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

Abstract

It is known that sectional-hyperbolic attracting sets, for a C2 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, as a consequence of the property that, for any physical measure whose support contains a singularity (which is necessarily a Lorenz-like singularity), almost every unstable manifold crosses the stable manifold of the singularity transversely. Examples of singular-hyperbolic attracting sets are provided showing that the bound is sharp.