Multidimensional Euclid s algorithms, numeration and substitutions

Transcription

1 Multidimensional Euclid s algorithms, numeration and substitutions V. Berthé LIAFA-CNRS-Paris-France berthe@liafa.jussieu.fr http :// Numeration Lorentz Center June 2010

2 Substitutions Substitutions on words and symbolic dynamical systems Substitutions on tiles : inflation/subdivision rules, tilings and point sets

3 in T. ling Substitutions T will be called andself-affine tilings with expansion map φ if it is φ-subdividing, repetitive ite local complexity. If φ is a similarity the tiling will be called self-similar. For self-s of R or R 2 = C there is an expansion constant λ for which φ(z) =λz. rule taking T T to the union of tiles in φ(t ) is called an inflate-and-subdivide rule be tes using Principle the expanding One takes map φ and then decomposes the image into the union of tiles o l scale. If T is φ-subdividing, then it will be invariant under this rule, therefore we sho a finite numer of tiles {T 1, T 2,..., T m} -and-subdivide rule rather than the tiling itself. The rule given in Figure 1 is an inflate ide rule with an expansive φ(z) = 3z. transformation However, the Q rule (the given inflation in Figure factor ) 3 is not an inflate-and-subd a rule that allows one to divide each QT i into copies of the T 1, T 2,..., T m ple 5. The L-triomino It isor a chair simple production substitution method uses four for tilings prototiles, each being an L fo ee unit squares. We have chosen to color the prototiles since they are inequivalent tion. The expansion map is φ(z) = 2z and in Figure 5 we show the substitution of th iles. Example Figure 5. The chair or L-triomino substitution.

4 ee unit squares. We have chosen to color the prototiles since they are inequivalent tion. The The chair expansion tiling map is φ(z) = 2z and in Figure 5 we show the substitution of th iles. 6 NATALIE PRIEBE FRANK This geometric substitution can be iterated simply by repeated application of φ followed by the appropriate subdivision. Parallel to the symbolic case, we call a tile that has been inflated and subdivided n times a level-n tile. In Figure 6 we show level-n tiles for n =2, 3, and 4. Figure 5. The chair or L-triomino substitution. Figure 6. Level-2, level-3, and level-4 tiles. Tilings Encyclopedia http ://tilings.math.uni-bielefeld.de/ E. Harriss, D. Frettlöh 2.2. A few important results. One of the earliest results was a characterization of the expansion constant λ C of a self-similar tiling of C. A primer on substitution tilings of the Euclidean plane -N. Priebe Frank- Expo. Math. 26 (2008), no. 4, Theorem 2.1. (Thurston [59], Kenyon [27]) A complex number λ is the expansion constant for some self-similar tiling if and only if λ is an algebraic integer which is strictly larger than all its Galois conjugates other than its complex conjugate.

5 An example of a substitution on words : Fibonacci substitution Definition A substitution σ is a morphism of the free monoid Example σ : 1 12, σ (1) is called the Fibonacci word σ (1) = The incidence matrix M σ of σ is defined by M σ = ( σ(j) i ) (i,j) A 2, where σ(j) i counts the number of occurrences of the letter i in σ(j) Unimodular substitution det M σ = ±1

6 The Tribonacci substitution [Rauzy 82] σ : 1 12, 2 13, 3 1 σ (1) : The incidence matrix of σ is M = It is primitive (there exists a power of M which contains only positive entries), unimodular and Pisot Its characteristic polynomial is X 3 X 2 X 1. Its dominant eigenvalue β > 1 is a Pisot number

7 Substitutive dynamical systems Let σ be a primitive substitution over A Let u A N be such that σ k (u) = u for some k 1 Let S be the shift : S(u n) = (u n+1 ) The symbolic dynamical system generated by σ is (X σ, S) with X σ := {S n (u); n N} A N Question Under which conditions is it possible to give a geometric representation of a substitutive dynamical system as a translation on the torus/ on a compact abelian group? [discrete spectrum] Remark Measure-theoretic discrete spectrum and topological discrete spectrum are equivalent for primitive substitutive dynamical systems [Host] Example Fibonacci substitution : (X σ, S) is isomorphic to (R/Z, R 1+ 5 ) 2

