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Bifurcation for families of Ahlfors island maps

Matthieu Astorg, Anna Miriam Benini, Nuria Fagella

TL;DR

This work generalizes the stability framework from rational and finite-type meromorphic maps to broad Ahlfors island maps, proving that stability notions align via holomorphic motions of backward orbits and the marking of singular values. It introduces and leverages the concepts of activity and passivity for singular values, including a shooting technique and backward-orbit holomorphy, to connect parameter perturbations with dynamical stability. For finite-type maps, the paper establishes density of the $J$-stable locus and shows attracting cycles can be forced or tuned near active parameters, while handling the complexities of general Ahlfors island maps through virtual cycles and tract geometry. Overall, the results extend classical MSS/Lyubich stability characterizations to a wider dynamical setting and provide robust tools for analyzing bifurcations in complex-analytic families, including meromorphic and transcendental cases.

Abstract

We extend Mañé-Sad-Sullivan and Lyubich's equivalent characterization of stability to the setting of Ahlfors island maps, which include notably all meromorphic maps. As a consequence we also obtain the density of $J$-stability for finite type maps in the sense of Epstein.

Bifurcation for families of Ahlfors island maps

TL;DR

This work generalizes the stability framework from rational and finite-type meromorphic maps to broad Ahlfors island maps, proving that stability notions align via holomorphic motions of backward orbits and the marking of singular values. It introduces and leverages the concepts of activity and passivity for singular values, including a shooting technique and backward-orbit holomorphy, to connect parameter perturbations with dynamical stability. For finite-type maps, the paper establishes density of the -stable locus and shows attracting cycles can be forced or tuned near active parameters, while handling the complexities of general Ahlfors island maps through virtual cycles and tract geometry. Overall, the results extend classical MSS/Lyubich stability characterizations to a wider dynamical setting and provide robust tools for analyzing bifurcations in complex-analytic families, including meromorphic and transcendental cases.

Abstract

We extend Mañé-Sad-Sullivan and Lyubich's equivalent characterization of stability to the setting of Ahlfors island maps, which include notably all meromorphic maps. As a consequence we also obtain the density of -stability for finite type maps in the sense of Epstein.
Paper Structure (14 sections, 43 theorems, 50 equations)

This paper contains 14 sections, 43 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Activity Locus of Ahlfors Island Maps
  3. Preliminaries: Ahlfors island maps, finite type maps and singular values
  4. Ahlfors island maps
  5. Activity of singular values and preliminary results
  6. Density Theorems
  7. Characterization of stability: Proof of Theorem \ref{['th:mssahlfors']}
  8. Generalities on $J$-stability: Proof of Proposition \ref{['prop:equivalent_definitions']}
  9. Independence of the notion of $J$-stability from the choice of $\varphi_\lambda$: Proof of Proposition \ref{['prop:indepe']}
  10. Holomorphic motion of backward orbits
  11. Finite type maps and attracting cycles
  12. Creation of attracting cycles near virtual cycles
  13. Creation of attracting cycles at active parameters
  14. Characterization of $J$-stability: Proof of Theorem \ref{['th:MSS']}

Key Result

Proposition 1.7

Let $\{f_\lambda = \varphi_\lambda \circ f_{\lambda_0} \circ \psi_\lambda^{-1}\}_{\lambda _in M}$ be a natural family of Ahlfors island maps satisfying i:item1 in Definition defi:jstab. Assume that either Then $\{f_\lambda = \varphi_\lambda \circ f_{\lambda_0} \circ \psi_\lambda^{-1}\}_{\lambda \in M}$ also satisfies i:item2 in Definition defi:jstab.

Theorems & Definitions (93)

  • Definition 1.1: Ahlfors island map
  • Definition 1.2: Finite type map
  • Definition 1.3: Natural families
  • Remark 1.4
  • Definition 1.5: Passive singular value
  • Definition 1.6: $J$-stability
  • Proposition 1.7
  • Proposition 1.8
  • Theorem 1.9: $J-$stability of Ahlfors Islands maps
  • Theorem 1.10: $J-$stability of finite type maps
  • ...and 83 more