Averaging of Hamiltonian Flows with an Ergodic Component
نویسندگان
چکیده
We consider a process on T2, which consists of the fast motion along the stream lines of an incompressible periodic vector field perturbed by white noise. It gives rise to a process on the graph naturally associated to the structure of the stream lines of the unperturbed flow. It has been shown by Freidlin and Wentzell that if the stream function of the flow is periodic, then the corresponding process on the graph weakly converges to a Markov process. We consider the situation when the stream function is not periodic, and the flow (when considered on the torus) has an ergodic component of positive measure. We show that if the rotation number is Diophantine then the process on the graph still converges to a Markov process, which spends a positive proportion of time in the vertex corresponding to the ergodic component of the flow.
منابع مشابه
Averaging of Hamiltonian Flows with an Ergodic Component by Dmitry
We consider a process on T2, which consists of fast motion along the stream lines of an incompressible periodic vector field perturbed by white noise. It gives rise to a process on the graph naturally associated to the structure of the stream lines of the unperturbed flow. It has been shown by Freidlin and Wentzell [Random Perturbations of Dynamical Systems, 2nd ed. Springer, New York (1998)] a...
متن کاملElliptic islands appearing in near-ergodic flows
It is proved that periodic and homoclinic trajectories which are tangent to the boundary of any scattering (ergodic) billiard produce elliptic islands in the ‘nearby’ Hamiltonian flows i.e. in a family of two-degrees-of-freedom smooth Hamiltonian flows which converge to the singular billiard flow smoothly where the billiard flow is smooth and continuously where it is continuous. Such Hamiltonia...
متن کاملAsymptotic Behaviour of Ergodic Integrals of ‘Renormalizable’ Parabolic Flows
Ten years ago A. Zorich discovered, by computer experiments on interval exchange transformations, some striking new power laws for the ergodic integrals of generic non-exact Hamiltonian flows on higher genus surfaces. In Zorich’s later work and in a joint paper authored by M. Kontsevich, Zorich and Kontsevich were able to explain conjecturally most of Zorich’s discoveries by relating them to th...
متن کاملLimited to Ergodic Bil l iards
Abs t rac t , Sufficient conditions are found so that a family of smooth Hamiltonian flows limits to a billiard flow as a parameter e --~ 0. This limit is proved to be C 1 near non-singular orbits and C o near orbits tangent to the billiard boundary. These results are used to prove that scattering (thus ergodic) billiards with tangent periodic orbits or tangent homoclinic orbits produce nearby ...
متن کاملOn Smooth Hamiltonian Flows Limiting to Ergodic Billiards
Suucient conditions are found so that a family of smooth Hamiltonian ows limits to a billiard ow as a parameter ! 0. This limit is proved to be C 1 near non-singular orbits and C 0 near orbits tangent to the billiard boundary. These results are used to prove that scattering (thus ergodic) billiards with tangent periodic orbits or tangent homoclinic orbits produce nearby Hamiltonian ows with ell...
متن کامل