8 Looking for suitable representations of dynamical systems in terms of... numeration systems geometric dynamical systems

9 The Dumont-Thomas numeration system It is based on the greedy algorithm and acts on words Let u = (u n) such that σ(u) = u We decompose prefixes of u 0 u N 1 into images by powers of σ of a finite number of words base σ Since σ(u) = u, there exists L such that σ(u 0 u L 1 ) u 0 u N 1 < σ(u 0 u L ) and thus a proper prefix p of σ(u L ) s.t. u 0 u N 1 = σ(u 0 u L 1 ) p with σ(u L ) = p u N s Hence, for every N, one has u 0 u N 1 = σ K (p K )σ K 1 (p K 1 ) σ(p 1 )p 0, the p i belong to a finite set of words that only depends on σ digits a numeration system on words... but also for integers and real numbers

10 The Dumont-Thomas numeration system Every prefix w of the Tribonacci word u can be uniquely expanded as w = σ n (p n)σ n 1 (p n 1 ) p 0, where the words p i are equal to the empy word or to the letter 1, and 111. Conversely every finite word that can be decomposed under this form is a prefix of the Tribonacci word. w = n ε i T i. i=0 Such a numeration exists for every primitive substitution

11 Tribonacci rotation [Rauzy 82] σ : 1 12, 2 13, 3 1 Theorem (X σ, S) is measure-theoretically isomorphic to the translation R β on the two-dimensional torus T 2 R β : T 2 T 2, x x + (1/β, 1/β 2 ) Idea of the proof The Tribonacci word codes the orbit of the point 0 under the action of the translation R β with respect to the partition of the following fundamental domain of T 2 One represents σ (1) as a broken line : l : {1, 2, 3} Z 3, 1 e 1, 2 e 2, 3 e 3, l(w) = w 1 e 1 + w 2 e 2 + w 3 e 3, that we will be projected according to the eigenspaces of M σ

12 How to reach nonalgebraic parameters? We have considered so far iterations of a single substitution. We now want to reach nonalgebraic parameters One can code a standard arithmetic discrete line (Freeman code) over the two-letter alphabet {0, 1}. One gets a Stumian word (u n) n N {0, 1} N What kind of representation can we give?

13 Numbers, sequences and lattices : dynamical representation of discrete lines We consider the substitutions One has Theorem σ 0 : 0 0, σ 0 : 1 10 σ 1 : 0 01, σ 1 : = σ 1 ( ) = σ 0 (011011) = σ 1 (0101) = σ 1 (00) A sturmian word can be written as an infinite composition of a finite number of substitutions (S-adic expansion) lim n + σa 1 0 σa 2 1 σa 2n 2n σa 2n+1 2n+1 (0) Continued fractions and Ostrowski numeration system Transformations acting on bases of a lattice

14 From Euclid s algorithm to lattice bases We consider the substitutions σ 0 : 0 0, σ 0 : 1 10 σ 1 : 0 01, σ 1 : 1 1 When one applies a substitution we perform a change of basis in the lattice Z 2 by applying the matrices [ 1 1 ] [ 1 0 ] One translates in a matricial way the additive steps of Euclid s algorithm by [ ] [ ] [ ] a 1 1 a1 = b 0 1 b 1 (a, b) (a 1, b 1 ) = (a b, b). Euclid s algorithm = action of SL(2, N)= substitution

15 S-adic expansions Theorem A sturmian word can be written as a finite composition of a finite number of substitutions S-adic expansion ai p i We are in the framework of lim n + σ 1σ 2...σ n(0) M σ1 M σn v Arithmetic dynamics [Sidorov-Vershik] arithmetic codings of dynamical systems that preserve their arithmetic structure Numeration dynamics [Keane]

