\'Etale dynamical systems and topological entropy

TL;DR

This paper explores defining topological entropy for dynamical systems using étale covers and analogs, aiming to unify topological and algebraic perspectives and propose new methods for measuring complexity.

Contribution

It introduces an étale framework for defining topological entropy in both topological and algebraic dynamical systems, providing foundational results and conjectures for future validation.

Findings

Proposed a new definition of topological entropy using étale covers.

Developed an étale analog for algebraic dynamical systems.

Established basic results and formulated a conjecture for further research.

Abstract

In this paper, we consider two questions about topological entropy of dynamical systems. We propose to resolve these questions by the same approach of using \'etale analogs of topological and algebraic dynamical systems. The first question is to define topological entropy for a topological dynamical system $(f,X,\Omega )$. The main idea is to make use - in addition to invariant compact subspaces of $(X,\Omega )$ - of compactifications of \'etale covers $\pi :(f',X',\Omega ')\rightarrow (f,X,\Omega )$, that is $\pi \circ f'=f\circ \pi$ and the fibers of $\pi $ are all finite. We prove some basic results and propose a conjecture, whose validity allows us to prove further results. The second question is to define topological entropy for algebraic dynamical systems, with the requirement that it should be as close to the pullback on cohomology groups as possible. To this end, we develop an \'etale analog of algebraic dynamical systems.