[Paper Review] When all closed subsets are recurrent?

Jie Li, Piotr Oprocha+2 more|arXiv (Cornell University)|Apr 30, 2015

Mathematical Dynamics and Fractals23 references3 citations

TL;DR

This paper investigates the conditions under which all closed subsets in a topological dynamical system (t.d.s.) are recurrent, focusing on the induced system on the hyperspace of compact subsets. It establishes that the hyperspace $K(X)$ is pointwise recurrent (i.e., all compact sets are recurrent) if and only if the original system $(X,T)$ is uniformly rigid, and constructs minimal, non-equicontinuous, distal, and uniformly rigid systems, resolving open questions in the literature on recurrence and rigidity in topological dynamics.

ABSTRACT

In the paper we study relations of rigidity, equicontinuity and pointwise recurrence between a t.d.s. $(X,T)$ and the t.d.s. $(K(X),T_K)$ induced on the hyperspace $K(X)$ of all compact subsets of $X$, and provide some characterizations. Among other examples, we construct a minimal, non-equicontinuous, distal and uniformly rigid t.d.s. and a t.d.s. which has dense small periodic sets but does not have dense distal points, solving that way open questions existing in the literature.

Motivation & Objective

• To characterize when all compact subsets in a t.d.s. are recurrent by studying the induced system on the hyperspace $K(X)$.
• To resolve open questions regarding the existence of minimal, non-equicontinuous, distal, and uniformly rigid systems.
• To clarify the relationship between recurrence, rigidity, equicontinuity, and distality in the induced hyperspace system $(K(X), T_K)$.
• To investigate the topological entropy of $(K(X), T_K)$ under pointwise recurrence and rigidity conditions.
• To examine the density of recurrent and distal points in hyperspace systems and their implications for disjointness from minimal systems.

Proposed method

• Analyzes the induced system $(K(X), T_K)$ on the hyperspace of compact subsets of a t.d.s. $(X,T)$.
• Uses ordinal rank analysis of recurrent sets $X^{(\alpha)}$ to study recurrence structure and derive contradictions via transfinite induction.
• Applies the concept of $n$-rigidity and uniform rigidity to characterize recurrence in $K(X)$.
• Constructs explicit examples of minimal, non-equicontinuous, distal, and uniformly rigid systems using symbolic dynamics and inverse limits.
• Employs topological and measure-theoretic tools, including the use of $\varepsilon$-neighborhoods and periodicity in compact sets, to analyze recurrence behavior.
• Leverages results from Bauer and Sigmund on hyperspace dynamics and extends them using new techniques to prove equivalence between recurrence in $K(X)$ and uniform rigidity in $X$.

Experimental results

Research questions

• RQ1 When is the induced system $(K(X), T_K)$ pointwise recurrent, i.e., when are all compact subsets recurrent?
• RQ2 What is the precise relationship between uniform rigidity of $(X,T)$ and pointwise recurrence in $(K(X), T_K)$?
• RQ3 Can a minimal, non-equicontinuous, distal, and uniformly rigid t.d.s. exist?
• RQ4 Does the existence of dense distal points in $(K(X), T_K)$ imply the same in $(X,T)$?
• RQ5 What is the topological entropy of $(K(X), T_K)$ when $K(X)$ is pointwise recurrent?

Key findings

• The hyperspace $(K(X), T_K)$ is pointwise recurrent if and only if the original system $(X,T)$ is uniformly rigid.
• For countable $X$, $K(X)$ is pointwise recurrent if and only if $(X,T)$ is uniformly rigid.
• The topological entropy of $(K(X), T_K)$ is zero whenever $K(X)$ is pointwise recurrent (i.e., 1-rigid).
• A minimal, non-equicontinuous, distal, and uniformly rigid t.d.s. exists, resolving an open problem in the literature.
• There exists a t.d.s. with dense small periodic sets but without dense distal points, answering an open question.
• An example is constructed where $(K(X), T_K)$ has dense distal points but $(X,T)$ does not, confirming a previously open question.