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Schmidt games and non-dense forward orbits of certain partially hyperbolic systems. (English) Zbl 1367.37035

Summary: Let \(f:M\to M\) 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 R. Broderick et al. [Ergodic Theory Dyn. Syst. 31, No. 4, 1095–1107 (2011; Zbl 1242.37018)], 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 \(M\).

MSC:

37D30 Partially hyperbolic systems and dominated splittings
37C45 Dimension theory of smooth dynamical systems
37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
91A44 Games involving topology, set theory, or logic
91A80 Applications of game theory

Citations:

Zbl 1242.37018
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References:

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