Möbius orthogonality for generalized Morse-Kakutani flows
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MÖBIUS ORTHOGONALITY FOR GENERALIZED MORSE-KAKUTANI FLOWS By WILLIAM A. VEECH Abstract. The Prouhet-Thue-Morse sequence and its generalizations have occured in many settings. “Morse-Kakutani flow” refers to Kakutani’s 1967 generalization of the Morse minimal flow (1922). These flows are Z2 skew products of almost one-to-one extensions of the adding machine (x → x+1 on the 2−adic completion of Z). “Generalized Morse-Kakutani flow” is a K skew product of similar construct, with base flow a factor of x → x + 1 on the profinite completion of Z and K any compact group, abelian or not, of countable density. The principal result is that Sarnak’s Möbius Orthogonality Conjecture holds for a restricted class of generalized Morse-Kakutani flows. Contents. 1. Introduction. 2. Skew products. 3. Representation of f(Λnp)(y). 4. Unique Ergodicity of Tp. 5. Minimality of (T,N). 6. Reduction of Tp ×Tq. 7. Ergodic Measures for Tp ×Tq. 8. Möbius orthogonality for (T,N). 9. Elements of K(Z) are strongly deterministic. 10. Digression on weakly almost periodic functions. References. 1. Introduction. Given f ∈ l∞(Z), Bf denotes the smallest translation invariant ∗-subalgebra of l∞(Z) that contains f and the constants. The maximal ideal space, X(f), of Bf is compact, metrizable and contains an image of Z as a dense subset. Translation by one determines a homeomorphism, T, of X(f). X(f) may be identified as the set of all pointwise limits of sequences of translates of f. Definition 1.1. f ∈ l∞(Z) shall be called strongly deterministic if the topological entropy of (X(f),T) is zero, i.e., htop(T) = 0. (1.1) Denote by SD the set of strongly deterministic functions. Manuscript received January 6, 2015; revised May 12, 2016. American Journal of Mathematics 139 (2017), 1157–1203. c  2017 by Johns Hopkins University Press. 1157 1158 W. A. VEECH The interest in SD stems from the Sarnak Möbius Orthogonality Conjecture, which may be stated to say CONJECTURE 1.2. (Sarnak, [S11]) If f ∈ SD, and if μ(·) is the Möbius function , then lim N→∞ 1 N N  n=1 μ(n)f(n) = 0. (1.2) See [G03, Lemma 18.15], for preservation of zero measure theoretic entropy under joinings and projective limits. These imply readily that SD is a translation invariant , norm-closed, ∗-subalgebra of l∞(Z). Viewing X(f) as a set of functions, it is evident that if g ∈ X(f), then X(g) is a closed, invariant set in X(f). Therefore, X(f) ⊂ SD when f ∈ SD. Our principal focus is on a set of three-step point-distal flows [V70], (T,N), introduced in [V70] and, in a less general setting, in [V69]. Leaving precise definitions for Section 2, (2.15), the general setup is a diagram of flow homomorphisms, (T,N) π1  (σ,M) π2   Rθ,X  (1.3) (Rθ,X) is addition by θ on a compact, metrizable group, X, containing Zθ as a dense subgroup. ((Rθ,X) ranges over a certain set of quotients of addition by one on the profinite completion of Z.) (σ,M) is a minimal homeomorphism of a compact metrizable space, M, and π2 : M → X is one-to-one on M\{π−1 2 {Zθ}}. (T,N) is a minimal skew product with possibly non-metrizable phase space N = M ×K, where K is an arbitrary compact topological group of countable density and π1((m,k)) = m. Denote the continuous skewing function by ϕ : M −→ K, so that T(m,k) = (σm,ϕ(m)k),(m,k) ∈ N. For any f ∈ C(N) and z ∈ N the sample function n → f(Tnz) belongs to SD. MÖBIUS ORTHOGONALITY FOR GENERALIZED MORSE-KAKUTANI FLOWS 1159 Preliminaries to the statement and proof of our main result, Theorem 8.5, will begin in Section 2. The theorem will state that, for a certain class of flows (1.3), if f ∈ C(N), then lim N→∞ 1 N N  n=1 μ(n)f(Tn z) = 0, z ∈ N. (1.4) The base flow has phase space any projective limit X = ← − lim n·∞ (Z/ΛnZ), where Λn = n j=1 λj, n ≥ 1, and {λj} is any...