Holomorphic Shadowing for H\'enon Maps Revisited: an Implicit Function Theorem Perspective

TL;DR

This paper revisits holomorphic shadowing in complex Hénon maps, offering an alternative proof of a key theorem using an implicit function theorem approach, enhancing understanding of bounded orbits and the solenoid locus.

Contribution

It provides a new proof of a fundamental theorem on Hénon maps using the implicit function theorem, differing from previous contraction mapping approaches.

Findings

Established an implicit function theorem framework for Hénon map analysis

Provided an alternative proof of Hubbard and Oberste-Vorth's theorem

Enhanced understanding of bounded orbits and solenoid locus in complex dynamics

Abstract

In studying the complex H\'enon maps, Mummert (in "Holomorphic shadowing for H\'enon maps" Nonlinearity 21 pp. 2887-2898, 2008) defined an operator the fixed points of which give rise to bounded orbits. This enabled him to obtain an estimate of the solenoid locus. Instead of the contraction mapping theorem, in this paper we present an implicit function theorem version of his result by providing an alternative proof of a theorem of Hubbard and Oberste-Vorth (in Real and Complex Dynamical Systems, pp.89-132, 1995).