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

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$M$.

DETERMINACY AND INDETERMINACY OF GAMES PLAYED ON COMPLETE METRIC SPACES

Schmidt’s game is a powerful tool for studying properties of certain sets which arise in Diophantine approximation theory, number theory and dynamics. Recently, many new results have been proven using this game. In this paper we address determinacy and indeterminacy questions regarding Schmidt’s game and its variations, as well as more general games played on complete metric spaces (for example, fractals). We show that, except for certain exceptional cases, these games are undetermined on certain sets. Judging by the vast numbers of papers utilising these games, we believe that the results in this paper will be of interest to a large audience of number theorists as well as set theorists and logicians.

Schmidt’s game, fractals, and orbits of toral endomorphisms

Given an integer matrix M∈GLn(ℝ) and a point y∈ℝn/ℤn, consider the set S. G. Dani showed in 1988 that whenever M is semisimple and y∈ℚn/ℤn, the set $ \tilde E(M,y)$ has full Hausdorff dimension. In this paper we strengthen this result, extending it to arbitrary M∈GLn(ℝ)∩Mn×n(ℤ) and y∈ℝn/ℤn, and in fact replacing the sequence of powers of M by any lacunary sequence of (not necessarily integer) m×n matrices. Furthermore, we show that sets of the form $ \tilde E(M,y)$ and their generalizations always intersect with ‘sufficiently regular’ fractal subsets of ℝn. As an application, we give an alternative proof of a recent result [M. Einsiedler and J. Tseng. Badly approximable systems of affine forms, fractals, and Schmidt games. Preprint, arXiv:0912.2445] on badly approximable systems of affine forms.