beta-numeration and Rauzy fractals for non-unit Pisot numbers

Transcription

1 beta-numeration and Rauzy fractals for non-unit Pisot numbers Anne Siegel (Irisa, CNRS, Rennes) Kanazawa, June 05

2 Overview β-expansion: expand a positive real number in a non-integer Pisot basis β Represent and understand β-integers (reals with no fractional part)? Which numbers have a purely periodic expansion? Answers for β Pisot unit (Thurston, Rauzy, Akiyama, Ito, Rao) Generalize to the non-unit case?

3 β-expansion β > 1 Pisot number of degree d β-transformation T β : x βx (mod 1) β-expansion u i = βt i 1 β (x) {0... [β]} d β : x [0, 1[ (u i ) i 1 {0... [β]} N x = i 1 u i β i Pseudo-expansion of 1 d β (1) = (t i) i 1 infinite and ultimately periodic will be used in the admissibility condition 1 = i 1 t i β i

4 β-shift Condition of admissibility: A one-sided sequence (u i ) i 1 is the β-expansion of x [0, 1[ iff all its shifted sequences are less that the expansion of d β (1). k 1, (u i ) i k < lex d β (1) Sofic language: the set of finite factors F β of admissible sequences is recognized by a finite graph [Bertrand]. Ultimately periodic expansions: elements of Q(β) [Schmidt].

5 Expansion on R + ; β-integers x R +. β M x < β M+1. d β (β M x) = ω M... ω 0 u 1... u n... x = ω M β M + + ω }{{} 0 + u 1 β u L β L +... }{{} integer part fractional part Reals with finite fractional part: Fin(β). Fin(β) = {ω M β M + + ω 0 + u 1 β u L β L ; ω M... ω 0 u 1... u L F β } β-integers: no fractional part Z + β = {ω Mβ M + + ω 0 ; ω M... ω 0 F β } Fin(β) How can we represent β-integers with respect to the symbolic topology?

6 Central tile: geometric representation of β-integers Canonical embedding: β-conjugates: β 2,..., β r, β r+1, β r+1,..., β r+s, β r+s ϕ β : P(β) Z[β 1 ] (P(β 2 ),... P(β r+s )) R r 1 C s R d 1 ϕ β (β i ) tends to 0; ϕ β (β i ) is expanding. Central tile: Representation of β-integers [Rauzy,Thurston] T β = ϕ β (Z + β ) basic tiles: natural covering of the central tile. They are defined thanks to the successor map in Z + β. β 3 = β 2 +β+1 β3 = 2β 2 +2β+1 β 3 = β + 1 β 3 = 3β 2 2β+2 β 2 = β + 1

7 Geometric β-tile: representation of the β-shift Expand every x R + in base β. Keep the fractional part in [0, 1[. Represent the integer part in the central tile. T β = {( ϕ β (ω M β M +... ω 0, }{{} representation of the integer part in the central tile ui β i }{{} fractional part in [0, 1] ), x R +, d β (x) = ω M... ω 0.u }

8 Geometric β-tile: properties in the unit case β Pisot unit The geometric β-tile T β has a cylinder shape basic tiles are non-overlaping Self-similarity: the central tile satisfies an IFS equation Non-empty interior Both have a nonzero Lebesgue measure Dynamics on T β [Rauzy, Thurston, Akiyama, Arnoux, Berthé, Canterini, Ito, Praggastis, Rao, S.,...]

9 Construction of the central tile Use symbolic dynamics (unit and non-unit case) Compute the β-expansion of 1; deduce a β-substitution σ β iterate the substitution to get a periodic point embed the periodic point as a stair in R d β-integers are the projections of the vertices on the expanding direction. The central tile is the projection of the vertices on the contracting plane

10 Purely periodic expansions? Let x [0, 1[. If d β (x) is purely periodic, the representation of some reals having x as a fractional part tends to the embedding of x (that is, (ϕ β (x), x)). Theorem (Ito-Rao) Let β a Pisot unit and x [0, 1[. The expansion of x is purely periodic iff (ϕ β (x), x) T β. β 3 3β has a purely periodic expansion does not have a purely periodic expansion

