arXiv subjects

Weisheng Wu

Publications and source records attributed to Weisheng Wu.

22 records · Page 2Linked to original sources

arXiv2015

On non-dense orbits of certain non-algebraic dynamical systems

In this paper, we manage to apply Schmidt games to certain non-algebraic dynamical systems. More precisely, we show that the set of points with non-dense forward orbit under a $C^2$-Anosov diffeomorphism with conformality on unstable manifolds is a winning set for Schmidt games. It is also proved that for a $C^{1+θ}$-expanding endomorphism the set of points with non-dense forward orbit is a winning set for certain variants of Schmidt games.

math.DS↗

arXiv2015

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

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 $\dim M$.

math.DS↗

arXiv2015

On the ergodicity of geodesic flows on surfaces of nonpositive curvature

Let $M$ be a smooth compact surface of nonpositive curvature, with genus $\geq 2$. We prove the ergodicity of the geodesic flow on the unit tangent bundle of $M$ with respect to the Liouville measure under the condition that the set of points with negative curvature on $M$ has finitely many connected components. Under the same condition, we prove that a non closed "flat" geodesic doesn't exist, and moreover, there are at most finitely many flat strips, and at most finitely many isolated closed "flat" geodesics.

math.DS↗

arXiv2013

Schmidt Games and Nondense forward Orbits of certain Partially Hyperbolic Systems

Let $f: M \to M$ be a partially hyperbolic diffeomorphism with conformality on unstable manifolds. Consider a set of points with nondense 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-Kleinbock, we show that $E_x(f,y)$ is a winning set of Schmidt games played on $W^u(x)$ which implies that $E_x(f,y)$ has full Hausdorff dimension equal to $\dim W^u(x)$. Furthermore we show that for any nonempty 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 $M$.

math.DS↗