Periodicity of certain integer sequences

Transcription

1 Periodicity of certain integer sequences Wolfgang Steiner (joint work with S. Akiyama, H. Brunotte and A. Pethő) CNRS, LIAFA (Paris 7) Monastir, August 2007

2 Shift radix systems r = (r 1,..., r d ) R d, a = (a 1,..., a d ) Z d : f (a) = (a 2,..., a d, a r ) d = {(r 1,..., r d ) : a Z d k f k (a) = (0,..., 0)} difficult to characterize, even for d = 2 (Akiyama, Brunotte, Pethő, Thuswaldner,... ) D d = {(r 1,..., r d ) : (f k (a)) k 0 eventually periodic a Z d } (1, 2) d = 2: f (a 1, a 2 ) = (a 2, r 1 a 1 + r 2 a 2 ) ( 1, 0) D 2 (1, 2)

3 Discretized rotations d = 2, r 1 = 1, r 2 = λ ( 2, 2): a k+1 = a k 1 + λa k 0 a k 1 + λa k + a k+1 < 1 ( ) ( ) ( ) ( ) ak 0 1 ak 1 0 = + a k+1 1 λ a k {λa k } eigenvalues of the matrix λ 2 ± λ = e±iα, λ = 2 cos α, elliptic rotation with angle α Conjecture (a k 1, a k ) k Z is periodic (=bounded) for all λ ( 2, 2) and all initial values (a 1, a 2 ) Z 2. In other words: the red line of the previous slide belongs to D 2 trivially true for λ = 1, 0, 1

4 α/π rational, λ quadratic: λ = ±1± 5 2, ± 2, ± 3 λ 2 = bλ + c, x = {λa k 1 }, y = {λa k } a k+1 = a k 1 + λa k = a k 1 λa k + y {λa k+1 } = { λa k 1 λ 2 a k + λy} = { x by + λy} = { x + (c/λ)y} = { x λ y} T λ : [0, 1) 2 [0, 1) 2, T λ (x, y) = (y, { x λ y}) T λ ({λa k 1 }, {λa k }) = ({λa k }, {λa k+1 }) For given λ, all sequences (a k ) k Z are periodic if and only if (T k λ (x, y)) k Z is periodic for all (x, y) (Z[λ] [0, 1)) 2. Adler, Kitchens, Tresser, Vivaldi, Lowenstein, Hatjispyros, Kouptsov, Goetz, Poggiaspalla,...

5 λ = γ = = 2 cos 4π 5 ( ) x T γ = y ( /γ ) ( ) ( x + y 0 x y/γ ), ( ) = 1 1/γ ( )

6 λ = γ = = 2 cos 4π 5 ( ) x T γ = y ( /γ ) ( ) ( x + y 0 x y/γ ), ( ) = 1 1/γ ( ) T γ

7 R T (R) T T ( ) T ( )

8 R T (R) T T ( ) T ( ) 0 0 γ 2

9 R T (R) T T ( ) T ( ) 0 0 γ 2 T ( T (

10 R T (R) T T ( ) T ( ) γ 2 γ 2 T (

11 R T (R) T T ( ) T ( ) T ( γ 2 γ 2 T (

12 R T (R) T T ( ) T ( ) T ( T 2 ( γ 2 γ 2 T (

13 R T (R) T T ( ) T ( ) T ( T 3 ( T 2 ( γ 2 γ 2 T (

14 R T (R) T T ( ) T ( ) T ( T 4 ( T 3 ( T 2 ( γ 2 γ 2 T (

15 R T (R) T T ( ) T ( ) T ( T 4 ( T 3 ( T 2 ( γ 2 γ 2 T 5 ( T (

16 R T (R) T T ( ) T ( ) T ( T 4 ( σ : T 3 ( T 2 ( γ 2 T 6 ( γ 2 T 5 ( T (

17 R T (R) T T ( ) T ( ) T ( D α T (D β ) T 4 ( T 3 ( R T 2 ( σ : γ 2 γ 2 D β T (D α ) T 2 (D β ) T 5 ( T 6 ( T (

18 Periodic points σ(l), l {0, 1}, codes the trajectory of z/γ 2 for z D l : T k 1 (z/ D σ(l)[k], 1 k σ(l) The previous slide shows T σ(l) (z/ = T (z)/γ 2. We obtain T k 1 (z/γ 2n ) D σ n (l)[k], 1 k σ n (l), T σn (l) (z/γ 2n ) = T (z)/γ 2n. If z R, z P = D α T (D α ) D β, then T m (z) [0, 1/ 2 for some m 5; set S(z) = γ 2 T m (z). Theorem (T k (z)) k Z is periodic iff S n (z) P for some n 0 or z R. Proof. If S n (z) P for all n 0, then S n (z)/γ 2n exists and is in the trajectory of z for all n 0, and π(z) = π(s n (z)/γ 2n ) σ n (1). (π(z) denotes the minimal period length of z.)

