Invariant Family of Leaf measures and The Ledrappier-Young Property for Hyperbolic Equilibrium States.

ABSTRACT: Let be a Riemannian, boundaryless, and compact manifold with , let be a () diffeomorphism of , and let be a Hölder continuous potential on . We construct an invariant and absolutely continuous family of measures (with transformation relations defined by ), which sit on local unstable leaves.

We present two main applications. First, given an ergodic homoclinic class , we prove that admits a local equilibrium state on if and only if is “recurrent on ” (a condition tested by counting periodic points), and one of the leaf measures gives a positive measure to a set of positively recurrent hyperbolic points; and if an equilibrium measure exists, the said invariant and absolutely continuous family of measures constitutes as its conditional measures. Second, we prove a Ledrappier-Young property for hyperbolic equilibrium states- if admits a conformal family of leaf measures, and a hyperbolic local equilibrium state, then the  leaf measures of the invariant family (respective to ) are equivalent to the conformal measures (on a full measure set). This extends the celebrated result by Ledrappier and Young for hyperbolic SRB measures, which states that a hyperbolic equilibrium state of the geometric potential (with pressure ) has conditional measures on local unstable leaves which are absolutely continuous w.r.t the Riemannian volume of these leaves.

Posted on Apr 8, 2021 in Dynamical Systems, Seminars