Simultaneous Dense and Nondense Orbits and the Space of Lattices

Abstract

We show that the set of points nondense under the ×n -map on the circle and dense for the geodesic flow after we identify the circle with a periodic horospherical orbit of the modular surface has full Haudorff dimension. We also show the analogous result for toral automorphisms on the 2 -torus and a diagonal flow. Our results can be interpreted in number-theoretic terms: the set of well-approximable numbers that are nondense under the ×n -map has full Hausdorff dimension. Similarly, the set of well-approximable 2 -vectors that are nondense under a hyperbolic toral automorphism has full Hausdorff dimension. Our result for numbers is the counterpart to a classical result of Kaufmann.