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Polynomial-like dynamics of analytic maps

Genadi Levin

TL;DR

We study backward invariant (BI) maps as a natural generalization of polynomial-like maps and prove that a polynomial-like restriction exists for BI maps if and only if there is a completely invariant full compact set $K$ containing the postcritical set $P_g$, with $K$ contained in the intersection of all pullbacks $\cap_{i\ge0} U_{-i}$. This result extends to generalized BI maps where $U_{-1}$ may have multiple Jordan components, yielding a generalized PL restriction with a (potentially disconnected) filled Julia set. The proofs combine complex-analytic techniques, circle-map conjugacies, and expansion arguments to establish a dichotomy: either a PL restriction exists and $K$ captures almost all repelling dynamics, or substantial repelling structure persists near $\partial K$ with no PL restriction. These findings broaden the straightening philosophy beyond classical polynomial-like maps and relate to invariant line fields and renormalization phenomena in holomorphic dynamics.

Abstract

The theory of polynomial-like maps is of fundamental importance in holomorphic dynamics. We study dynamical properties of a larger class of maps. Our main result is that, under some natural conditions, a map of this class has a completely invariant compact set if and only if this set is the filled Julia set of a polynomial-like restriction of the map. We also generalize this result to include maps with non-connected domains of definition.

Polynomial-like dynamics of analytic maps

TL;DR

We study backward invariant (BI) maps as a natural generalization of polynomial-like maps and prove that a polynomial-like restriction exists for BI maps if and only if there is a completely invariant full compact set containing the postcritical set , with contained in the intersection of all pullbacks . This result extends to generalized BI maps where may have multiple Jordan components, yielding a generalized PL restriction with a (potentially disconnected) filled Julia set. The proofs combine complex-analytic techniques, circle-map conjugacies, and expansion arguments to establish a dichotomy: either a PL restriction exists and captures almost all repelling dynamics, or substantial repelling structure persists near with no PL restriction. These findings broaden the straightening philosophy beyond classical polynomial-like maps and relate to invariant line fields and renormalization phenomena in holomorphic dynamics.

Abstract

The theory of polynomial-like maps is of fundamental importance in holomorphic dynamics. We study dynamical properties of a larger class of maps. Our main result is that, under some natural conditions, a map of this class has a completely invariant compact set if and only if this set is the filled Julia set of a polynomial-like restriction of the map. We also generalize this result to include maps with non-connected domains of definition.

Paper Structure

This paper contains 16 sections, 13 theorems, 35 equations.

Table of Contents

  1. Introduction and main results
  2. Polynomial-like maps
  3. Main result
  4. Generalized BI maps
  5. Description of the paper
  6. Examples of BI maps
  7. On BI maps
  8. Proof of Theorem \ref{['thm-mainstep']}
  9. The case of connected $K$
  10. Proof of Theorem \ref{['thm-m']} for disconnected $K$
  11. Proof of Theorem \ref{['thm-m']}
  12. Proof of Corollary \ref{['c-uniq']} and its Complement
  13. Proof of Theorem \ref{['thm-gen-bi']}
  14. Entropy of locally analytic maps
  15. Uniformization
  16. ...and 1 more sections

Key Result

Theorem 1.1

Suppose that $g: U_{-1}\to U_0$ is a proper holomorphic map of some degree $d\ge 2$ which extends to a BI map $g: U_{-i-1}\to U_{-i}$, $i=1,2,...$. Let $K\subset \cap_{i\ge 0} U_{-i}$ be a compact set as follows: Then $g$ has a polynomial-like restriction and $K$ is its filled Julia set. In particular, $K$ is a continuum,

Theorems & Definitions (30)

  • Definition 1.1
  • Remark 1.2
  • Example 1.3
  • Definition 1.4
  • Theorem 1.1
  • Corollary 1.5
  • Remark 1.6
  • Remark 1.7
  • Theorem 1.2
  • Theorem 1.3
  • ...and 20 more