Simultaneous dense and non-dense orbits for toral diffeomorphisms. (28th January 2016) - TSENG, JIMMY
Abstract : We show that, for pairs of hyperbolic toral automorphisms on the 2 -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 C^2 -Anosov diffeomorphisms on the 2 -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.