On the number of ergodic physical/SRB measures of singular-hyperbolic attracting sets

TL;DR

This paper establishes a sharp upper bound on the number of ergodic physical measures supported on connected singular-hyperbolic attracting sets for 3-dimensional flows, depending solely on the number of Lorenz-like equilibria.

Contribution

It provides the first explicit upper bound for ergodic physical measures in singular-hyperbolic sets, linking the bound to the number of Lorenz-like equilibria.

Findings

The bound depends only on the number of Lorenz-like equilibria.

Examples show the bound is sharp.

The result extends understanding of ergodic measures in hyperbolic dynamics.

Abstract

It is known that sectional-hyperbolic attracting sets, for a $C^2$ flow on a finite dimensional compact manifold, have at most finitely many ergodic physical invariant probability measures. We prove an upper bound for the number of distinct ergodic physical measures supported on a connected singular-hyperbolic attracting set for a $3$-flow. This bound depends only on the number of Lorenz-like equilibria contained in the attracting set. Examples of singular-hyperbolic attracting sets are provided showing that the bound is sharp.