\'Etale dynamical systems and topological entropy

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.