Symbolic dynamics and Arnold diffusion
The author considers Graff tori of three degrees of freedom initially hyperbolic Hamiltonian systems [\textit{S. M. Graff}, J. Differ. Equations 15, 1-69 (1974; Zbl 0257.34048)] and extends the classical Birkhoff-Smale theorem for hyperbolic points or normally hyperbolic tori to partially hyperbolic Graff tori. The main result: if the stable and unstable manifold of a partially hyperbolic Graff tori intersect transversally, then there exists a hyperbolic invariant set near a homoclinic orbit on which the dynamics is conjugated to a Bernoulli shift. The existence of infinitely many hyperbolic periodic orbits near the given torus is deduced. Then the relevance of these results for the instability of near-integrable Hamiltonian systems is discussed.
scientific article; zbMATH DE number 1884222
scientific article