16 Multidimensional continued fractions If we start with two parameters (α, β), one looks for two rational sequences (p n/q n) et (r n/q n) with the same denominator that satisfy Geometrically lim p n/q n = α, lim r n/q n = β. Arithmetically and dynamically translation on the torus : R α,β : T 2 T 2, (x, y) x + (α, β) action of Z 2 on T : (m, n).(x, y) = mα + nβ

17 Continued fractions Euclid s algorithm Starting with two numbers, one subtracts the smallest to the largest Unimodularity [ pn+1 q det n+1 p n q n ] = ±1 Rem SL(2, N) is a finitely generated free monoid. It is generated by [ ] [ ] and Best approximation property Theorem A rational number p/q is a best approximation of the real number α if every p /q wth 1 q q, p/q p /q satifies qα p < q α p Every best approximation of α is a convergent

18 From SL(2, N) to SL(3, N) SL(2, N) is a free and finitely generated monoid SL(3, N) is not free SL(3, N) is not finitely generated. Consider the family of matrices 1 0 n 1 n n 1 These matrices are undecomposable for n 3 [Rivat]

19 Multidimensional continued fractions There is no canonical generalization of continued fractions to higher dimensions Several approaches are possible best simultaneous approximations but we then loose unimodularity, and the sequence of best approximations heavily depends on the chosen norm [Lagarias] Klein polyhedra and sails [Arnold] unimodular multidimensional Euclid s algorithms Fibered systems e.g., Jacobi-Perron algorithm, Brun algorithm [Brentjes, Schweiger] sequences of nested cones approximating a direction [Nogueira] lattice reduction / geodesic flow (LLL), [Lagarias],[Ferguson-Forcade], [Just], [Grabiner-Lagarias][Smeets]

20 What is expected? We are given (α 1,, α d ) which produces a sequence of basis (B (k) ) of Z d+1 and/or a sequence of approximations (p (k 1 ),, p(k) d, q(k) ) Arithmetics A two-dimensional continued fraction algorithm is expected to detect integer relations for (1, α 1,, α d ) give algebraic characterizations of periodic expansions converge sufficiently fast max dist(b (k) i, (α, 1)R) k 0 i and provide good rational approximations Good means with respect to Dirichlet s theorem : there exist infinitely many (p i /q) 1 i d such that max α i p i /q 1 i q 1+1/d

21 What is expected? We are given (α 1,, α d ) which produces a sequence of basis (B (k) ) of Z d+1 and/or a sequence of approximations (p (k 1 ),, p(k) d, q(k) ) Arithmetics A two-dimensional continued fraction algorithm is expected to detect integer relations for (1, α 1,, α d ) give algebraic characterizations of periodic expansions converge sufficiently fast max dist(b (k) i, (α, 1)R) k 0 i and provide good rational approximations Good means with respect to Dirichlet s theorem : there exist infinitely many (p i /q) 1 i d such that max α i p i /q 1 i q 1+1/d Dynamics We also want... reasonable ergodic properties (ergodic invariant measure, natural extension) to be able to control the number of executions, the depth if the parameters are rational etc. to be able to perform a dynamical analysis à la Brigitte Vallée...

