Pesin theory for transcendental maps and applications

TL;DR

This paper extends Pesin theory to certain transcendental maps, analyzing Lyapunov exponents and inverse branches, with applications to boundary dynamics and periodic points in complex analysis.

Contribution

It develops Pesin theory for transcendental maps' boundary dynamics, providing new insights into inverse branches and Lyapunov exponents under specific conditions.

Findings

Generic inverse branches are well-defined and conformal.

Lyapunov exponents with respect to harmonic measure are characterized.

Density of periodic boundary points is established for certain Fatou components.

Abstract

In this paper, we develop Pesin theory for the boundary map of some Fatou components of transcendental functions, under certain hyptothesis on the singular values and the Lyapunov exponent. That is, we prove that generic inverse branches for such maps are well-defined and conformal. In particular, we study in depth the Lyapunov exponents with respect to harmonic measure, providing results which are of independent interest. As an application of our results, we describe in detail generic inverse branches for centered inner functions, and we prove density of periodic boundary points for a large class of Fatou components. Our proofs use techniques from measure theory, ergodic theory, conformal analysis, and inner functions, as well as estimates on harmonic measure.