2015/04/08 by Weisheng Wu, Wu, Weisheng · 1 citation
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems #math.DS
paper · pdf · doi:10.48550/arxiv.1504.01835
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
openalex publication_date 2015/04/08 · arxiv created 2015/04/11 · arxiv updated 2015/04/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let f: M → M be a C1+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 \citeWu as follows. Consider a set of points with non-dense forward orbit: E(f, y) := \ z∈ M: y∉ \fk(z), k ∈ ℕ\\ for some y ∈ M and Ex(f, y) := E(f, y) ∩ Wu(x) for any x∈ M. We show that Ex(f,y) is a winning set for such modified Schmidt games played on Wu(x), which implies that Ex(f,y) has Hausdorff dimension equal to dim Wu(x). Then for any nonempty open set V ⊂ M we show that E(f, y) ∩ 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.