A Mechanism of Ergodicity in Standard-Like Maps

Abstract

The main statement of KAM-theory says that small perturbations of integrable non-degenerate Hamiltonian flows with compact phase spaces have subsets of large relative measure filled with invariant tori carrying quasi-periodic motions. One of the most important problems in the theory of Hamiltonian chaos is to understand the structure of dynamics on the remaining sets. The hypothesis which naturally came out of the entropy theory around thirty years ago says that these sets generically consist of ergodic components of positive measure with non-zero Lyapunov exponents. In particular, A. N. Kolmogorov formulated this hypothesis stressing at the same time that quasi periodic flows on invariant tori and unstable flows with non-zero Lyapunov exponents exhaust all possible stable types of dynamics.