[Paper Review] Mean equicontinuity, almost automorphy and regularity

Felipe García‐Ramos, Tobias Jäger+1 more|arXiv (Cornell University)|Aug 14, 2019

Mathematical Dynamics and Fractals25 references4 citations

TL;DR

This paper establishes a complete classification of strictly ergodic topological dynamical systems with discrete spectrum by characterizing the structure of their maximal equicontinuous factors. It proves that for minimal mean equicontinuous systems, the factor map is almost 1-1 (i.e., the system is almost automorphic) if and only if the system is frequently stable, and regular (trivial fibres on a full Haar measure set) if and only if it is diam-mean equicontinuous. The work resolves a long-standing question by constructing a transitive system with positive topological entropy that is almost diam-mean equicontinuous, and provides a partial positive answer to Furstenberg's multiple recurrence conjecture for mean equicontinuous systems.

ABSTRACT

The aim of this article is to obtain a better understanding and classification of strictly ergodic topological dynamical systems with discrete spectrum. To that end, we first determine when an isomorphic maximal equicontinuous factor map of a minimal topological dynamical system has trivial (one point) fibres. In other words, we characterize when minimal mean equicontinuous systems are almost automorphic. Furthermore, we investigate another natural subclass of mean equicontinuous systems, so-called diam-mean equicontinuous systems, and show that a minimal system is diam-mean equicontinuous if and only if the maximal equicontinuous factor is regular (the points with trivial fibres have full Haar measure). Combined with previous results in the field, this provides a natural characterization for every step of a natural hierarchy for strictly ergodic topological models of ergodic systems with discrete spectrum. We also construct an example of a transitive almost diam-mean equicontinuous system with positive topological entropy, and we give a partial answer to a question of Furstenberg related to multiple recurrence.

Motivation & Objective

• To classify strictly ergodic topological dynamical systems with discrete spectrum by analyzing the structure of their maximal equicontinuous factor maps.
• To determine when minimal mean equicontinuous systems are almost automorphic, i.e., when the maximal equicontinuous factor map has almost surely trivial fibres.
• To characterize regularity of the maximal equicontinuous factor in terms of a new notion, diam-mean equicontinuity, and show it is equivalent to the factor map being regular (trivial fibres on a full Haar measure set).
• To construct a transitive system that is almost diam-mean equicontinuous but has positive topological entropy, demonstrating that this class is strictly larger than Banach diam-mean equicontinuous systems.
• To provide a partial answer to Furstenberg's question on multiple recurrence by proving that for minimal mean equicontinuous systems, generic points satisfy the recurrence condition for all powers of the shift.

Proposed method

• Introduces the concept of frequent stability, a local property related to Lyapunov stability, and proves that for minimal mean equicontinuous systems, the maximal equicontinuous factor map is almost 1-1 if and only if the system is frequently stable.
• Defines diam-mean equicontinuity as a strengthening of mean equicontinuity, based on the average diameter of separable sets over time, and proves that this property is equivalent to the regularity of the maximal equicontinuous factor.
• Uses the pointwise multiple ergodic theorem and properties of invariant measures to analyze the recurrence behavior of generic points in mean equicontinuous systems.
• Applies the Halmos-Von Neumann theorem and spectral theory to show that mean equicontinuous systems with isomorphic maximal equicontinuous factors have discrete spectrum.
• Constructs a transitive system via symbolic dynamics on a subshift of finite type, showing it is almost diam-mean equicontinuous but not Banach diam-mean equicontinuous, and has positive topological entropy.
• Leverages the isomorphism between the original system and its maximal equicontinuous factor (established by Li, Tu, and Ye) to transfer measure-theoretic properties to the topological setting.

Experimental results

Research questions

• RQ1 When is the maximal equicontinuous factor map of a minimal mean equicontinuous system almost 1-1, i.e., when is the system almost automorphic?
• RQ2 What is the topological and measure-theoretic characterization of systems for which the maximal equicontinuous factor map is regular (trivial fibres on a full Haar measure set)?
• RQ3 Can a transitive system be almost diam-mean equicontinuous while having positive topological entropy, and how does this relate to Banach diam-mean equicontinuity?
• RQ4 Does Furstenberg's multiple recurrence conjecture hold for minimal mean equicontinuous systems, and if so, on what set of points?
• RQ5 What is the relationship between the hierarchy of mean equicontinuity properties (Banach, diam, mean, frequent) and their implications for topological and measure-theoretic structure?

Key findings

• A minimal mean equicontinuous system is almost automorphic if and only if it is frequently stable, establishing a topological characterization of almost 1-1 fibres in the maximal equicontinuous factor.
• The maximal equicontinuous factor map is regular if and only if the system is diam-mean equicontinuous, providing a complete characterization of regularity in terms of a local topological property.
• There exists a transitive system that is almost diam-mean equicontinuous but has positive topological entropy, showing that this class strictly contains the class of Banach diam-mean equicontinuous systems.
• The system constructed in the paper has zero Banach mean equicontinuity but positive topological entropy, demonstrating that the hierarchy of mean equicontinuity properties is strict and non-trivial.
• For minimal mean equicontinuous systems, a full measure set of points satisfies Furstenberg's multiple recurrence condition: for every d ∈ ℕ, the diagonal point (x,x,…,x) is minimal under the product system T × T² × … × Tᵈ.
• The average diameter of the set A₁ in the construction tends to zero as N → ∞, indicating a stronger form of equicontinuity than almost diam-mean equicontinuity.