Topological pressure for one-dimensional holomorphic dynamical systems

TL;DR

This paper extends Bowen's classical result to a class of one-dimensional holomorphic maps, showing that topological pressure can be computed from repelling periodic points for a broad class of potentials.

Contribution

It demonstrates that for certain holomorphic maps, the topological pressure is fully determined by repelling periodic points, generalizing Bowen's theorem to non-uniform hyperbolic settings.

Findings

Topological pressure determined by repelling periodic points

Extension of Bowen's classical result to holomorphic maps

Applicable to a wide class of potentials

Abstract

For a class of one-dimensional holomorphic maps f of the Riemann sphere we prove that for a wide class of potentials h the topological pressure is entirely determined by the values of h on the repelling periodic points of f. This is a version of a classical result of Bowen for hyperbolic diffeomorphisms in the holomorphic non-uniformly hyperbolic setting.