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Invariant escaping Fatou components with two rank-one limit functions for automorphisms of ${\mathbb C}^2$

Part of: Holomorphic mappings and correspondences Complex dynamical systems

Published online by Cambridge University Press:  22 November 2021

ANNA MIRIAM BENINI Open the ORCID record for ANNA MIRIAM BENINI [Opens in a new window] ,
ALBERTO SARACCO Open the ORCID record for ALBERTO SARACCO [Opens in a new window]  and
MICHELA ZEDDA Open the ORCID record for MICHELA ZEDDA [Opens in a new window]
ANNA MIRIAM BENINI*
Affiliation:
Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università degli studi di Parma, Parco Area delle Scienze, 53/A, 43124 Parma PR, Italy
ALBERTO SARACCO
Affiliation:
Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università degli studi di Parma, Parco Area delle Scienze, 53/A, 43124 Parma PR, Italy
MICHELA ZEDDA
Affiliation:
Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università degli studi di Parma, Parco Area delle Scienze, 53/A, 43124 Parma PR, Italy
*
e-mail: ambenini@gmail.com
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Abstract

We construct automorphisms of ${\mathbb C}^2$ , and more precisely transcendental Hénon maps, with an invariant escaping Fatou component which has exactly two distinct limit functions, both of (generic) rank one. We also prove a general growth lemma for the norm of points in orbits belonging to invariant escaping Fatou components for automorphisms of the form $F(z,w)=(g(z,w),z)$ with $g(z,w):{\mathbb C}^2\rightarrow {\mathbb C}$ holomorphic.

Keywords

transcendental Henon mapsBaker domainsFatou components with rank one limit manifoldsescaping Fatou components

MSC classification

Primary: 32H50: Iteration problems 37F10: Polynomials; rational maps; entire and meromorphic functions
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 43 , Issue 2 , February 2023 , pp. 401 - 416
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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