22 Multidimensional continued fraction algorithms Allowed operations on numbers +,, /,, [ ], Allowed operations on matrices : elementary basis transformations interchanging two vectors permutation matrices adding an integer multiple of one basis vector to another basis vector transvection matrices Ex. LLL algorithm size reduction steps and exchange steps

23 How does LLL produce good approximations? Let M t := α α α d 0 0 t

24 How does LLL produce good approximations? Let M t := α α α d 0 0 t We take t small One has det(m t) = t Rem : One changes the lattice at each step instead of changing the bases of a fixed lattice

25 How does LLL produce good approximations? Let α α 2 M t := α d 0 0 t LLL produces in polynomial time a vector (b 1,... b d+1 ) such that j, b j 1 b d/4 det(m t) 1/d+1 = 2 d/4 t 1/d+1 But b 1 = p 1 e 1 + p 2 e p d e d + q( α 1 e 1 α d e d + te d+1 ) b 1 = (p 1 qα 1 )e (p d qα d e d ) + qte d+1 One deduces that i, p i α i q 2 d/4 t 1/d+1 and qt 2 d/4 t 1/d+1 We deduce that for all i p i α i q 2 (d+1)/4 1/q 1/d

26 How does LLL produce good approximations? Let M t := α α α d 0 0 t Towards continued fractions? One has a priori to recompute everything from the beginning when one changes t For a dynamical version, see [Smeets]

27 Multidimensional Euclid s algorithms : a zoo of algorithms Jacobi-Perron : we subtract the first one to the two other ones with 0 x 1, x 2 x 3 (x 1, x 2, x 3 ) (x 2 [ x 2 x 1 ]x 1, x 3 [ x 3 x 1 ]x 1, x 1 ) Brun : we subtract the second largest and we reorder with x 1 x 2 x 3 (x 1, x 2, x 3 ) (x 1, x 2, x 3 x 2 ) Poincaré : we subtract the previous one and we reorder with x 1 x 2 x 3 (x 1, x 2, x 3 ) (x 1, x 2 x 1, x 3 x 2 ) Selmer : we subtract the smallest to the largest and we reorder with x 1 x 2 x 2 (x 1, x 2, x 3 ) (x 1, x 2, x 3 x 1 ) Fully subtractive : we subtract the smallest one to all the largest ones and we reorder with x 1 x 2 x 3 (x 1, x 2, x 3 ) (x 1, x 2 x 1, x 3 x 1 )

28 Poincaré algorithm [Nogueira 95] (x 1, x 2, x 3 ) (x 1, x 2 x 1, x 3 x 2 ), x 1 x 2 x 3 1/ϕ 2 + 1/ϕ = 1 1/ϕ 2 1/ϕ 100 1/ϕ 3 1/ϕ /ϕ 1/ϕ 4 1/ϕ /ϕ 1/ϕ 2 1/ϕ k+1 1/ϕ k 100 i<k 1/ϕi

29 Jacobi-Perron versus Ostrowski Jacobi-Perron Its linear version is defined on X = {(a, b, c) R 3 0 a, b < c} by We set (a, b, c) (b b/a a, c c/a a, a) α := a/c, β := b/c Its projective version is defined on [0, 1) [0, 1) by ( β β (α, β) α, 1 ) 1 α α = ({β/α}, {1/α}) α

30 Jacobi-Perron versus Ostrowski Jacobi-Perron Its linear version is defined on X = {(a, b, c) R 3 0 a, b < c} by We set (a, b, c) (b b/a a, c c/a a, a) α := a/c, β := b/c Its projective version is defined on [0, 1) [0, 1) by ( β β (α, β) α, 1 ) 1 α α = ({β/α}, {1/α}) α Ostrowski (α, β) ({1/α}, { β α })

31 Jacobi-Perron versus Ostrowski Jacobi-Perron Its linear version is defined on X = {(a, b, c) R 3 0 a, b < c} by We set (a, b, c) (b b/a a, c c/a a, a) α := a/c, β := b/c Its projective version is defined on [0, 1) [0, 1) by ( β β (α, β) α, 1 ) 1 α α = ({β/α}, {1/α}) α Theorem There exists δ > 0 s.t. for a.e. (α, β), there exists n 0 = n 0 (α, β) s.t. for all n n 0 α p n/q n < 1, β r n/q n < 1, q 1+δ n where p n, q n, r n are given by Brun/Jacobi-Perron qn 1+δ Brun [Ito-Fujita-Keane-Ohtsuki ] ; Jacobi-Perron [Broise-Guivarc h 99]

