Aperiodic homeomorphisms approximate chain mixing endomorphisms on the Cantor set

TL;DR

This paper demonstrates that aperiodic homeomorphisms on the Cantor set can approximate chain mixing endomorphisms under certain conditions, expanding understanding of topological conjugacy and approximation in dynamical systems.

Contribution

It establishes that homeomorphisms conjugate to aperiodic homeomorphisms can approximate chain mixing endomorphisms on the Cantor set, given a necessary condition on periodic points.

Findings

Homeomorphisms conjugate to g approximate f under certain conditions.

Approximation holds for chain mixing endomorphisms with fixed points.

Topological conjugacy plays a key role in approximation results.

Abstract

Let $f$ be a chain mixing continuous onto mapping from the Cantor set onto itself.Let $g$ be an aperiodic homeomorphism on the Cantor set. We show that homeomorphisms that are topologically conjugate to g approximate $f$ in the topology of uniform convergence if a trivial necessary condition on periodic points is satisfied.In particular, let $f$ be a chain mixing continuous onto mapping from the Cantor set onto itself with a fixed point and g, an aperiodic homeomorphism on the Cantor set. Then, homeomorphisms that are topologically conjugate to $g$ approximate $f$.