$(\omega, \alpha, n)$-sensitivity and limit sets of zero entropy homeomorphisms on the square

TL;DR

This paper introduces a new form of chaos called $(\omega, \alpha, n)$-sensitivity for homeomorphisms, constructs examples on the square with zero entropy exhibiting this chaos, and analyzes their limit sets using boundary permeating techniques.

Contribution

It defines $(\omega, \alpha, n)$-sensitivity, constructs zero entropy square homeomorphisms with this property, and deeply investigates their limit sets via boundary permeating methods.

Findings

Existence of zero entropy homeomorphisms with $(\omega, \alpha, n)$-sensitivity.

Construction of homeomorphisms with prescribed limit sets.

Analysis of limit set structures using boundary permeating techniques.

Abstract

For a homeomorphism $f$ of a compact metric space $X$ and a positive integer $n\geq 2$, we introduce the notion of $(\omega, \alpha, n)$-sensitivity of $f$, which describes such a kind of chaos: there is some $c>0$ such that for any $x\in X$ and any open neighborhood $U$ of $x$, there are points $\{x_i\}_{i=1}^n$ and $\{y_i\}_{i=1}^n$ in $U$ such that both the collection of $\omega$-limit sets $\omega(x_i, f)$ and that of the $\alpha$-limit sets $\alpha(y_i, f)$ are pairwise $c$-separated. Then we construct a class of homeomorphisms of the square $[-1, 1]^2$ which are $(\omega, \alpha, n)$-sensitive for any $n\geq 2$ and have zero topological entropies. To investigate further the complexity of zero entropy homeomorphisms by using limit sets, we analyze in depth the limit sets of square homeomorphisms by the boundary permeating technique. Specially, we prove that for any given set of points $Y\equiv\{y_{n1}, y_{n2}:n\in\mathbb N\}$ in $(-1, 1)^2$ which satisfies some loosely technical conditions, and for any given family of pairwise disjoint countable dense subsets $\{W_n:n\in\mathbb N\}$ of $(-1, 1)^2-Y$, there is a zero entropy homeomorphism $f$ on the square $[-1, 1]^2$ such that $\omega(x, f)=\{y_{n1}\}$ and $\alpha(x, f)=\{y_{n2}\}$ for any $n$ and any $x\in W_n$.