Unstable pressure and u-equilibrium states for partially hyperbolic diffeomorphisms

Unstable pressure and u-equilibrium states are introduced and investigated for a partially hyperbolic diffeomorphism f. We define the unstable 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 Fréchet differentiability and their relations to u-equilibrium states, are also considered.

On Fixed Point Property under Lipschitz and Uniform Embeddings

We first present a generalization of ω⁎-Gâteaux differentiability theorems of Lipschitz mappings from open sets to those closed convex sets admitting nonsupport points and then show that every nonempty bounded closed convex subset of a Banach space has the fixed point property for isometries if it Lipschitz embeds into a super reflexive space. With the application of Baudier-Lancien-Schlumprecht’s theorem, we finally show that every nonempty bounded closed convex subset of a Banach space has the fixed point property for continuous affine mappings if it uniformly embeds into the Tsirelson space T⁎.