenmS =
∣∣N(β− 1)∣∣ ·mB
RS(x)Φ←−−−
n→∞ βnT−nS x = βnψ−1T−nB ψx = βn((β− 1)T−nB ψx− ?
)= (β− 1)βnT−nB ψx− βn? Φ−−−→
n→∞ Φ(β− 1)×RB(ψx) if y := ψx ∈ Z[β]
Therefore:RS((β− 1)y mod 1))︸ ︷︷ ︸this is 1
|N(β−1)|of all tiles
= RS(ψ−1(y)) = Φ(β− 1)︸ ︷︷ ︸regular linear transformation in Rd−1
× RB(y)
Tom Hejda (LIAFA, Paris & CTU in Prague) Multiple tilings for symmetricβ-expansions 10
![Page 17: Multipletilings forsymmetric -expansionsfan/uploads/Main/Hejda_Fan15.pdfBasicdefinitions—multipletilings ThecollectionfR(x)g x2X S\Z[ ] isamultipletilingofdegreem m itisaunionofmtilings:](https://reader033.vdocuments.pub/reader033/viewer/2022060306/5f098b447e708231d427559b/html5/thumbnails/17.jpg)
The idea — Theorem B
Theorem BLet d > 2 and βd = βd−1 + · · ·+ β+ 1. Then the symmetric β-transformationinduces a multiple tiling of Rd−1 with covering degree d− 1.
I The norm of β− 1 is N(β− 1) = ±(d− 1)
I Show that for d-Bonacci numbers, {RB(y)}y∈XB∩Z[β] is a tiling
I We show that all z ∈ Z[β] satisfy TkB(z) = 1/β for some k(using Property (F) for greedy expansions [Frougny, Solomyak, 1992])
I Arithmetic condition [Kalle, Steiner, 2012]:Φ(z) ∈ R(x)⇔ there exists y ∈ Z[β] such that:
1 Tp(y) = y for some p2 x = Tk(y+ β−kz) for some large enough k
I In the end, it’s just addition of 1
Tom Hejda (LIAFA, Paris & CTU in Prague) Multiple tilings for symmetricβ-expansions 11
![Page 18: Multipletilings forsymmetric -expansionsfan/uploads/Main/Hejda_Fan15.pdfBasicdefinitions—multipletilings ThecollectionfR(x)g x2X S\Z[ ] isamultipletilingofdegreem m itisaunionofmtilings:](https://reader033.vdocuments.pub/reader033/viewer/2022060306/5f098b447e708231d427559b/html5/thumbnails/18.jpg)
Fact or Fiction?
Answer “Yes” or “No”Let β be a Pisot unit, β > 2.Then the symmetric β-transformation induces a single tiling, i.e., mS = 1.
Answer “Yes” or “No”Let β be a Pisot unit, β ∈ [golden mean, 2).Then the balanced β-transformation induces a single tiling, i.e., mB = 1.
QuestionWhat happens for small β?(Example: for minimal Pisot β ≈ 1.324 we have mS = mB = 2.)
Tom Hejda (LIAFA, Paris & CTU in Prague) Multiple tilings for symmetricβ-expansions 12
![Page 19: Multipletilings forsymmetric -expansionsfan/uploads/Main/Hejda_Fan15.pdfBasicdefinitions—multipletilings ThecollectionfR(x)g x2X S\Z[ ] isamultipletilingofdegreem m itisaunionofmtilings:](https://reader033.vdocuments.pub/reader033/viewer/2022060306/5f098b447e708231d427559b/html5/thumbnails/19.jpg)
image taken from Wikimedia Commons
We announce that
Numeration 2016will be held in
PragueonMay 23–27 (Monday–Friday)
at the Czech Technical University
and the satellite workshop
SAGE DaysonMay 21–22 (Saturday–Sunday)
![Page 20: Multipletilings forsymmetric -expansionsfan/uploads/Main/Hejda_Fan15.pdfBasicdefinitions—multipletilings ThecollectionfR(x)g x2X S\Z[ ] isamultipletilingofdegreem m itisaunionofmtilings:](https://reader033.vdocuments.pub/reader033/viewer/2022060306/5f098b447e708231d427559b/html5/thumbnails/20.jpg)
image taken from Wikimedia Commons
We announce that
Numeration 2016will be held in
PragueonMay 23–27 (Monday–Friday)
at the Czech Technical University
and the satellite workshop
SAGE DaysonMay 21–22 (Saturday–Sunday)