Seminario de Sistemas Dinámicos de Santiago (April 12, 2021)

DAY / TIME April 12, 2021 / 4:30 PM – 5:30 PM (Santiago Time, GMT-4)
LOCATION Zoom meeting (ID 948 5787 9126, password required; ask for it at sistemasdinamicoscl@gmail.com)
SPEAKER Snir Ben Ovadia (Weizmann Institute of Science)
TITLE Invariant Family of Leaf measures and The Ledrappier-Young Property for Hyperbolic Equilibrium States
ABSTRACT 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.