SARNAK S CONJECTURE FOR SOME AUTOMATIC SEQUENCES. S. Ferenczi, J. Ku laga-przymus, M. Lemańczyk, C. Mauduit

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1 SARNAK S CONJECTURE FOR SOME AUTOMATIC SEQUENCES S. Ferenczi, J. Ku laga-przymus, M. Lemańczyk, C. Mauduit 1

2 ARITHMETICAL NOTIONS Möbius function µp1q 1,µpp 1...p s q p 1q s if p 1,...,p s are distinct prime numbers and µpnq 0 if n is divisible by the square of a prime number. Von Mangoldt function Λpnq lnp if n p k, p prime and k ě 1, Λpnq 0 otherwise. Möbius orthogonality A sequence u n,n P N of complex numbers of modulus less than 1 is (asymptotically) orthogonal to the Möbius function if ř nďx µpnqu n opxq. Prime number theorem u satisfies a PNT if ř N n 1 Λpnqu n ř N i 1 u n `opnq. 2

3 DYNAMICAL NOTIONS Symbolic dynamical system For a (minimal) sequence u on an alphabet A, the one-sided shift Spx 0 x 1 x 2 q x 1 x 2 on the subset X u of A N made with the infinite sequences such that for every t ă s, x t x s is a factor of u. Sarnak s conjecture X compact metric space, T continuous transformation of X, of zero topological entropy, f continuous X Ñ C. Any sequence fpt n xq is orthogonal to the Möbius function. 3

4 CLASSICAL SEQUENCES Thue - Morse Möbius orthogonality and prime number theorem : deduced from Mauduit - Rivat, or Dartyge - Tenenbaum, or Indlekofer - Katai. Sarnak s conjecrture for the associated symbolic system: El Abdalaoui- Kasian- Lemańczyk. Rudin - Shapiro Möbius orthogonal and prime number theorem : Mauduit - Rivat. Invertible automatic sequences Möbius orthogonal and prime number theorem : deduced from Drmota. 4

5 SUBSTITUTIONS σ is an application from an alphabet A into the set A of finite words on A; it extends to a morphism of A for the concatenation. Afixed pointofσ isaninfinitesequenceuwithσu u.σ hasconstant lengthlif σa l for all ai n A. The associated symbolic dynamical system px σ,sq is px u,sq for a fixed point. The Perron-Frobenius eigenvalue is the largest eigenvalue of the matrix giving the number of occurrences of j in σi. σ is primitive if a power of this matrix is strictly positive. 5

6 SUBSTITUTIONS WITH REPETITIONS Multiplicative length of σ It is smaller than r if for all i P A, σi pj 1 piqq a 1piq...pj qi piqq a q i piq,, a1 piq P A,..., a qi piq P A, q i ď r. Proposition 1 There exists a constant Cprq such that, if σ is a primitive substitution of multiplicative length smaller than r and Perron-Frobenius eigenvalue strictly larger than Cprq, px σ,sq, if it is aperiodic, satisfies Sarnak s conjecture. If px σ,sq is weakly mixing, the fixed points satisfy a prime number theorem. In particular, Sarnak holds for (primitive) substitutions of multiplicative length smaller than r and constant length larger than Cprq. 6

7 Examples 0 Ñ 0 a 1 b, 1 Ñ 1 a 0 b, a `b ą Cp2q; 0 Ñ 0 k`1 12, 1 Ñ 12, 2 Ñ 0 k 12, k `2 ą Cp3q, weakly mixing. INGREDIENTS OF PROOF 1 Recursion rules σ n i pσ n 1 j 1 piqq a 1piq...pσ n 1 j qi piqq a q i piq. These give estimates of Bourgain on ř x n µpnq for a fixed point x, using that σ n i close to λ t σ n t j and thus larger than Cprq t σ n t j for all i,j if t is large enough. 7

8 ROKHLIN TOWERS By recognizability, we build towers with names σ n i : for each n P N, disjoints sets S j F i,n, i P A,0 ď j ď σ n i 1,withF i,n inthecylinderset rσ n is,madebycutting and stacking following the recursion rules. S 4 F 1,n S 3 F 1,n S 2 F 1,n σ n i SF 1,n F 1,n F 2,n For a function f which is constant on the levels of the towers, Bourgain s estimates work for ř fpnqµpnq; these f approximate every continuous function. 8

9 BIJECTIVE SUBSTITUTIONS Substitution σ of constant length l such that σ j defined by σ j paq pσaq j is a bijection of A, j 1,...,l. Then σ defines an invertible automaton. Proposition 2 Each aperiodic dynamical system px σ,sq associated to a bijective substitution satisfies Sarnak s conjecture. 9

10 INGREDIENTS OF PROOF 2 Katai - Bourgain - Sarnak - Ziegler criterion If pa n q Ă C is bounded and 1 ÿ a nr a ns Ñ 0 N nďn for all sufficiently large different prime numbers r, s. Then(for Möbius or every multiplicative function) 1 ÿ a n µpnq Ñ 0 N nďn We want to apply it to a n fps n xq. Thus we consider Joinings of S r and S s = ergodic measures ν invariant by U S r ˆ S s, with the right marginals. Generic points for these measures = points for which 1 N ř nďn δ U n x Ñ ν. 10

11 We need to define a few more sequences and systems A group substitution S is the group of permutations of A. Define rσ: S Ñ S l by setting rσpτq pσ 1 τqpσ 2 τq...pσ l τq rσpidq τ for each τ P S. We restrict it to G Ă S, the subgroup generated by σ 1,σ 2,...,σ l. px σ,sq is a topological factor of px rσ,sq A generalized Morse sequence Each group substitution θpgq θpeq g for each g P G generates the same symbolic system as the generalized Morse sequence x θpeq ˆ θpeq ˆ..., where b ˆ c pb c 1 qpb c 2 q...pb c c q and b g pb 1 g,...,b b gq for b,c words over G. 11

12 Odometers T px 0 x 1 x 2 q ppx 0 `1qx 1 x 2, q with carrying of the remainder to the right, on X Π tě0 t0,1,...,l t 1u for integers l t ě 2, t ě 0. (Not a symbolic system, but a translation on a compact group). Compact group extensions T ψ px,gq : ptx,ψpxqgq, T ergodic automorphism, G compact metric group, ψ: X Ñ G measurable map called a cocycle. 12

13 The generalized Morse sequence system px x,sq is an extension of the odometer (X,T q by a cocycle ψ. Unfortunately ψ is not continuous!. So we need one more object A Toeplitz sequence px n x n`1 x 1 n px x,sq is an extension of px px,sq by a continuous cocycle φ. 13

14 Topological (plain lines) and measure-theoretical (dashed line) maps Morse system given by x px x,sq»» continuous cpt. group extension px px ˆG,S φ, q» Morse system given by Morse cocycle (cpt. group extension) px ˆG,T ψ q Toeplitz system px px,sq» odometer px,t q 14

15 And thus... The joining of T r ψ and Ts ψ are (more or less) product measures. All points are generic for S r φ ˆSs φ and the corresponding measure. We conclude by KBSZ. Result also valid for every regular Morse sequence x b 0 ˆb 1 ˆ..., b t P t0,1u l t, t ě 0, where the set tp : p is prime and p l t for some tu is finite. Result also valid for Rudin - Shapiro and some generalizations. 15