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$M$ is a Riemannian, boundaryless, and compact manifold with $\dim M\geq 2$,and $f$ is a $C^{1+\beta}$ ($\beta>0$) diffeomorphism of $M$. $\varphi$ is aH\"older continuous potential on $M$. We construct an invariant and absolutelycontinuous family of measures (with transformation relations defined by$\varphi$), which sit on local unstable leaves. We present two mainapplications. First, given an ergodic homoclinic class $H_\chi(p)$, we provethat $\varphi$ admits a local equilibrium state on $H_\chi(p)$ if and only if$\varphi$ is "recurrent on $H_\chi(p)$" (a condition tested by countingperiodic points), and one of the leaf measures gives a positive measure to aset of positively recurrent hyperbolic points; and if an equilibrium measureexists, the said invariant and absolutely continuous family of measuresconstitutes as its conditional measures. An immediate corollary is the localproduct structure of hyperbolic equilibrium states.  Second, we prove a Ledrappier-Young property for hyperbolic equilibriumstates -- if $\varphi$ admits a conformal family of leaf measures, and ahyperbolic local equilibrium state, then the leaf measures of the invariantfamily (respective to $\varphi$) are equivalent to the conformal measures (on afull measure set). This extends the celebrated result by Ledrappier and Youngfor hyperbolic SRB measures, which states that a hyperbolic equilibrium stateof the geometric potential (with pressure 0) has conditional measures on localunstable leaves which are absolutely continuous w.r.t the Riemannian volume ofthese leaves.

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