Non-archimedean connected Julia sets with branching

TL;DR

This paper constructs the first examples of non-archimedean rational functions with connected Julia sets exhibiting branching, and analyzes their entropy properties, answering a question about entropy differences.

Contribution

It introduces new examples of non-archimedean rational functions with connected, branched Julia sets and computes their measure-theoretic and topological entropy.

Findings

Julia sets are connected but not contained in a line segment

Measure-theoretic entropy is strictly less than topological entropy for some examples

Answers a question of Favre and Rivera-Letelier regarding entropy differences

Abstract

We construct the first examples of rational functions defined over a non-archimedean field with certain dynamical properties. In particular, we find such functions whose Julia sets, in the Berkovich projective line, are connected but not contained in a line segment. We also show how to compute the measure-theoretic and topological entropy of such maps. In particular, we show for some of our examples that the measure-theoretic entropy is strictly smaller than the topological entropy, thus answering a question of Favre and Rivera-Letelier.