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: $E(f,y):=\{z\in M:y\mathrm{\notin }\overline{\{f^k(z),k\in \mathbb{N}\}}\}$ 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 $E(f, y)\cap V$withlowerpointwisedimensionconvergingto$\dim M$ 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 $M$.