Measure-theoretic mean equicontinuity and bounded complexity

Abstract

Let (X, B, µ, T ) be a measure preserving system. We say that a function f ∈ L 2 (X, µ) is µ-mean equicontinuous if for any ε > 0 there is k ∈ N and measurable

Mean equicontinuity, complexity and applications

“…It is well known that the set of all almost periodic functions, denoted by H ap , is spanned by the set of eigenfunctions. Following this line, also inspired by the study of bounded complexity functions in [32,28] (will be introduced in Subsection 6.3), Yu [63] showed that Theorem 4.12 holds even without the assumption of ergodicity. 63]).…”

“…We have the corresponding dichotomy property. The measure-functional version of mean equicontinuity was implied in [59] and explicitly stated in [18] (or see [63]), which will have value in the study of discrete spectrum (see Section 4). The opposite side is referred to [18].…”

“…be a semi-metric, then the Hamming distance can also be expressed as a mean metric w.r.t. to H α as follows: With the equivalence of measure-theoretic mean equicontinuity and discrete spectrum at hand (Theorem 6.13), Yu [63] showed that Ferenczi's result still holds when removing the assumption of ergodicity. That is, we have Theorem 6.15 ([63]).…”

“…In [28] Huang et al proved the converse is also true, and at same time it is also equivalent to mean equicontinuity of this invariant measure. Yu in [63] first generalized the result of Ferenczi by dropping the ergodicity assumption, and then with coauthors in [64] further generalized the result to countable discrete amenable group action. We need to mention that a study on measure complexity on Lebesgue space with respect to admissible metrics was given by Vershik, Zatitskiy and Petrov [61], some results of which cover a part of those mentioned ones, but were proved by different approaches.…”

Discrete spectrum for amenable group actions

“…In fact, this may go back to [19] via scaled entropy and even [18] by Vershik. Different from the one by Katok, in [1] Ferenczi characterized an ergodic invariant measure with discrete spectrum via measure-theoretic complexity using names of a partition and the Hamming distance (we remark that this kind of measure-theoretic complexity was also discussed in [10]), which was extended recently to non-ergodic case by Yu in [23].…”