  $M$ 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 a
H\"older 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. 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 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.
