Invariant family of leaf measures and the Ledrappier–Young property for hyperbolic equilibrium states


 The manifold M is a Riemannian, boundaryless, and compact manifold with 
 
 
 
$\dim M\geq 2$

 
 , and f is a 
 
 
 
$C^{1+\beta }$

 
 (
 
 
 
$\kern0.3pt\beta>0$

 
 ) diffeomorphism of M. 
 
 
 
$\varphi $

 
 is 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 constitute 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 
 
 
 
$\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 Sinai–Ruelle–Bowen measures, which states that a hyperbolic equilibrium state of the geometric potential (with pressure zero) has conditional measures on local unstable leaves which are absolutely continuous with respect to the Riemannian volume of these leaves.