[Paper Review] Eventual nonsensitivity and tame dynamical systems

Eli Glasner, Michael Megrelishvili|arXiv (Cornell University)|May 11, 2014

Mathematical Dynamics and FractalsMathematics45 references18 citations

TL;DR

This paper introduces a novel characterization of tame dynamical systems using eventual nonsensitivity and eventual fragmentability, establishing that a symbolic system $X \subset \{0,1\}^\mathbb{Z}$ is tame if and only if for every infinite $L \subseteq \mathbb{Z}$, there exists an infinite $K \subseteq L$ such that $\pi_K(X)$ is countable. The key contribution is a hierarchy of 'smallness' properties for families of continuous functions, linking them to representability on Asplund and Rosenthal Banach spaces, with applications to subshifts, Sturmian-like sequences, and order-preserving systems.

ABSTRACT

In this paper we characterize tame dynamical systems and functions in terms of eventual non-sensitivity and eventual fragmentability. As a notable application we obtain a neat characterization of tame subshifts $X \subset \{0,1\}^{\mathbb Z}$: for every infinite subset $L \subseteq {\mathbb Z}$ there exists an infinite subset $K \subseteq L$ such that $π_{K}(X)$ is a countable subset of $\{0,1\}^K$. The notion of eventual fragmentability is one of the properties we encounter which indicate some "smallness" of a family. We investigate a "smallness hierarchy" for families of continuous functions on compact dynamical systems, and link the existence of a "small" family which separates points of a dynamical system $(G,X)$ to the representability of $X$ on "good" Banach spaces. For example, for metric dynamical systems the property of admitting a separating family which is eventually fragmented is equivalent to being tame. We give some sufficient conditions for coding functions to be tame and, among other applications, show that certain multidimensional analogues of Sturmian sequences are tame. We also show that linearly ordered dynamical systems are tame and discuss examples where some universal dynamical systems associated with certain Polish groups are tame.

Motivation & Objective

• To develop a new characterization of tame dynamical systems using the concepts of eventual nonsensitivity and eventual fragmentability.
• To establish a hierarchy of 'smallness' for families of continuous functions on compact dynamical systems.
• To link the existence of separating, eventually fragmented families to representability on Rosenthal Banach spaces.
• To provide a combinatorial characterization of tame subshifts in $\{0,1\}^\mathbb{Z}$, showing that projections onto infinite subsets are countable.
• To extend the theory to order-preserving systems and universal systems associated with Polish groups, proving their tameness.

Proposed method

• Introduces the notion of eventual fragmentability as a refinement of fragmentability, capturing 'smallness' in function families.
• Uses the evaluation map $F \times X \to \mathbb{R}$ to relate bounded $S$-invariant families $F \subset C(X)$ to canonical bilinear maps on Banach spaces $V \times V^*$.
• Applies Rosenthal's $\ell_1$-sequence characterization via independent families of sets to prove that $\beta\mathbb{N}$ is not $w^*$-embeddable into duals of Rosenthal spaces.
• Employs the pigeonhole principle and infinite Ramsey-type arguments to extract independent subfamilies from uncountable collections of sets.
• Leverages the structure of symbolic systems to reduce tameness to combinatorial properties of projections onto infinite subsets of $\mathbb{Z}$.
• Uses the theory of enveloping semigroups and WAP/Asplund/Rosenthal representability to classify dynamical systems by function family complexity.

Experimental results

Research questions

• RQ1 When is a compact dynamical system $X$ representable on a Rosenthal Banach space, and how does this relate to the structure of its separating function families?
• RQ2 What combinatorial condition on a subshift $X \subset \{0,1\}^\mathbb{Z}$ ensures that it is tame?
• RQ3 How does eventual fragmentability of a function family relate to the absence of $\ell_1$-sequences and representability on Asplund spaces?
• RQ4 Under what conditions is a linearly ordered dynamical system tame, and how does this relate to the structure of its enveloping semigroup?
• RQ5 Can the universal minimal system $M(G)$ or the universal affine system $IA(G)$ of a Polish group $G$ be tame, and when does this occur?

Key findings

• A symbolic system $X \subset \{0,1\}^\mathbb{Z}$ is tame if and only if for every infinite $L \subseteq \mathbb{Z}$, there exists an infinite $K \subseteq L$ such that $\pi_K(X)$ is countable.
• A bounded $S$-invariant family $F \subset C(X)$ is eventually fragmented if and only if the associated dynamical system is Rosenthal representable.
• The existence of a separating, eventually fragmented family $F$ characterizes tame metric dynamical systems.
• The system $(H_+(\mathbb{T}), \mathbb{T})$ of orientation-preserving circle homeomorphisms is tame, and its universal minimal system $M(H_+(\mathbb{T}))$ is tame.
• The Stone-Čech compactification $\beta\mathbb{N}$ is not $w^*$-embeddable into the dual of any Rosenthal Banach space, implying that not all compacta are Rosenthal-representable.
• Multidimensional analogues of Sturmian sequences are shown to be tame using sufficient conditions on coding functions.