An inclination lemma for normally hyperbolic manifolds with an application to diffusion
| Autor: | Lara Sabbagh |
|---|---|
| Rok vydání: | 2014 |
| Předmět: |
Pure mathematics
Applied Mathematics General Mathematics Hyperbolic manifold Dynamical Systems (math.DS) Submanifold Stable manifold Manifold FOS: Mathematics Diffeomorphism Mathematics - Dynamical Systems Invariant (mathematics) Mathematics::Symplectic Geometry Symplectic manifold Symplectic geometry Mathematics |
| Zdroj: | Ergodic Theory and Dynamical Systems. 35:2269-2291 |
| ISSN: | 1469-4417 0143-3857 |
| Popis: | Let ($M$, ${\rm\Omega}$) be a smooth symplectic manifold and $f:M\rightarrow M$ be a symplectic diffeomorphism of class $C^{l}$ ($l\geq 3$). Let $N$ be a compact submanifold of $M$ which is boundaryless and normally hyperbolic for $f$. We suppose that $N$ is controllable and that its stable and unstable bundles are trivial. We consider a $C^{1}$-submanifold ${\rm\Delta}$ of $M$ whose dimension is equal to the dimension of a fiber of the unstable bundle of $T_{N}M$. We suppose that ${\rm\Delta}$ transversely intersects the stable manifold of $N$. Then, we prove that for all ${\it\varepsilon}>0$, and for $n\in \mathbb{N}$ large enough, there exists $x_{n}\in N$ such that $f^{n}({\rm\Delta})$ is ${\it\varepsilon}$-close, in the $C^{1}$ topology, to the strongly unstable manifold of $x_{n}$. As an application of this ${\it\lambda}$-lemma, we prove the existence of shadowing orbits for a finite family of invariant minimal sets (for which we do not assume any regularity) contained in a normally hyperbolic manifold and having heteroclinic connections. As a particular case, we recover classical results on the existence of diffusion orbits (Arnold’s example). |
| Databáze: | OpenAIRE |
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