Modified Schmidt games and non-dense forward orbits of partially hyperbolic systems
Modified Schmidt games and non-dense forward orbits of partially hyperbolic systems
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DOI：
10.3934/dcds.2016.36.3463
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发表时间：
2015-04
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影响因子：
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通讯作者：
Weisheng Wu
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文献类型：
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作者：
Weisheng Wu
设f：M \to M$是C^{1+\theta}$-部分双曲同态.引入了一类由f$诱导的在任意不稳定流形上进行的修正的施密特对策。利用它我们推广了\cite{Wu}的一些结果如下。考虑一组具有非稠密前向轨道的点：$$E（f，y）：= \{ z\in M：y\notin \overline{\{f^k（z），k \in \mathbb{N}\}}$$对于M$中的某个$y \，$$E_{x}（f，y）：= E（f，y）\cap W^u（x）$$对于M$中的任何$x\。我们证明了$E_x（f，y）$是在$W^u（x）$上进行的这种修改的施密特对策的获胜集，这意味着$E_x（f，y）$的Hausdorff维数等于$\dim W^u（x）$。然后对任何非空开集V \子集M$，我们证明了E（f，y）\cap V$的全Hausdorff维数等于$\dim M$，通过构造在E（f，y）$上支持的测度，使其较低的点态维数近似于$\dim M$。
Let $f: M \to M$ be a $C^{1+\theta}$-partially hyperbolic diffeomorphism. We introduce a type of modified Schmidt games which is induced by $f$ and played on any unstable manifold. Utilizing it we generalize some results of \cite{Wu} as follows. Consider a set of points with non-dense forward orbit: $$E(f, y) := \{ z\in M: y\notin \overline{\{f^k(z), k \in \mathbb{N}\}}\}$$ for some $y \in M$ and $$E_{x}(f, y) := E(f, y) \cap W^u(x)$$ for any $x\in M$. We show that $E_x(f,y)$ is a winning set for such modified Schmidt games played on $W^u(x)$, which implies that $E_x(f,y)$ has Hausdorff dimension equal to $\dim W^u(x)$. Then for any nonempty open set $V \subset M$ we show that $E(f, y) \cap V$ has full Hausdorff dimension equal to $\dim M$, by using a technique of constructing measures supported on $E(f, y)$ with lower pointwise dimension approximating $\dim M$.