Cantor subsystems on the Gehman dendrite

TL;DR

This paper constructs specific dynamical systems on the Gehman dendrite that replicate any given surjective Cantor system on its endpoints, demonstrating diverse mixing and exact behaviors.

Contribution

It introduces methods to embed arbitrary surjective Cantor systems as subsystems on the Gehman dendrite with prescribed dynamical properties.

Findings

Constructed mixing but not exact maps on the Gehman dendrite.

Constructed exact maps on the Gehman dendrite.

Subsystems on endpoints are conjugate to any given surjective Cantor system.

Abstract

In the present note we focus on dynamics on the Gehman dendrite $\mathcal{G}$. It is well-known that the set of its endpoints is homeomorphic to a standard Cantor ternary set. For any given surjective Cantor system $\mathcal{C}$ we provide constructions of (i) a mixing but not exact and (ii) an exact map on $\mathcal{G}$, such that in both cases the subsystem formed by $\text{End}(\mathcal{G})$ is conjugate to the initially chosen system on $\mathcal{C}$.