Density of the set of symbolic dynamics with all ergodic measures supported on periodic orbits

TL;DR

This paper demonstrates that within the space of homeomorphisms and endomorphisms on the Cantor set, one can find maps arbitrarily close to any given map where all orbits have periodic orbits as their limit sets, highlighting the density of such dynamics.

Contribution

The authors prove the density of maps with all orbits supported on periodic points in the space of homeomorphisms and endomorphisms on the Cantor set.

Findings

Existence of maps with all orbits approaching periodic orbits.

Density of such maps in the space of homeomorphisms.

Approximation of any map by maps with purely periodic orbit dynamics.

Abstract

Let $K$ be the Cantor set. We prove that arbitrarily close to a homeomorphism $T:K\rightarrow K$ there exists a homeomorphism $\widetilde T:K\rightarrow K$ such that the $\alpha$-limit and the $\omega$-limit of every orbit is a periodic orbit. We also prove that arbitrarily close to an endomorphism $T:K\rightarrow K$ there exists an endomorphism $\widetilde T:K\rightarrow K$ close to $T$ such that every orbit is finally periodic.