# Metric-measure boundary and geodesic flow on Alexandrov spaces

## TL;DR

This paper explores the relationship between geodesic flow and volume growth in Alexandrov spaces, establishing conditions for the existence and measure-preserving properties of geodesic flow using analytic and integral geometry tools.

## Contribution

It introduces a new analytic approach linking geodesic flow existence to volume growth, with implications for measure preservation in Alexandrov spaces.

## Key findings

- Geodesic flow exists and preserves Liouville measure in key cases.
- Volume growth of balls relates to the abundance of infinite geodesics.
- Analytic tools connect integral geometry with geodesic flow analysis.

## Abstract

We relate the existence of many infinite geodesics on Alexandrov spaces to a statement about the average growth of volumes of balls. We deduce that the geodesic flow exists and preserves the Liouville measure in several important cases. The developed analytic tool has close ties to integral geometry.