April 12, 2021 / 4:30 PM – 5:30 PM (Santiago Time, GMT-4)

     Zoom meeting (ID 948 5787 9126, password required; ask for it at sistemasdinamicoscl@gmail.com)

    Snir Ben Ovadia (Weizmann Institute of Science)

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

    Let M be a Riemannian, boundaryless, and compact manifold with \dim M\geq 2, and let f be a C^{1+\beta} (\beta>0) diffeomorphism of M. Let \varphi be a Hölder continuous potential on M. We construct an invariant and absolutely continuous family of measures (with transformation relations defined by \varphi), which sit on local unstable leaves.

    We present two main applications. First, given an ergodic homoclinic class H_\chi(p), we prove that \varphi admits a local equilibrium state on H_\chi(p) if and only if \varphi is «recurrent on H_\chi(p)» (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 \varphi admits a conformal family of leaf measures, and a hyperbolic local equilibrium state, then the  leaf measures of the invariant family (respective to \varphi) 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 0) has conditional measures on local unstable leaves which are absolutely continuous w.r.t the Riemannian volume of these leaves.