Pesin theory for transcendental maps and applications

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.