Abstract
We generalize the notion of Schmidt games to the setting of the general Caratheódory construction. The winning sets for such generalized Schmidt games usually have large corresponding Caratheódory dimensions (e.g., Hausdorff dimension and topological entropy). As an application, we show that for every $C^{1+\unicode[STIX]{x1D703}}$-partially hyperbolic diffeomorphism $f:M\rightarrow M$ satisfying certain technical conditions, the topological entropy of the set of points with non-dense forward orbits is bounded below by the unstable metric entropy (in the sense of Ledrappier–Young) of certain invariant measures. This also gives a unified proof of the fact that the topological entropy of such a set is equal to the topological entropy of $f$, when $f$ is a toral automorphism or the time-one map of a certain non-quasiunipotent homogeneous flow.
Publisher
Cambridge University Press (CUP)
Subject
Applied Mathematics,General Mathematics
Reference40 articles.
1. On orbits of endomorphisms of tori and the Schmidt game
2. [16] L. Guan and W. Wu . Bounded orbits of certain diagonalizable flows on $\text{SL}_{n}(\mathbb{R})/\text{SL}_{n}(\mathbb{Z})$ . Trans. Amer. Math. Soc., to appear.
3. Schmidt’s game, fractals, and orbits of toral endomorphisms
4. Bounded orbits of nonquasiunipotent flows on homogeneous spaces;Kleinbock;Amer. Math. Soc. Trans.,1996
5. Badly approximable systems of linear forms
Cited by
4 articles.
订阅此论文施引文献
订阅此论文施引文献，注册后可以免费订阅5篇论文的施引文献，订阅后可以查看论文全部施引文献