A CRITERION FOR ERGODICITY AND NONUNIFORM HYPERBOLICITY FOR BILLIARDS

In this thesis we give an introduction to the study of mathematical billiards and proved a criterion using ergodic homoclinic class to establish ergodicity and nonuniform hyperbolicity for smooth maps with singularities, as a generalization of the result by F. Rodriguez Hertz, M. A. Rodriguez Hertz, A. Tahzibi and R. Ures.

Firstly, we review several concepts and theorems in ergodic theory. We survey the settings and results in the Pesin theory for smooth maps with singularities, especially the existence of Pesin blocks, the Stable Manifold Theorem and the absolute continuity of stable and unstable partitions. The original criterion, applied to diffeomorphisms on compact connected manifolds, is also introduced.

Secondly, we give an self-contained introduction to the general billiard theory, in particular the chaotic billiards. The basic definitions, facts and examples are surveyed, most of them with proofs. The most important classes of chaotic billiards, i.e., the Sinai billiards and Bunimovich billiards are discussed. Also we describe the class of billiards that the Pesin theory for smooth maps with singularities is applicable, i.e., the billiards of class $\Pi$.

Finally, the generalized criterion under the setting of smooth maps with singularities is proved with Hopf argument. As a corollary, we prove that in the hyperbolic case, the ergodic components in the Spectral Decomposition Theorem are exactly the ergodic homoclinic classes. This provided the possibility to apply the theory of ergodic homoclinic class to the study of billiards.
 

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