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Pesin theory for transcendental maps and applications

Anna Jové

TL;DR

The paper extends Pesin theory to the boundary dynamics of transcendental maps, proving that generic inverse branches on boundary points of attracting Fatou basins are well-defined and contracting under a finite-entropy Lyapunov condition and a sparsity condition on singular values with respect to harmonic measure. It then adapts the construction to entire maps via first return maps and Rokhlin's natural extension, obtaining analogous Pesin-type results in this broader setting. As applications, it develops Pesin theory for centered inner functions and establishes density of periodic boundary points under suitable postsingular conditions, while also providing Lyapunov-exponent results (integrability and non-negativity) for transcendental maps, including parabolic basins and Baker domains. The methods integrate measure-theoretic ergodic theory, distortion estimates, conformal analysis, and harmonic-measure techniques to overcome non-compactness and infinite singularity sets inherent in the transcendental context.

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.

Pesin theory for transcendental maps and applications

TL;DR

The paper extends Pesin theory to the boundary dynamics of transcendental maps, proving that generic inverse branches on boundary points of attracting Fatou basins are well-defined and contracting under a finite-entropy Lyapunov condition and a sparsity condition on singular values with respect to harmonic measure. It then adapts the construction to entire maps via first return maps and Rokhlin's natural extension, obtaining analogous Pesin-type results in this broader setting. As applications, it develops Pesin theory for centered inner functions and establishes density of periodic boundary points under suitable postsingular conditions, while also providing Lyapunov-exponent results (integrability and non-negativity) for transcendental maps, including parabolic basins and Baker domains. The methods integrate measure-theoretic ergodic theory, distortion estimates, conformal analysis, and harmonic-measure techniques to overcome non-compactness and infinite singularity sets inherent in the transcendental context.

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.

Paper Structure

This paper contains 25 sections, 45 theorems, 113 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Distortion estimates for univalent maps
  4. Abstract Ergodic Theory
  5. Rokhlin's natural extension
  6. Harmonic measure
  7. Inner functions. Inverse branches and distortion
  8. On the setting and the hypotheses
  9. Fatou components of functions of class $\mathbb{K}$
  10. On the condition $\int_{\partial U} \log\left| x-SV\right|^{-1} d\omega_U(x)<\infty$
  11. Pesin theory for attracting basins. Theorem \ref{['teo:A']}
  12. Proof of Theorem \ref{['thm-pesin']}
  13. Entire functions and the first return map. Theorem \ref{['teo:B']}
  14. The first return map and Rokhlin's natural extension
  15. Pesin theory for the first return map. Proof of Theorem \ref{['teo:B']}
  16. ...and 10 more sections

Key Result

Theorem 1

(Pesin theory for attracting basins of transcendental maps) Let $f\colon\mathbb{C}\to\widehat{\mathbb{C}}$ be a meromorphic function, and let $U$ be a simply connected attracting basin, with fixed point $p\in U$. Let $\omega_U$ be the harmonic measure on $\partial U$ with base point $p$. Assume $f$

Figures (4)

  • Figure 8.1: Situation after Lemma \ref{['sublemma-puntperiodic1']}.
  • Figure 8.2: The construction of the curve $\gamma$ in $\mathbb{D}$, and its image $\varphi(\gamma)$ in the dynamical plane.
  • Figure 9.1: A visual representation of the definition of the sector $S_{\alpha, r} (x, \xi)$.
  • Figure 9.2: A visual scheme to approximate harmonic measure of sectors.

Theorems & Definitions (86)

  • Theorem 1
  • Theorem 2
  • Remark
  • Corollary 3
  • Corollary 4
  • Definition
  • Proposition 5
  • Proposition 6
  • Remark
  • Remark
  • ...and 76 more