Schmidt games and Markov partitions
IOP Publishing / London Mathematical Society
© 2009 IOP Publishing Ltd and London Mathematical Society
Let T be a C2-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 x0 in M, the set of points whose forward orbit closures miss x0 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.
This is the author accepted manuscript. The final version is available from IOP Publishing via the DOI in this record.
Vol. 22, pp. 525 - 543