Quantitative Estimates on Invariant Manifolds for Surface Diffeomorphisms
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866929599113330688 |
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| author | Crovisier, Sylvain Lyubich, Mikhail Pujals, Enrique Yang, Jonguk |
| author_facet | Crovisier, Sylvain Lyubich, Mikhail Pujals, Enrique Yang, Jonguk |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_13286 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quantitative Estimates on Invariant Manifolds for Surface Diffeomorphisms Crovisier, Sylvain Lyubich, Mikhail Pujals, Enrique Yang, Jonguk Dynamical Systems 37C05 (primary) 37E30 (secondary) 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. |
| title | Quantitative Estimates on Invariant Manifolds for Surface Diffeomorphisms |
| topic | Dynamical Systems 37C05 (primary) 37E30 (secondary) |
| url | https://arxiv.org/abs/2411.13286 |