Orders of accumulation of entropy on manifolds

TL;DR

This paper constructs continuous self-maps on manifolds with prescribed countable ordinal orders of entropy accumulation, extending the understanding of entropy structures on compact manifolds.

Contribution

It provides a method to realize any countable ordinal as the order of entropy accumulation on any compact manifold, including homeomorphisms in higher dimensions.

Findings

Any countable ordinal can be realized as the order of entropy accumulation.

Constructs include surjective maps on manifolds with prescribed entropy properties.

Homeomorphisms with arbitrary entropy accumulation orders exist on manifolds of dimension at least 2.

Abstract

For a continuous self-map $T$ of a compact metrizable space with finite topological entropy, the order of accumulation of entropy of $T$ is a countable ordinal that arises in the theory of entropy structure and symbolic extensions. Given any compact manifold $M$ and any countable ordinal $\al$, we construct a continuous, surjective self-map of $M$ having order of accumulation of entropy $\al$. If the dimension of $M$ is at least 2, then the map can be chosen to be a homeomorphism.