19 Proposition For every (integer) denominator Q 1, it is sufficient to check the periodicity of the points z [0, 1) 2 1 Q (Z[γ])2 with z [ γ, γ] 2 in order to determine if all points z 1 Q (Z[γ])2 are periodic. Proof. Let π(z) =, i.e. S n (z) P for all n 0. We have ( ) 0 1 T (z) = T (x, y) = (x, y)a + (0, x y/γ ) with A =, 1 1/γ therefore S(z) = γ 2 T m (z) = γ 2 (za m + t) for some m 5 and t {(0, 0), (0, 1), (1, 1/γ), (1/γ 2, 1/, (1/γ 2, 1/γ), (1/γ, 0)}.

20 Hence we have some m k and t k, k 0, such that S n (z) = γ 2n z A m m n + n γ 2(n k+1) t k A m k+1+ +m n k=1 and t k A m { 1, 1/γ, 1/γ 2, 0, 1/γ 2, 1/γ, 1} 2. (S n (z)) = z A m m n γ 2n + (S n (z)) z γ 2n + n k=1 n k=1 z is aperiodic if and only if S n (z) is aperiodic. t k A m k+1 + +m n γ 2(n k+1) γ 2 γ 2(n k+1) < z γ 2n + γ Theorem If z (Z[γ]) 2 or z ( Z[γ] 2 )2, then π(z) <, but π(1/3, 0) =.

21 Theorem If λ = γ, then the minimal period length π(z) of (T k (z)) k Z is 1 if z = (0, 0) or z = (γ/ 5, γ/ 5) 5 for the other points of R (5 4 n + 1)/3, 5(5 4 n + 1)/3 if S n (z) D α T (D α ), n 0 (10 4 n 1)/3, 5(10 4 n 1)/3 if S n (z) D β, n 0 if S n (z) P for all n 0 The minimal period length of (a k ) k Z is π({γa k 1 }, {γa k }).

22 400 Long orbits of (a k 1, a k ) k Z, λ = γ:

23 Aperiodic points, λ = γ Aperiodic points, λ = 1/γ.

24 λ = 1/γ = = 2 cos π 5 T 1/γ ( ) ( x 0 1 T 1/γ = y 1 γ ) ( ) ( x + y 0 x + γy )

25 First return map ˆT on the scaling domain, λ = 1/γ: T 3 ( ) ˆT T 4 ( ) T ( ) D B D A T 2 ( ) T ( )

26 λ = 1/γ : T 4 ( ) T ( ) ˆT D α D β contracting map: U(z) = z/γ 2 + (1/γ, 1/γ) l k stands for ˆT k U(D l ) σ :

27 λ = 1/γ : T 4 ( ) T ( ) ˆT D α D β contracting map: U(z) = z/γ 2 + (1/γ, 1/γ) l k stands for ˆT k U(D l ) σ :

28 λ = 1/γ : T 4 ( ) T ( ) ˆT D α D β contracting map: U(z) = z/γ 2 + (1/γ, 1/γ) l k stands for ˆT k U(D l ) σ : run lengths of the Thue-Morse sequence!

29 250 Long orbits of (a k 1, a k ) k Z, λ = 1/γ:

30 First return map ˆT on the scaling domain D 2 D 3, λ = 2: (l k stands for T k (D l )) B A 1 A B B B B 5 A A B ˆT

31 ˆT D α λ = 2 : U(x, y) = ( 2 1)(y, x) σ : ˆT (D β ) D ζ D β

32 400 Long orbits of (a k 1, a k ) k Z, λ = 2:

33 Aperiodic points, λ = 2. Aperiodic points, λ = 2.

34 400 Long orbits of (a k 1, a k ) k Z, λ = 1/γ:

35 Aperiodic points, λ = 1/γ Aperiodic points, λ = γ.

36 Aperiodic points, λ = 3.

37 Aperiodic points, λ = 3.

38 Higher degree of λ = 2 cos pπ q λ d = b 1 λ d b d, x j = {λ j a k 1 }, y j = {λ j a k }, 1 j < d a k+1 = a k 1 + λa k = a k 1 λa k + y 1 {λa k+1 } = { λa k 1 λ 2 a k + λy 1 } = { x 1 y 2 + λy 1 }. {λ d 2 a k+1 } = { x d 2 y d 1 + λ d 2 y 1 } {λ d 1 a k+1 } = { λ d 1 a k 1 λ d a k + λ d 1 y 1 } = { x d 1 b 1 y d 1 b d 1 y 1 + λ d 1 y 1 } T (x 1,..., x d 1, y 1,..., y d 1 ) = (y 1,..., y d 1, { x 1 y 2 +λy 1 },...) cubic λ: works by Goetz and Poggiaspalla on planar rotations rational λ, irrational α/π: related works by Bosio, Vivaldi, Vladimirov