Simultaneous dense and non-dense orbits for toral diffeomorphisms

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.