arXiv 1504.018352015-04-11math.DS

修正的Schmidt对策与部分双曲系统的非稠密前向轨道

Modified Schmidt games and non-dense forward orbits of partially hyperbolic systems

Weisheng Wu

备注：: 19 pages. Remark 4.10 is corrected. We have followed the proof scheme in \cite{Wu}. arXiv admin note: text overlap with arXiv:1311.5309

摘要：

设$f: M \to M$是一个$C^{1+θ}$ -部分双曲同态.引入了一类由$f$导出的在任意不稳定流形上进行的修正Schmidt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：对于某些$y \in M$为$$E(f, y) := \{ z\in M: y\notin \overline{\{f^k(z), k \in \mathbb{N}\}}\}$$，对于任何$x\in M$为$$E_{x}(f, y) := E(f, y) \cap W^u(x)$$。我们发现，$E_x(f,y)$是一个胜利的设置，这样修改施密特游戏上$W^u(x)$，这意味着$E_x(f,y)$的Hausdorff维数等于$\dim W^u(x)$。然后，对任意非空开集$V \subset M$，通过构造$E(f, y)$上的测度，证明了$E(f, y) \cap V$的全Hausdorff维数等于$\dim M$，且$E(f, y)$的低逐点维数逼近$\dim M$ .

英文摘要：

Let $f: M \to M$ be a $C^{1+θ}$-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 $\dim M$.

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