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Backward-iteration sequences with bounded hyperbolic steps for analytic self-maps of the disk

Pietro Poggi-Corradini

TL;DR

This paper addresses backward-iteration sequences for analytic disk self-maps, focusing on sequences with bounded hyperbolic steps (BISBS) to reveal map structure near the Denjoy–Wolff point. It develops a parabolic-case analysis, classifies BISBS by height (type 1 non-zero height vs type 2 zero height), and constructs a conjugation to an upper-half-plane model that governs BISBS behavior. Key results include existence and uniqueness of the conjugating map in non-zero-height cases, precise asymptotics for the conjugation, and nonexistence results for certain type I parabolic maps, supported by explicit examples. The work provides a framework for understanding backward dynamics and complements the classical forward Denjoy–Wolff theory with concrete conjugacies and limiting behavior.

Abstract

A lot is known about the forward iterates of an analytic function which is bounded by 1 in modulus on the unit disk. The Denjoy-Wolff Theorem describes their convergence properties and several authors, from the 1880's to the 1980's, have provided conjugations which yield very precise descriptions of the dynamics. Backward-iteration sequences are of a different nature because a point could have infinitely many preimages as well as none. However, if we insist in choosing preimages that are at a finite hyperbolic distance each time, we obtain sequences which have many similarities with the forward-iteration sequences, and which also reveal more information about the map itself. In this note we try to present a complete study of backward-iteration sequences with bounded hyperbolic steps for analytic self-maps of the disk.

Backward-iteration sequences with bounded hyperbolic steps for analytic self-maps of the disk

TL;DR

This paper addresses backward-iteration sequences for analytic disk self-maps, focusing on sequences with bounded hyperbolic steps (BISBS) to reveal map structure near the Denjoy–Wolff point. It develops a parabolic-case analysis, classifies BISBS by height (type 1 non-zero height vs type 2 zero height), and constructs a conjugation to an upper-half-plane model that governs BISBS behavior. Key results include existence and uniqueness of the conjugating map in non-zero-height cases, precise asymptotics for the conjugation, and nonexistence results for certain type I parabolic maps, supported by explicit examples. The work provides a framework for understanding backward dynamics and complements the classical forward Denjoy–Wolff theory with concrete conjugacies and limiting behavior.

Abstract

A lot is known about the forward iterates of an analytic function which is bounded by 1 in modulus on the unit disk. The Denjoy-Wolff Theorem describes their convergence properties and several authors, from the 1880's to the 1980's, have provided conjugations which yield very precise descriptions of the dynamics. Backward-iteration sequences are of a different nature because a point could have infinitely many preimages as well as none. However, if we insist in choosing preimages that are at a finite hyperbolic distance each time, we obtain sequences which have many similarities with the forward-iteration sequences, and which also reveal more information about the map itself. In this note we try to present a complete study of backward-iteration sequences with bounded hyperbolic steps for analytic self-maps of the disk.

Paper Structure

This paper contains 15 sections, 19 theorems, 138 equations.

Table of Contents

  1. Introduction
  2. Boundary repelling fixed points
  3. Backward-iteration sequences with bounded steps
  4. Parabolic forward iteration
  5. Parabolic backward iteration
  6. Uniqueness for BISBS of type 1
  7. More on BISBS of type 1
  8. BISBS for type I parabolic maps
  9. Convergence properties
  10. The parabolic case
  11. Proof of Theorem \ref{['thm:ratios']}, Theorem \ref{['thm:conj']} and Corollary \ref{['cor:ratios']}
  12. Uniqueness in the non-zero-height case
  13. Further properties for BISBS of type 1
  14. Non-existence results for parabolic maps of type I
  15. Examples

Key Result

Theorem 1.1

Suppose $\phi$ is an analytic map of the disk with $\phi(\mathbb{D})\subset\mathbb{D}$, and $\zeta,\xi \in\partial \mathbb{D}$. If there is a sequence $\{p_{n} \}\subset \mathbb{D}$ such that $p_{n}\rightarrow \zeta$, $\phi (p_{n})\rightarrow \xi$, and then

Theorems & Definitions (41)

  • Theorem 1.1: Julia and Carathéodory,sh Chap. 4
  • Corollary 1.2
  • Theorem 1.3: Cowen-Pommerenke, see Theorem 4.1 of cp
  • Theorem 1.4: finn Thm. 1.2
  • Definition 1.5
  • Theorem 1.6
  • Example 1.7
  • Theorem 1.8: Pommerenke, pom (3.17)
  • Lemma 1.9
  • proof : Proof of Lemma \ref{['lem:sum']}
  • ...and 31 more