Attracting measures

Abstract Under mild assumptions, the SRB measure μ associated to an Axiom A attractor A has the following properties: (i) the empirical measure starting at a typical point near A converges weakly to μ ; (ii) the pushforward of any Lebesgue-absolutely continuous probability measure supported near A converges weakly to μ . In general, a measure with the first property is called a ‘physical measure’, and physical measures are recognised by many authors as being important in their own right. Comparatively little has been written about the second property, but we highlight that this is also important in its own right as it characterises the attraction of general absolutely continuous measures to a measure μ —in such a case, we say μ is an ‘attracting measure’. Attracting measures represent a kind of decay of correlation of observables that is more accessible than classical mixing when the measure is singular with respect to the Lebesgue measure. We prove a result that serves as a topological abstraction of the original result establishing that mixing SRB measures on Axiom A attractors are attracting measures.