Normal forms and Misiurewicz renormalization for dissipative surface diffeomorphisms

TL;DR

This paper introduces a hyperbolic renormalization framework for dissipative surface diffeomorphisms, enhancing existing methods to better understand Hénon-like map dynamics and parameter space geometry.

Contribution

It develops an improved renormalization and normal form technique specifically for small determinant maps, extending the Palis-Takens approach.

Findings

Provides uniform bounds for large periods in hyperbolic renormalization

Enhances understanding of Hénon-like map dynamics

Offers new tools for analyzing parameter space geometry

Abstract

We define a hyperbolic renormalizations suitable for maps of small determinant, with uniform bounds for large periods. The techniques involve an improvement of the celebrated Palis-Takens renormalization and normal forms (fibered linearizations). These techniques are useful to study the dynamics of H\'enon like maps and the geometry of their parameter space.