Large deviation principles for non-uniformly hyperbolic rational maps

TL;DR

This paper establishes large deviation principles for certain non-uniformly hyperbolic rational maps, specifically those satisfying the Topological Collet-Eckmann condition, by analyzing equilibrium states and pressure functions.

Contribution

It proves large deviation principles for iterated preimages, periodic points, and Birkhoff averages in non-uniform hyperbolic rational maps, extending existing theories.

Findings

Unique equilibrium states for Hölder continuous potentials

Characterization of pressure functions via preimages, periodic points, and averages

Large deviation principles for multiple dynamical distributions

Abstract

We show some level-2 large deviation principles for rational maps satisfying a strong form of non-uniform hyperbolicity, called "Topological Collet-Eckmann". More precisely, we prove a large deviation principle for the distribution of iterated preimages, periodic points, and Birkhoff averages. For this purpose we show that each H{\"o}lder continuous potential admits a unique equilibrium state, and that the pressure function can be characterized in terms of iterated preimages, periodic points, and Birkhoff averages. Then we use a variant of a general result of Kifer.