  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.
