Persistence of laminations

TL;DR

This paper extends the Hirsch-Pugh-Shub theorem on the persistence of normally hyperbolic laminations to include endomorphisms, complex analytic cases, and non-compact laminations, with applications in complex dynamics.

Contribution

It provides new persistence results for complex and non-compact laminations, expanding the classical theory to broader dynamical systems.

Findings

Persistence of complex laminations in dynamics of several complex variables

Construction of laminar structures on stable and unstable manifolds

Extensions to endomorphisms and non-compact laminations

Abstract

We present a modern proof of some extensions of the celebrated Hirsch-Pugh-Shub theorem on persistence of normally hyperbolic compact laminations. Our extensions consist of allowing the dynamics to be an endomorphism, of considering the complex analytic case and of allowing the laminations to be non compact. To study the analytic case, we use the formalism of deformations of complex structures. We present various persistent complex laminations which appear in dynamics of several complex variables: Henon maps, fibered holomorphic maps... In order to proof the persistence theorems, we construct a laminar structure on the stable and unstable of the normally hyperbolic laminations.