99-53 Sza'sz D.
Ball-Avoiding Theorems (98K, amstex) Feb 16, 99
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Abstract. Consider a nice hyperbolic dynamical system (singularities not excluded). Statements about the topological smallness of the subset of orbits, which avoid an open subset of the phase space --- for every moment of time, or just for a not too small subset of times --- play a key role in showing hyperbolicity or ergodicity of semi-dispersive billiards, in particular, of hard ball systems. Beside surveying the characteristic results, called ball-avoiding theorems, and giving an idea of the methods of their proofs, their applications are also illustrated. Further we also discuss analogous questions (which had arisen, for intance, in number theory), when Hausdorff dimension is taken instead of the topological one. The answers strongly depend on the notion of dimension which is used. Finally, ball avoiding subsets are naturally related to repellers extensively studied by physicists. For the interested reader we also sketch some analytic and rigorous results about repellers and escape times.

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