11 Purely periodic expansions of rationals? p q is purely periodic iff (ϕ β( p q ), p q ) = ( p q, p q, p q,... ) T β. γ(β) = sup{ɛ < 1, [0, ɛ[ Q is purely periodic } γ(β) = A diagonal[0, A] T β. Application If Fin(β) = Z[ 1 β ] then 0 is an inner point of T β [Akiyama] If β d = a d 1 β d a 1 β + 1 with a d 1 a d 2... a 1 1 then γ(β) > 0 2 dimensional unit case: γ(β) = 0 or γ(β) = 1 [Schmidt?] β 2 = β + 1 γ(β) = 1 β 2 = 3β 1 γ(β) = 0 β 3 = β 2 + β + 1, γ(β) > 2/3

12 Non-unit case? The basic tiles in the central tile have overlaps. 1 N(β) k does not have a purely periodic expansion for all k. β 3 = 3β 2 2β+2 β 2 = 2β + 2 β 2 = 4β + 3 γ(β) = sup{ɛ < 1, p q < ɛ and gcd(n(β), q) = 1 = d β( p q ) pure per. } β 2 2β 2: γ(β) = 1 β 2 4β 3: γ(β) 1/200. γ(β) = 0? Add a p-adic representation?

13 Central adic-tile: geometric representation of integers Canonical embedding: β-conjugates: β 2,..., β r, β r+1, β r+1,..., β r+s, β r+s prime ideals of O Q(β) which contain β: βo Q(β) = ν i=1 I i n i. ϕ β : P(β) Z[β 1 ] (P(β 2 ),... P(β r+s ), P(β),..., P(β) )) K σ }{{}}{{}}{{} R r 1 C s K I1 K Iν Global representation of β-integers = ϕ β (Z + β ) K σ R d 1 T adic β p N(β) Q dp p β 2 = 4β + 2 K σ = R Z 2 2 β 2 = 4β + 3 K σ = R Z 3

14 Drawing pictures? When β has degree 2: β 2 = aβ + b. Let p a divisor of b. Compute the ramifications of p in O Q(β). Find a Q p basis of the completion of O Q(β) relatively to a prime ideal that contains p. Compute the p-adic expansions of 1 and b/β in the base. Project each coordinate a i p i Q p to a i p i. The projection is continuous and preserves the measure. Multiplication by β commutes with the multiplication by 1/p. Addition commutes with a non-continuous-map. Apply the stair projection method to draw the tile.

15 Properties of the central adic tile The unit case properties can be generalized to the non-unit case: Basic tiles do not overlap Self-similarity: The basic tiles satifies an IFS equation The tile T adic β has a nonzero Haar measure β 2 = 4β + 3 Theorem (Akiyama,Barat,Berthé,S.) If Fin(β) = Z[ 1 β ] then 0 is an inner point of the tiles T β, T β and T adic β (central tile, full geometric tile, adic tiles).

16 Purely periodic expansions Global tile: T β = {( ϕ β (ω M β M +... ω 0, ui β i ), x R }{{} + } }{{} central adic tile fractional part Theorem (Berthé,S.) Let β a Pisot non-unit and x [0, 1[. The β-expansion of x is purely periodic iff (ϕ β (x), x) T adic β. Corollary if Fin(β) = Z[ 1 β ] then there exists ɛ and k such that ɛ, k, ( p q < ɛ and N(β)k p) = d β ( p ) purely periodic q

17 Game: guess the value of γ(β) Proposition If β 2 = aβ + b with b prime, then γ(β) iff [0, ] Zb Tβadic. gcd(a, b) = 1: iff [0, ] [0, 1] is included in the rep. of Tβadic. gcd(a, b) = p: iff [0, ] [0, 1] {0} is included in the rep. of Tβadic. β 2 = 4β + 3, γ(β) = 0? β 2 = 10β + 3, γ(β) > 1/4? β 2 = 4β + 2, γ(β) = 1? Chalenge. Use self-similarity and ergodic properties of additions in p-adic spaces to get the value of γ(β)?