Simultaneous dense and non-dense orbits for toral diffeomorphisms
Ergodic Theory and Dynamical Systems
Cambridge University Press
© Cambridge University Press, 2016
We show that, for pairs of hyperbolic toral automorphisms on the -torus, the points with dense forward orbits under one map and non-dense forward orbits under the other is a dense, uncountable set. The pair of maps can be non-commuting. We also show the same for pairs of -Anosov diffeomorphisms on the -torus. (The pairs must satisfy slight constraints.) Our main tools are the Baire category theorem and a geometric construction that allows us to give a geometric characterization of the fractal that is the set of points with forward orbits that miss a certain open set.
This is the author accepted manuscript. The final version is available from Cambridge University Press via the DOI in this record.
Vol. 37, Iss. 4, pp. 1308 - 1322