Thermodynamic formalism methods in one-dimensional real and complex dynamics

TL;DR

This paper surveys thermodynamic formalism in one-dimensional real and complex dynamics, exploring hyperbolicity, pressure, equilibrium states, and their connections to Hausdorff dimension and growth rates of derivatives.

Contribution

It provides a comprehensive overview of thermodynamic methods and their applications to geometric and dynamical properties in one-dimensional dynamics.

Findings

Relations between geometric pressure and Hausdorff dimension.

Analysis of fluctuations in potential sums and derivative growth.

Connections between thermodynamic formalism and geometric coding trees.

Abstract

We survey some results on non-uniform hyperbolicity, geometric pressure and equilibrium states in one-dimensional real and complex dynamics. We present some relations with Hausdorff dimension and measures with refined gauge functions of limit sets for geometric coding trees for rational functions on the Riemann sphere. We discuss fluctuations of iterated sums of the potential $-t\log |f'|$ and of radial growth of derivative of univalent functions on the unit disc and the boundaries of range domains preserved by a holomorphic map $f$ repelling towards the domains.