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 $M$.

We prove that for typical rotation numbers $0<{\it\theta}<1$, the boundary of the Siegel disk of $f_{{\it\theta}}(z)=e^{2{\it\pi}i{\it\theta}}\sin (z)$ centered at the origin is a Jordan curve which passes through exactly two critical points ${\it\pi}/2$ and $-{\it\pi}/2$.