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Extremal rate of convergence in discrete hyperbolic and parabolic dynamics

Francisco J. Cruz-Zamorano, Konstantinos Zarvalis

TL;DR

This work characterizes the extremal rate of convergence for non-elliptic self-maps of the upper half-plane with Denjoy–Wolff point at infinity, distinguishing hyperbolic and parabolic cases. It leverages the Herglotz representation, Koenigs function conformality at infinity, asymptotic boundary behavior, and hyperbolic distance to provide equivalent criteria for extremality, including integral conditions and distance limits. A key contribution is tying extremality to finite-parameter representations (Herglotz triplets) and to operator-theoretic consequences via composition operators on classical function spaces after disc-translation. The results give precise, computable criteria for extremal rate and unify several classical notions (finite shift, Koenigs conformality, and hyperbolic distance asymptotics) under the umbrella of extremal dynamics.

Abstract

This paper investigates the dynamical behaviour of holomorphic self-maps of the upper half-plane. More precisely, we focus on the hyperbolic and parabolic self-maps whose orbits approach the Denjoy--Wolff point with the slowest possible rate. We characterize self-maps of such extremal rate using various tools, like the Herglotz representation, the conformality of the Koenigs function at the Denjoy--Wolff point and the hyperbolic distance.

Extremal rate of convergence in discrete hyperbolic and parabolic dynamics

TL;DR

This work characterizes the extremal rate of convergence for non-elliptic self-maps of the upper half-plane with Denjoy–Wolff point at infinity, distinguishing hyperbolic and parabolic cases. It leverages the Herglotz representation, Koenigs function conformality at infinity, asymptotic boundary behavior, and hyperbolic distance to provide equivalent criteria for extremality, including integral conditions and distance limits. A key contribution is tying extremality to finite-parameter representations (Herglotz triplets) and to operator-theoretic consequences via composition operators on classical function spaces after disc-translation. The results give precise, computable criteria for extremal rate and unify several classical notions (finite shift, Koenigs conformality, and hyperbolic distance asymptotics) under the umbrella of extremal dynamics.

Abstract

This paper investigates the dynamical behaviour of holomorphic self-maps of the upper half-plane. More precisely, we focus on the hyperbolic and parabolic self-maps whose orbits approach the Denjoy--Wolff point with the slowest possible rate. We characterize self-maps of such extremal rate using various tools, like the Herglotz representation, the conformality of the Koenigs function at the Denjoy--Wolff point and the hyperbolic distance.
Paper Structure (10 sections, 25 theorems, 116 equations)

This paper contains 10 sections, 25 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hyperbolic distance
  4. Discrete dynamics
  5. Extremal Rates
  6. Herglotz representation
  7. Conformality
  8. Asymptotic behaviour near infinity
  9. Hyperbolic distance
  10. Composition operators

Key Result

Theorem A

Let $f\vcentcolon \mathbb{H} \to \mathbb{H}$ be hyperbolic with Denjoy--Wolff point infinity and Herglotz representation Then, $f$ is of extremal rate if and only if

Theorems & Definitions (51)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem 2.1
  • Theorem 2.2
  • Proposition 3.1
  • proof
  • Remark 3.2
  • Definition 3.3
  • Proposition 3.4
  • ...and 41 more