Schmidt games and Markov partitions

Let T be a C ²-expanding self-map of a compact, connected, C ^∞, Riemannian manifold M. We correct a minor gap in the proof of a theorem from the literature: the set of points whose forward orbits are nondense has full Hausdorff dimension. Our correction allows us to strengthen the theorem. Combining the correction with Schmidt games, we generalize the theorem in dimension one: given a point x ₀ ∈ M, the set of points whose forward orbit closures miss x ₀ is a winning set. Finally, our key lemma, the no matching lemma, may be of independent interest in the theory of symbolic dynamics or the theory of Markov partitions.