Unstable pressure and u-equilibrium states for partially hyperbolic diffeomorphsims

Unstable pressure and u-equilibrium states are introduced and investigated for a partially hyperbolic diffeomorphsim $f$. We define the u-pressure $P^u(f, varphi)$ of $f$ at a continuous function $varphi$ via the dynamics of $f$ on local unstable leaves. A variational principle for unstable pressure $P^u(f, varphi)$, which states that $P^u(f, varphi)$ is the supremum of the sum of the unstable entropy and the integral of $varphi$ taken over all invariant measures, is obtained. U-equilibrium states at which the supremum in the variational principle attains and their relation to Gibbs u-states are studied. Differentiability properties of unstable pressure, such as tangent functionals, Gateaux differentiability and Fr{e}chet differentiability and their relations to u-equilibrium states, are also considered.