Exceptional sets in homogeneous spaces and Hausdorff dimension

TL;DR

This paper investigates the Hausdorff dimension of certain dynamical sets in homogeneous spaces, showing it is bounded above by a function of the ball radius, and introduces a method for estimating open cover cardinalities.

Contribution

It provides new bounds on the Hausdorff dimension of sets in homogeneous spaces and develops a general volume-based method for open cover estimation in dynamical systems.

Findings

Hausdorff dimension of sets missing a ball is bounded by \\dim X + C r^{\\dim X}/\\log r

Exponential mixing results are used to derive dimension bounds

A volume estimate method for open covers of dynamical sets is introduced

Abstract

In this paper we study the dimension of a family of sets arising in open dynamics. We use exponential mixing results for diagonalizable flows in compact homogeneous spaces $X$ to show that the Hausdorff dimension of set of points that lie on trajectories missing a particular open ball of radius $r$ is at most $$\dim X + C\frac{r^{\dim X}}{\log r},$$ where $C>0$ is a constant independent of $r>0$. Meanwhile, we also describe a general method for computing the least cardinality of open covers of dynamical sets using volume estimates.