Accumulation of periodic points for local uniformly quasiregular mappings

TL;DR

This paper studies how periodic points accumulate in local uniformly quasiregular mappings on Riemannian manifolds, showing that Julia sets are densely accumulated by periodic points under certain conditions.

Contribution

It establishes new results on the accumulation and density of periodic points in local uniformly quasiregular dynamics, especially near Julia sets and singularities.

Findings

Julia sets are accumulated by periodic points.

Periodic points are dense in Julia sets under certain conditions.

Conditions for the density of repelling periodic points are provided.

Abstract

We consider accumulation of periodic points in local uniformly quasiregular dynamics. Given a local uniformly quasiregular mapping $f$ with a countable and closed set of isolated essential singularities and their accumulation points on a closed Riemannian manifold, we show that points in the Julia set are accumulated by periodic points. If, in addition, the Fatou set is non-empty and connected, the accumulation is by periodic points in the Julia set itself. We also give sufficient conditions for the density of repelling periodic points.