Ergodic automorphisms with simple spectrum and rapidly decreasing correlations

Abstract

In this note, we prove the existence of invertible measure-preserving transformations T which act on a Lebesgue probability space (X,A , μ), are of spectral multiplicity 1 (i.e., have a simple spectrum), and admit the following estimates of the rate of decrease of correlations for a dense family functions f ∈ L2(X,A , μ): 〈f(T x), f(x)〉 = O(|k|−1/2+e) for all e > 0. Interestingly, according to the identity 〈f(T kx), f(x)〉 = σf (k), where σf is the spectral measure corresponding to the function f , the rate of decrease of the Fourier coefficients σf (k) specified above is the limiting value for all singular Borel measures on [0, 1]. In the ergodic theory of transformations with invariant measure, the following problem posed by Banach is well known: Does there exist a measure-preserving transformation with simple Lebesgue spectrum? In Ulam’s book [1, Chap. VI, Sec. 6], this problem was stated as follows: Does there exist an f ∈ L2(X,A , μ) and an invertible measure-preserving transformation T : X → X for which the sequence of functions {f(T kx) : k ∈ Z} is a complete orthogonal family in the Hilbert space L2(X,A , μ)? It is easy to give an example of a dynamical system with this property on a space with infinite measure. Indeed, take the set of integers Z with the standard counting measure ν({j}) ≡ 1 for the phase spaceX and consider the map T : j → j + 1. Obviously, the functions δk = δ0 ◦ T , which are defined by δk(j) = 1 if j = k and δ0(j) = 0 otherwise, form a basis in L2(Z, ν). For spaces with finite measure, Banach’s question is open. Kirillov [2] considered the following natural generalization of Banach’s question: Describe Abelian topological groups G for which there exists a continuous G-action with finite invariant measure having simple spectrum or finite spectral multiplicity and, moreover, a measure of maximal spectral type is equivalent to the Haar measure on the dual group G. In this note, we concentrate on the case G = Z. A survey of basic constructions and main results of the spectral theory of dynamical systems can be found in [3] and [4]. The problems of Banach and Kirillov are related to the rate of decrease at infinity of the correlation functions Rf (k) = 〈f(T x), f(x)〉, f ∈ L(X,A , μ).