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

Abstract

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$M$.