Unstable pressure and equilibrium states of diffeomorphisms along invariant laminations

Let $ f: M\to M $ be a $ C^{1+r} (r>0) $ diffeomorphism on a closed smooth manifold $ M $. Given an $ f $-invariant lamination $ {{\mathcal L}}^d $ on $ M $, we define the unstable topological pressure $ P^u(f, \varphi, {{\mathcal L}}^d) $ of $ f $ at a continuous potential function $ \varphi $ via the dynamics of $ f $ on local unstable leaves inside the lamination $ {{\mathcal L}}^d $. A variational principle for unstable topological pressure, which states that $ P^u(f, \varphi, {{\mathcal L}}^d) $ is the supremum of the sum of the unstable metric entropy along the lamination and the integral of $ \varphi $ taken over all invariant measures, is obtained. Basic properties of u-equilibrium states at which the supremum in the variational principle is attained and their relation to Sinai-Ruelle-Bowen measures are studied. We also present examples for which the unstable pressure and variational principle may have further applications.