Shrinking targets in fast mixing flows and the geodesic flow on negatively curved manifolds

TL;DR

This paper investigates the relationship between hitting times and measure dimensions in rapidly mixing flows, applying the findings to geodesic flows on negatively curved manifolds to establish a logarithm law.

Contribution

It introduces a connection between hitting times and measure dimensions in fast mixing flows and applies this to geodesic flows on negatively curved manifolds to prove a logarithm law.

Findings

Hitting times relate to measure dimensions in rapid mixing flows.

Established a logarithm law for geodesic flows on negatively curved manifolds.

Demonstrated the applicability of the theoretical results to specific geometric flows.

Abstract

We show that in a rapidly mixing flow with an invariant measure, the time which is needed to hit a given section is related to a sort of conditional dimension of the measure at the section. The result is applied to the geodesic flow of compact variable negative sectional curvature manifolds, establishing a logarithm law for such kind of flow.