Quantitative Estimates on Invariant Manifolds for Surface Diffeomorphisms

TL;DR

This paper provides a detailed quantitative analysis of the geometry of invariant manifolds in two-dimensional dissipative systems, establishing explicit bounds and regularity properties that support renormalization theory for Hénon-like maps.

Contribution

It introduces explicit inequalities for orbit regularity, relates local dynamics to quasi-linearization, and proves the existence of smooth invariant manifolds with uniform bounds.

Findings

Explicit inequalities for orbit regularity

Bounded geometries for stable and center manifolds

Foundation for renormalization theory of Hénon-like maps

Abstract

We carry out a detailed quantitative analysis on the geometry of invariant manifolds for smooth dissipative systems in dimension two. We begin by quantifying the regularity of any orbit (finite or infinite) in the phase space with a set of explicit inequalities. Then we relate this directly to the quasi-linearization of the local dynamics on regular neighborhoods of this orbit. The parameters of regularity explicitly determine the sizes of the regular neighborhoods and the smooth norms of the corresponding regular charts. As a corollary, we establish the existence of smooth stable and center manifolds with uniformly bounded geometries for regular orbits independently of any pre-existing invariant measure. This provides us with the technical background for the renormalization theory of H\'enon-like maps developed in the sequel papers.