Nondense orbits on homogeneous spaces and applications to geometry and number theory

TL;DR

This paper establishes conditions under which points with trajectories avoiding certain submanifolds in homogeneous spaces form a large, stable set, with applications to geometry and number theory.

Contribution

It introduces new criteria for hyperplane absolute winning sets in homogeneous spaces, linking dynamics to geometric and number-theoretic properties.

Findings

Set of points avoiding submanifolds is hyperplane absolute winning.

Results apply to constructing exceptional geodesics.

Shows non-density of certain function values at integers.

Abstract

Let $G$ be a Lie group, $\Gamma\subset G$ a discrete subgroup, $X=G/\Gamma$, and $f$ an affine map from $X$ to itself. We give conditions on a submanifold $Z$ of $X$ guaranteeing that the set of points $x\in X$ with $f$-trajectories avoiding $Z$ is hyperplane absolute winning (a property which implies full Hausdorff dimension and is stable under countable intersections). A similar result is proved for one-parameter actions on $X$. This has applications to constructing exceptional geodesics on locally symmetric spaces, and to non-density of the set of values of certain functions at integer points.