Unstable pressure and u-equilibrium states for partially hyperbolic diffeomorphisms

Abstract

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.

Unstable Topological Pressure for Partially Hyperbolic Diffeomorphisms with Sub-additive Potentials

“…Remark 2.3. With a quite similar argument as the proof of Lemma 2.1 in [8], we can show that the unstable topological pressure for subadditive potentials we defined above is independent of the choice of δ.…”

“…A fundamental result is due to Hu, Hua, and Wu ( [7]), they introduced the so called unstable topological and unstable metric entropy, obtained the corresponding Shannon-McMillan-Breiman theorem and variational principle. Along this line, Hu, Wu, and Zhu investigated the unstable topological pressure for additive potentials in [8].…”

Unstable Metric Pressure of Partially Hyperbolic Diffeomorphisms with sub-additive Potentials

“…Proof. Note that the distance d u on the unstable manifold is equivalent to the Riemannian metric d (see the observation in front of Proposition 2.4 of [9] ), so any unstable local neighborhood W u (x, δ) is compact under d u . Then one can get the desired result using a similar argument of Lemma 2.2 of [7].…”

“…In [15], Tian and Wu generalize the above result with additional consideration of an arbitrary subset (not necessarily compact or invariant). In [9], Hu, Wu, and Zhu investigated the unstable topological pressure for additive potentials, and obtained a variational principle.…”

Sub-additive unstable topological pressure on saturated subsets

“…In [12], Tian and Wu generalize the result above with additional consideration of an arbitrary subset (not necessarily compact or invariant). In [6], Hu, Wu, and Zhu investigated the unstable topological pressure for additive potentials. In [15] and [16], we worked on both unstable topological and measure theoretical pressure for C 1 -smooth partially hyperbolic diffeomorphisms with sub-additive potentials, the expected variational principle was established.…”