Hausdorff dimension of divergent trajectories on homogeneous space

TL;DR

This paper proves that the set of points with divergent on average trajectories in a homogeneous space has Hausdorff dimension strictly less than the space's dimension, confirming a conjecture by Y. Cheung.

Contribution

It establishes a Hausdorff dimension bound for divergent on average trajectories, advancing understanding of dynamical behavior on homogeneous spaces.

Findings

Hausdorff dimension of divergent on average set is less than space dimension

Confirms Y. Cheung's conjecture on divergence sets

Provides new bounds on the size of divergent trajectories

Abstract

For one parameter subgroup action on a finite volume homogeneous space, we consider the set of points admitting divergent on average trajectories. We show that the Hausdorff dimension of this set is strictly less than the manifold dimension of the homogeneous space. As a corollary we know that the Hausdorff dimension of the set of points admitting divergent trajectories is not full, which proves a conjecture of Y. Cheung.