Chain mixing endomorphisms are approximated by subshifts on the Cantor set

TL;DR

This paper demonstrates that chain mixing endomorphisms on the Cantor set can be approximated by subshifts, providing conditions under which such approximations are possible, especially focusing on topological conjugacy and periodic points.

Contribution

It establishes new approximation results for chain mixing endomorphisms on the Cantor set using subshifts, highlighting the role of periodic points and conjugacy.

Findings

Homeomorphisms conjugate to subshifts approximate chain mixing endomorphisms.

Approximation depends on the presence of fixed points and periodic points.

Homeomorphisms conjugate to full shifts can approximate any chain mixing endomorphism.

Abstract

Let f be a chain mixing continuous onto mapping from the Cantor set onto itself. Let g be a homeomorphism on the Cantor set that is topologically conjugate to a subshift. Then, homeomorphisms that are topologically conjugate to g approximate f in the topology of uniform convergence if a trivial necessary condition on the periodic points holds. In particular, if f is a chain mixing continuous onto mapping from the Cantor set onto itself with a fixed point, then homeomorphisms on the Cantor set that are topologically conjugate to a subshift approximate f in the topology of uniform convergence. In addition, homeomorphisms on the Cantor set that are topologically conjugate to a subshift without periodic points approximate any chain mixing continuous onto mappings from the Cantor set onto itself. In particular, let f be a homeomorphism on the Cantor set that is topologically conjugate to a full shift. Let g be a homeomorphism on the Cantor set that is topologically conjugate to a subshift. Then, a sequence of homeomorphisms that is topologically conjugate to g approximates f.