2024/11/20 by Sylvain Crovisier, Crovisier, Sylvain, Mikhail Lyubich +5
Mathematics · Physics and Astronomy · #37C05 (primary) 37E30 (secondary) #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.2411.13286
openalex publication_date 2024/11/20 · openalex created_date 2024/11/24 · openalex updated_date 2026/07/28
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 Hénon-like maps developed in the sequel papers.