Simultaneous dense and nondense orbits and the space of lattices

TL;DR

This paper demonstrates that certain sets of points are both nondense under specific dynamical systems and dense under others, having full Hausdorff dimension, with implications in number theory and dynamics.

Contribution

It establishes that sets of points nondense under the $ imes n$-map and dense under geodesic flow or toral automorphisms have full Hausdorff dimension, extending classical results.

Findings

Sets of well-approximable numbers nondense under $ imes n$-map have full Hausdorff dimension.

Analogous results hold for well-approximable 2-vectors under hyperbolic toral automorphisms.

The results unify and extend classical theorems in number theory and dynamical systems.

Abstract

We show that set of points nondense under the $\times n$-map on the circle and dense for the geodesic flow under the induced map on the circle corresponding to the expanding horospherical subgroup 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 $\times n$-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 and gives a comprehensive understanding.