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On symmetries of the IFS attractor

Genadi Levin

TL;DR

The paper addresses symmetry structures of attractors for holomorphic IFS in the plane by transferring methods from complex dynamics to the study of IFS. It introduces J-like, non-laminar plane compacts and proves that nontrivial local holomorphic symmetries are finite in this setting, with the group of local conformal automorphisms likewise finite. It then provides a comprehensive framework (SSC/OSC, complex inverse dynamics, box-like restrictions) and proves necessary and sufficient conditions for two SSC holomorphic IFS to share the same attractor via infinite and finite functional equations, along with a description of all holomorphic IFS sharing an attractor and implications for multiplier spectra. The work combines geometric complex analysis (Koebe distortion, Schwarz lemma) with inverse-dynamics techniques to yield finiteness, equivalence, and structural results that extend Julia-set type symmetry phenomena to holomorphic IFS.

Abstract

We apply some methods and technique of complex dynamics to study the set of symmetries of attractors of holomorphic Iterated Function Systems (IFS), as well as relations between IFS sharing the same attractor.

On symmetries of the IFS attractor

TL;DR

The paper addresses symmetry structures of attractors for holomorphic IFS in the plane by transferring methods from complex dynamics to the study of IFS. It introduces J-like, non-laminar plane compacts and proves that nontrivial local holomorphic symmetries are finite in this setting, with the group of local conformal automorphisms likewise finite. It then provides a comprehensive framework (SSC/OSC, complex inverse dynamics, box-like restrictions) and proves necessary and sufficient conditions for two SSC holomorphic IFS to share the same attractor via infinite and finite functional equations, along with a description of all holomorphic IFS sharing an attractor and implications for multiplier spectra. The work combines geometric complex analysis (Koebe distortion, Schwarz lemma) with inverse-dynamics techniques to yield finiteness, equivalence, and structural results that extend Julia-set type symmetry phenomena to holomorphic IFS.

Abstract

We apply some methods and technique of complex dynamics to study the set of symmetries of attractors of holomorphic Iterated Function Systems (IFS), as well as relations between IFS sharing the same attractor.
Paper Structure (16 sections, 17 theorems, 60 equations)

This paper contains 16 sections, 17 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Local (holomorphic) symmetries of plane compacta
  3. Holomorphic IFS sharing the same attractor: preliminaries and main results
  4. Holomorphic IFS
  5. Separation conditions and the associated inverse dynamics
  6. Symmetries of the attractor under separation conditions
  7. Main results
  8. Some results in complex analysis
  9. The Koebe distortion theorem
  10. Schwarz's lemma and the slit complex plane
  11. Proofs of the main results
  12. Theorem \ref{['t-homeo']}
  13. Theorem \ref{['t-finite']}+Corollary \ref{['c-finite']}
  14. Theorems \ref{['t-symmifs']}-\ref{['t-finitegen']}: the key lemma
  15. Proof of Theorem \ref{['t-symmifs']} and Corollary \ref{['c-preper']}
  16. ...and 1 more sections

Key Result

Theorem 2.3

Let a compact $K\subset\mathbb{C}$ with no isolated points be $J$-like relative to a domain $U$. Then either

Theorems & Definitions (31)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Definition 3.1
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • Remark 3.5
  • ...and 21 more