Topological entropy of a rational map over a complete metrized field

TL;DR

This paper extends the understanding of topological entropy for dominant rational maps to non-Archimedean fields, establishing an upper bound related to dynamical degrees and introducing the e-reduction concept.

Contribution

It generalizes a key entropy bound from complex to non-Archimedean settings and introduces the e-reduction of Berkovich spaces.

Findings

Topological entropy is bounded by maximum dynamical degrees in non-Archimedean fields.

Regular maps with extensions over valuation rings have zero entropy.

Introduces the e-reduction of Berkovich spaces as a new tool.

Abstract

We prove that the topological entropy of any dominant rational self-map of a projective variety defined over a complete non-Archimedean field is bounded from above by the maximum of its dynamical degrees, thereby extending a theorem of Gromov and Dinh-Sibony from the complex to the non-Archimedean setting. We proceed by proving that any regular self-map which admits a regular extension to a projective model defined over the valuation ring has necessarily zero entropy. To this end we introduce the e-reduction of a Berkovich analytic space, a notion of independent interest.