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

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

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

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

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

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

Tom Hejda (LIAFA, Paris & CTU in Prague) Multiple tilings for symmetricβ-expansions 7

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

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

Tom Hejda (LIAFA, Paris & CTU in Prague) Multiple tilings for symmetricβ-expansions 7

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

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

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

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

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

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

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

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

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)

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)