Substitutions and symbolic dynamical systems, Lecture 7: The Rudin-Shapiro, Fibonacci and Chacon sequences

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1 Substitutions and symbolic dynamical systems, Lecture 7: The Rudin-Shapiro, Fibonacci and Chacon sequences 18 mars 2015

2 The Rudin-Shapiro sequence The Rudin-Shapiro sequence is the fixed point u beginning by a of the substitution a ab,,b ac, c db, d dc We have u n = a or b if and only if there is an even number of 11 in the expansion of n in base 2. Let v be the sequence defined from u replacing a,b by 1 and c,d by 1. It is the limit of the sequence A n defined by A n+1 = A n B n and B n+1 = A n ( B n ). (set A n = φ(σ n (a)) and B n = φ(σ n (b)). It is a 2 automatic sequence (see Lecture 3).

3 The complexity of u is 8n 8 for n 2 and v has asymptotically the same complexity. The systems X u and X v are topologically conjugate. The system (X u,s) has rank at most four and is a four-point extension of the dyadic rotation. One has S(w,e) = (Sw,φ(w)e) for a map φ of Y (the dyadic rotation) into the symmetric group on four points.

4 The Fibonacci sequence The Fibonacci sequence is the fixed point u of the substitution a ab, b a. The system (X u,s) is uniquely ergodic as a consequence of the following result. Proposition If u is a fixed point of a primitive substitution, then (X u,s) is uniquely ergodic. We use, as for the Morse sequence, the fact that if (X u,s) has uniform frequencies, i.e. if N(W,u k u k+n 1 )/n tends to a limit uniformly in k, then it is uniformly ergodic (see Lecture 6).

5 Te Fibonacci sequence has complexity n+1 (it is Sturmian). More generally, one has Proposition The fixed points of a primitive substitution have at most linear complexity. For every n there is p such that inf a A σp 1 (a) n inf a A σp (a) Every word of length n can then be included in σ p (ab). Since there are at most K = Card(A) 2 possible choices for a,b, p u (n) 2Kdα p < 2K d c αn where (by Perron-Frobenius), c, d are such that cα p < inf a A σ p (a) < sup σ p (a) < dα p a A

6 The ternary Chacon sequence The Chacon substitution O 0012, 1 12, is primitive. Thus the system (X 0,S) associated with the fixed point v beginning with 0 is uniquely ergodic. Note that v begins with the word w n defined by w 0 = 0 and w n+1 = w n w n 1w n where w n is obtained from w n by changing the first letter from 0 to 2.

7 The binary Chacon sequence Let u be the sequence beginning with b n, where b 0 = 0 and b n+1 = b n b n 1b n. The system (X,S) is called the Chacon map. Note that u is the fixed point of the non-primitive morphism , 1 1. We send X 0 onto X by replacing every 2 by 0. To come back, we replace 0 by 2 whenever there is a 1 just before. The two systems are conjugate and thus (X,S) also is uniquely ergodic. Let µ be its invariant measure.

8 The system (X,S) is of rank 1, with the following constuction. We cut the interval [0,1[ into two intervals P 0,P 1 of lengths 2/3 and 1/3 and we take F 0 = P 0. We cut F 0 into three intervals of equal length. The transformation sends the fist one to the second one, the second one to a subinterval a 1 of P 1 beginnning at the left end of P 1 with the corresponding length 2/9 and this interval is send to the third piece of P 0. At step n, we cut the n-th stack into three equal columns and send the top of the first one to the bottom of the second one, the top of the second one to a subinterval a n+1 of P 1 of corresponding length and this subinterval to the bottom of the last column.

9 4/9 2/3 F 0 = P 0 a 1 2/9 4/9 0 2/3 1 P 0 P 1 0 2/9 a 1 F 1

10 The P-name of a point w X is the sequence P(w) such that P(w) n = i whenever S n w P i. By associating to a point its P-name, we send the basis F n of the n-th stack to the cylinder [u 0,...,u hn 1] where h 0 = 1 and h n+1 = 3h n +1, and thus h n = 3n Note that n = 1 a n = P 0 and that the Lebesgue measure of the n-th stack hn 1 i=0 Si F n tends to 1.

11 The word b n = u 0 u hn 1 is called the n-block. The symbols 1 are called spacers. Proposition The Chacon system is weakly mixing : there are no eigenfunctions in L 2 (X,µ) except the constants, which are simple. Let f be an eigenfuction for the eigenvalue λ, of norm 1. We approximate f in L 2 (X,µ) by a sequence of functions f n of norm 1 constant on the levels of the n-stack. Let C 1, C 2 be the first and second column of the n-stack. Their measure is close to 1/3, larger than 1/4 and f n (S hn ) λf n 2 2( f n (S hn ) f(s hn ) 2 + λ hn f λ hn f n 2 ) C 1 C 1 C 1 < 4ε for n large enough. But as f n is constant on the levels of the n-stack, S hn f n = f n on C 1 and so λ hn 1 2 < 8ε if n is large enough.

12 But since f n is constant on the levels of the levels of the n-stack, S hn f n = f n on C 1 and so λ hn 1 2 < 16ε if n is large enough. In the same way, S h n+1 f n = f n on C 2 implies that λ hn < 16ε if n is large enough. We conclude that λ = 1 and it is a simple value because unique ergodicity implies ergodicity.

13 Ergodicity and mixing A transformation T is ergodic if and only if for every pair of measurable sets A and B lim n k=0 n 1 µ(a T k B) = µ(a)µ(b). The transformation T is weakly mixing if and only if for every pair of measurable sets A and B 1 n 1 lim µ(a T k B) µ(a)µ(b) = 0. n n k=0 The transformation T is strongly mixing if and only if for every pair of measurable sets A and B lim n µ(a Tn B) = µ(a)µ(b).

