Simultaneous dense and nondense orbits for toral diffeomorphisms

Abstract

We show that, for pairs of hyperbolic toral automorphisms on the $2$-torus, the points with dense forward orbits under one map and nondense forward orbits under the other is a dense, uncountable set. The pair of maps can be noncommuting. 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.