Abstract
We assume that a symplectic real-analytic map has an invariant normally hyperbolic cylinder and an associated transverse homoclinic cylinder. We prove that generically in the real-analytic category the boundaries of the invariant cylinder are connected by trajectories of the map.
Highlights
A Hamiltonian dynamical system is defined with the help of a Hamilton function H : M → R on a symplectic manifold M of dimension 2n
On the other hand a generic Hamiltonian system is nearly integrable in a neighbourhood of a totally elliptic equilibrium or totally elliptic periodic orbit
According to this shadowing lemma (Lemma 4), in order to show that two open sets are connected by a forward orbit of the map Φ, it is sufficient to show that the intersections of these sets with A are connected by orbits of the iterated function system {F0, . . . , FN }
Summary
A Hamiltonian dynamical system is defined with the help of a Hamilton function H : M → R on a symplectic manifold M of dimension 2n. The fact that the large cylinder A is an invariant domain for the area-preserving map F0 is crucial, as we use the Poincare Recurrence Theorem in an essential way (we first prove a certain weak shadowing result, Lemma 2, that holds without this assumption on the map F0, Lemma 4 is deduced from it in the case of area-preserving F0) According to this shadowing lemma (Lemma 4), in order to show that two open sets are connected by a forward orbit of the map Φ, it is sufficient to show that the intersections of these sets with A are connected by orbits of the iterated function system {F0, .
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