Exponential volume limits

TL;DR

This paper proves that on a closed Riemannian manifold, measures obtained from exponential volume decay under a diffeomorphism are SRB measures, linking volume decay rates to statistical properties of dynamical systems.

Contribution

It establishes a new criterion connecting exponential volume decay to the existence of SRB measures in smooth dynamical systems.

Findings

Exponential volume decay implies the measure is SRB.

The result applies to $C^{1+eta}$ diffeomorphisms on closed manifolds.

Provides a new method to identify SRB measures via volume decay rates.

Abstract

Let $M$ be a $d$-dimensional closed Riemannian manifold, let $f\in\mathrm{Diff}^{1+\beta}(M)$, and denote by $m$ the Riemannian volume form of $M$. We prove that if $m\circ f^{-n}\xrightarrow[n\to\infty]{}\mu$ exponentially fast, then $\mu$ is an SRB measure.