Hyperbolicity and Quasi-hyperbolicity in Polynomial Diffeomorphisms of ${\Bbb C}^2$

TL;DR

This paper investigates the conditions under which complex Henon maps, which are quasi-hyperbolic, become uniformly hyperbolic, focusing on the role of tangencies between stable and unstable manifolds.

Contribution

It establishes a precise criterion linking the absence of tangencies to the hyperbolicity of quasi-hyperbolic polynomial diffeomorphisms in A7C^2.

Findings

Quasi-hyperbolic maps are uniformly hyperbolic if and only if stable and unstable manifolds do not have tangencies.

Provides a characterization of hyperbolicity in polynomial diffeomorphisms of A7C^2.

Clarifies the relationship between hyperbolicity and manifold tangencies in complex dynamics.

Abstract

We consider complex Henon maps which are quasi-hyperbolic. We show that a quasi-hyperbolic map is uniformly hyperbolic if and only if there are no tangencies between stable and unstable manifolds.