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Backward dynamics of non-expanding maps in Gromov hyperbolic metric spaces

Leandro Arosio, Matteo Fiacchi, Lorenzo Guerini, Anders Karlsson

Abstract

We study the interplay between the backward dynamics of a non-expanding self-map $f$ of a proper geodesic Gromov hyperbolic metric space $X$ and the boundary regular fixed points of $f$ in the Gromov boundary. To do so, we introduce the notion of stable dilation at a boundary regular fixed point of the Gromov boundary, whose value is related to the dynamical behaviour of the fixed point. This theory applies in particular to holomorphic self-maps of bounded domains $Ω\subset\subset \mathbb{C}^q$, where $Ω$ is either strongly pseudoconvex, convex finite type, or pseudoconvex finite type with $q=2$, and solves several open problems from the literature. We extend results of holomorphic self-maps of the disc $\mathbb{D}\subset \mathbb{C}$ obtained by Bracci and Poggi-Corradini. In particular, with our geometric approach we are able to answer a question, open even for the unit ball $\mathbb{B}^q\subset \mathbb{C}^q$, namely that for holomorphic parabolic self-maps any escaping backward orbit with bounded step always converges to a point in the boundary.

Backward dynamics of non-expanding maps in Gromov hyperbolic metric spaces

Abstract

We study the interplay between the backward dynamics of a non-expanding self-map of a proper geodesic Gromov hyperbolic metric space and the boundary regular fixed points of in the Gromov boundary. To do so, we introduce the notion of stable dilation at a boundary regular fixed point of the Gromov boundary, whose value is related to the dynamical behaviour of the fixed point. This theory applies in particular to holomorphic self-maps of bounded domains , where is either strongly pseudoconvex, convex finite type, or pseudoconvex finite type with , and solves several open problems from the literature. We extend results of holomorphic self-maps of the disc obtained by Bracci and Poggi-Corradini. In particular, with our geometric approach we are able to answer a question, open even for the unit ball , namely that for holomorphic parabolic self-maps any escaping backward orbit with bounded step always converges to a point in the boundary.
Paper Structure (10 sections, 38 theorems, 160 equations)

This paper contains 10 sections, 38 theorems, 160 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Gromov hyperbolicity
  4. Horofunctions
  5. Forward dynamics of elliptic maps
  6. Behaviour of the dilation at a BRFP under iterates
  7. The case of equivalent compactifications
  8. Stable dilation at a BRFP
  9. Backward orbits with bounded step converging to a repelling BRFP
  10. Backward Denjoy--Wolff theorem

Key Result

Theorem 1.1

Ab1988 Let $f\colon \Omega\to \Omega$ be a holomorphic self-map of a bounded strongly pseudoconvex domain $\Omega\subset \mathbb{C}^q$. If $f$ admits an escaping orbit, then there exists a point $\zeta\in \partial \Omega$ such that every forward orbit converges to $\zeta$.

Theorems & Definitions (96)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3: PC1
  • Theorem 1.4: BrPoggiCorradini
  • Theorem 1.5
  • Theorem 1.6
  • Definition 2.1
  • Definition 2.2: Gromov compactification
  • Remark 2.3
  • Definition 2.4
  • ...and 86 more