# A note on invariant measures for Filippov systems

## TL;DR

This paper investigates invariant measures for Filippov systems on compact manifolds, focusing on those preserving volume and exploring the relationship between invariant measures and trajectory uniqueness breaks.

## Contribution

It defines invariant measures for Filippov systems, proves a conjecture relating invariant measures to trajectory uniqueness, and analyzes measure existence in various examples.

## Key findings

- Volume-preserving Filippov systems are refractive and piecewise volume preserving.
- Invariant measures do not see trajectories with non-uniqueness breaks, proven for Lipschitz differential inclusions.
- Analysis of invariant measures in multiple Filippov system examples on compact manifolds.

## Abstract

We are interested in Filippov systems which preserve a probability measure on a compact manifold. We define a measure to be invariant for a Filippov system as the natural analogous definition of invariant measure for flows. Our main result concerns Filippov systems which preserve a probability measure equivalent to the volume measure. As a consequence, the volume preserving Filippov systems are the refractive piecewise volume preserving ones. We conjecture that if a Filippov system admits an invariant probability measure, this measure does not see the trajectories where there is a break of uniqueness. We prove this conjecture for Lipschitz differential inclusions. Then, in light of our previous results, we analyze the existence of invariant measures for many examples of Filippov systems defined on compact manifolds.