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Schmidt games and non-dense forward orbits of certain partially hyperbolic systems
Published online by Cambridge University Press: 11 February 2015
Abstract
Let $f:M\rightarrow M$ be a partially hyperbolic diffeomorphism with conformality on unstable manifolds. 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
$y\in M$. Define
$E_{x}(f,y):=E(f,y)\cap W^{u}(x)$ for any
$x\in M$. Following a method of Broderick, Fishman and Kleinbock [Schmidt’s game, fractals, and orbits of toral endomorphisms. Ergod. Th. & Dynam. Sys.31 (2011), 1095–1107], we show that
$E_{x}(f,y)$ is a winning set for Schmidt games played on
$W^{u}(x)$ which implies that
$E_{x}(f,y)$ has Hausdorff dimension equal to
$\dim W^{u}(x)$. Furthermore, we show that for any non-empty open set
$V\subset M$,
$E(f,y)\cap V$ has full Hausdorff dimension equal to
$\dim M$, by constructing measures supported on
$E(f,y)\cap V$ with lower pointwise dimension converging to
$\dim M$ and with conditional measures supported on
$E_{x}(f,y)\cap V$. The results can be extended to the set of points with forward orbit staying away from a countable subset of
$M$.
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- Research Article
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- © Cambridge University Press, 2015
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