32 Back to substitutions Motivation Rauzy fractals associated with nonalgebraic parameters Discrete Geometry (discrete planes and lines) Strategy An elementary step for a MCF algorithm matrix incidence matrix substitution Nonalgebraic parameters MCF algorithm periodic expansions finite products of MCF substitutions Main tool A formalism which associates a geometric substitution/generalized substitution with a unimodular matrix, and which produces approximations of the Rauzy fractal

33 e 3 e 3 e 2 e 2 e 1 e 1 e 3 e 2 e 2 e 1 e 1 e 3 e 3 e 2 e 2 e 1 e 1

34 Generalized substitutions Abelianisation Let d be the cardinality of A. Let l : A N d be the abelianisation map l(w) = t ( w 1, w 2,, w d ) Generalized substitutions [Arnoux-Ito][Ei] Let σ be a unimodular substitution. E1 (σ)( x, i ) = j A P, σ(j)=pis ( M 1 σ ( x ) l(p), j )

35 e 3 e 3 e 2 e 2 e 1 e 1 e 3 e 2 e 2 e 1 e 1 e 3 e 3 e 2 e 2 e 1 e 1

36 Some iterations

37 Generation

38 Generation

39 Generation

40 Generation

41 Generation

42 Generation

43 Generation

44 From generalized substitutions to Rauzy fractals Theorem [Arnoux-Ito] Let σ be a Pisot irreducible substitution Let π c be the projection onto the contracting plane of M σ along its expanding line Let R σ be the Rauzy fractal associated with σ Let U be the upper part of the unit cube One has lim n + Mn σπ c(e 1 (σ)) n (U) = R σ

45 From generalized substitutions to Rauzy fractals Theorem [Arnoux-Ito] Let σ be a Pisot irreducible substitution Let π c be the projection onto the contracting plane of M σ along its expanding line Let R σ be the Rauzy fractal associated with σ Let U be the upper part of the unit cube One has lim n + Mn σπ c(e 1 (σ)) n (U) = R σ

46 From generalized substitutions to Rauzy fractals Theorem [Arnoux-Ito] Let σ be a Pisot irreducible substitution Let π c be the projection onto the contracting plane of M σ along its expanding line Let R σ be the Rauzy fractal associated with σ Let U be the upper part of the unit cube One has lim n + Mn σπ c(e 1 (σ)) n (U) = R σ

47 Changing the order of letters 1 12, 2 23, , 2 32, 3 231

48 Jacobi-Perron substitutions Let σ B,C be the substitution σ B,C (1) = 3, σ B,C (2) = 13 B, σ B,C (3) = 23 C Its incidence matrice M σ satisfies M σ = B C The linear Jacobi-Perron is defined on {(a, b, c) R 3 0 a, b < c} as Let If F(a, b, c) = (a 1, b 1, c 1 ), then F(a, b, c) = (b b/a a, c c/a a, a) B = b/a and C = c/a. (a 1, b 1, c 1 ) = t Mσ 1 (a, b, c) with B = b/a and C = c/a

49 Generalized Jacobi-Perron substitutions Let σ B,C be the substitution σ B,C (1) = 3, σ B,C (2) = 13 B, σ B,C (3) = 23 C

50 Jacobi-Perron Rauzy fractals Let σ B,C be the substitution σ B,C (1) = 3, σ B,C (2) = 13 B, σ B,C (3) = 23 C. Its incidence matrice M σ satisfies M σ = B C Theorem [Dubois, Farhane, Paysant-Le Roux] Every finite product of Jacobi-Perron matrices is Pisot irreducible (d = 3) Theorem [B.-Jolivet-Siegel] The Rauzy fractal associated with a finite product of Jacobi-Perron matrices is connected and satisfies the tiling property [discrete spectrum]

51 Concatenation rules