Topological and geometric hyperbolicity criteria for polynomial automorphisms of

Eric Bedford,Romain Dujardin

doi:10.1017/etds.2021.47

Abstract

AbstractWe prove that uniform hyperbolicity is invariant under topological conjugacy for dissipative polynomial automorphisms of $\mathbb {C}^2$ . Along the way we also show that a sufficient condition for hyperbolicity is that local stable and unstable manifolds of saddle points have uniform geometry.

Highlights

Hyperbolicity here is understood in the sense of [BS1], that is, we say that a complex Hénon map f is hyperbolic if its Julia set J = Jf is a hyperbolic set, which must be of saddle type
We further show that the transversality assumption is superfluous (Theorems 2.12 and 2.13), and that, as might be expected, in the dissipative case it is enough to control the geometry of unstable manifolds (Theorem 2.18)
If J − is globally laminated one might expect that the additional assumption that Wu(x) ≃ C for every x in Theorem 2.18 would follow from the density of unstable manifolds of saddle points

Summary

Introduction

Following [BS8], a complex Hénon map f is said to be quasi-hyperbolic if there exist positive constants r and B such that for every saddle periodic point p: (i) Wrs/u(p) is closed in B(p, r); and (ii) the area of Wrs/u(p) is bounded by B. Under the assumptions of Proposition 2.4, there exist a neighborhood N of J ⋆ and a lamination Wu of N by Riemann surfaces which extends the family of local unstable manifolds of saddle points.

Results

Conclusion