2022/03/07 by Snir Ben Ovadia, Ovadia, Snir Ben
Mathematics · #Mathematical Dynamics and Fractals #Geometric Analysis and Curvature Flows
paper · pdf · doi:10.48550/arxiv.2203.03492
M is a Riemannian, boundaryless, and compact manifold with dim M≥ 2, and f is a C1+β (β>0) diffeomorphism of M. φ is a Hölder continuous potential on M. We construct an invariant and absolutely continuous family of measures (with transformation relations defined by φ), which sit on local unstable leaves. We present two main applications. First, given an ergodic homoclinic class Hχ(p), we prove that φ admits a local equilibrium state on Hχ(p) if and only if φ is "recurrent on Hχ(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. An immediate corollary is the local product structure of hyperbolic equilibrium states. Second, we prove a Ledrappier-Young property for hyperbolic equilibrium states -- if φ admits a conformal family of leaf measures, and a hyperbolic local equilibrium state, then the leaf measures of the invariant family (respective to φ) 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.