14 Proposition The Chacon map is not strongly mixing. Indeed, µ(s hp F n F n ) > 1/4µ(F n ) if p n while µ(f n ) < 1/4 if n 2.

15 Proposition The complexity of u is p n (u) = 2n 1 if n 2 and the complexity of v is p v (n) = 2n+1 for n 0. Two pairs of factors of length n of v give the same factor of u : the left extensions of the two left special factors of length n 1. For n = 4 : the pair (0012,2012) gives 0010 and the pair (0120, 2102) gives The proof that v has complexity 2n+1 can follow the path below (distinct from the P. Fogg).

16 Special and neutral words For a word x is the set S of factors of v, let l(x) = Card{a A ax S}, e(x) = Card{(a,b) A A axb S}, r(x) = Card{a A xa S}, m(x) = e(x) l(x) r(x)+1. A word is left-special if l(x) 2, right-special if r(x) 2, bispecial if it is left and right special. A word is neutral if m(x) = 0. Note that m(x) 1 is the number of edges minus the number of vertices of the graph E(x) = {(a,b) A axb S}.

17 We set s n = p n+1 p n, b n = s n+1 s n. Proposition One has b n = x S A n m(x). This is easy since m(x) = (e(x) l(x) r(x)+1) x S A n x S A n = p n+2 p n+1 p n+1 p n = s n+1 s n = b n.

18 Proposition For each length n 0, we have either 1 No bispecial factor, 2 One bispecial factor which is neutral, 3 Two bispecial factors of multiplicity 1 and 1. The neutral bispecial factors are the words τ n (0) where τ(x) = 012σ(0). Their length is a multiple of 3. The pairs of bispecial factors of opposite mutiplicity are the (τ n (012),τ n (120)). The first one has multiplicity 1, the second one 1.

19 The right-special words

20 Proposition The induced map of Chacon s transformation on the cylinder w 0 = 0 is semitopologically conjugate to the triadic rotation. We consider the substitution τ : a aab,b bab. Its fixed point beginning with a is deduced from the binary Chacon sequence by 0 a,10 b. Let T be the induced map of S on the cylinder Y = [0]. We have Tw = Sw if w 0 = w 1 = 0 and Tw = S 2 w if w 0 = w 1 = 1. Call Q a,q b these cylinders. Code the trajectory of a point w under T by w n = a if T n w Q a and w n = b if T n w Q b. The dynamical system ossociated with τ is topologically conjugate with (Y,T). The structure of the system associated with τ can be analysed in the same way as for the Morse system, showing that it is a coding of the triadic rotation.

21 Representation by Rokhlin stacks A sequence of partitions P n = {P n 1,...,Pn k n } generates a measure-theoretic dynamical system (X,T,µ) if there is a set E with µ(e) = 0 such that for every pair (x,x ) (X \E) 2 if x,x are in the same set of the partition P n for all n 0, then x = x.

22 A system (X,T,µ) has rank one if 1 there exits sequences of positive integers (q n ) n 0 and (a n,i ) n 0,1 i qn 1 such that the sequence of integers h n defined by h 0 = 1 and h n+1 = q n h n + q n 1 j=1 a n,i satisfies n=0 h n+1 q n h n h n+1 < 2 There exist subsets of X denoted by (F n ) n 0, by (F n,i ) n 0,1 i qn and by (C n,i,j ) n 0,2 i qn,1 j a n,i such that for every n,

23 1 (F n,i ) 1 i qn is a partition of F n, 2 The sets (T k F n ) 0 k hn 1 are disjoint, 3 T hn F n,i = Cn,i,1 if a n,i 0 and i < q n, 4 T hn F n,i = F n,i+1 if a n,i = 0 if i < q n. 5 TC n,i,j = C n,i,j+1 if j < a n,i, 6 TC n,i,an,i = F n,i+1 if i < q n 7 F n+1 = F n,1.

24 3 the sequence of partitions P n = {F n,tf n,...,t hn 1 F n,x \ hn 1 k=0 Tk F n } generates the system (X,T,µ). The union of the (T k F n ) 1 k hn 1 is called a Rokhlin stack of base F n. We say that the system is generated by the sequence of Rokhlin stacks with base F n. If we replace the sequence of partitions P n by {F n,0,tf n,0,...,t hn,0 1 F n,0,...,f n,1,tf n,1,...,t hn,1 1 F n,1,x \ p=0,1 hn,p 1 k=0 T k F n,p }, we have a system of rank two. The sets C n,i,j correspond to letters called the spacers in the associated symbolic sequence. We will show that the Morse system has rank two with F n,e = σ n [e], h n = 2 n and q n = 2 (and thus q n = 2 and a n,i = 0).

25 Proposition We have for all n 0 X u = e=0,1 2n 1 k=0 Sk σ n [e] Proposition We have for all n 0 and e = 0 or 1, σ n [e] = [σ n (e)σ n (0)] [σ n (e)σ n (1)] and σ n [e] = σ n+1 [e] S 2n σ n+1 [e ]

26 We set F n,e = σ n [e] and at stage n, we have two Rokhlin stacks 2n 1 k=0 Sk F n,e with bases F n,0 and F n,1, with height 2 n whose levels S k F n,e are disjoint sets of measure 2 n 1. The shift map sends each level of each stack, except the top ones, onto the level immediately above. We cut F n,e into two subsets of equal measure F n+1,e = σ n+1 [e] and H n+1,e = S 2n σ n+1 [e]. The shift map sends S 2n 1 F n+1,0 onto H n+1,1. This defines the (n+1)-stacks, which will have height 2 n+1.

27 F n,0 F n,1 F n+1,0 F n+1,1 F n+1,0 F n+1,1