# Finitely many physical measures for sectional-hyperbolic attracting sets and statistical stability

## TL;DR

This paper proves that sectional-hyperbolic attracting sets have finitely many physical measures with basins covering almost all points, and these measures depend continuously on the flow, establishing statistical stability.

## Contribution

It establishes the finiteness and statistical stability of physical measures for sectional-hyperbolic attracting sets in $C^1$ flows.

## Key findings

- Finitely many physical measures cover almost all points in the basin.
- Physical measures depend continuously on the flow in the $C^1$ topology.
- Central-unstable disks expand to contain uniformly sized balls.

## Abstract

We show that a sectional-hyperbolic attracting set for a H\"older-$C^1$ vector field admits finitely many physical/SRB measures whose ergodic basins cover Lebesgue almost all points of the basin of topological attraction. In addition, these physical measures depend continuously on the flow in the $C^1$ topology, that is, sectional-hyperbolic attracting sets are statistically stable. To prove these results we show that each central-unstable disk in a neighborhood of this class of attracting sets is eventually expanded to contain a ball whose inner radius is uniformly bounded away from zero.