Unstable pressure and u-equilibrium states for partially hyperbolic diffeomorphsims

Abstract

Unstable pressure and u-equilibrium states are introduced and investigated for a partially hyperbolic diffeomorphsim $f$. We define the u-pressure $P^u(f,\phi )$ of $f$ at a continuous function $\phi$ via the dynamics of $f$ on local unstable leaves. A variational principle for unstable pressure $P^u(f,\phi )$, which states that $P^u(f,\phi )$ is the supremum of the sum of the unstable entropy and the integral of $\phi$ 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 Fréchet differentiability and their relations to u-equilibrium states, are also considered.

arXiv commentA draft version

References in corpus (1)

• Entropy along expanding foliations

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