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.

中文翻译：

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部分双曲微分同胚的不稳定压力和 u 平衡状态

引入不稳定压力和 u 平衡状态并研究了部分双曲微分同胚F. 我们定义不稳定压力$P^{u}(f, \varphi )$的F在连续函数$\varphi $通过动力学F在局部不稳定的叶子上。不稳定压力的变分原理$P^{u}(f, \varphi )$，其中指出$P^{u}(f, \varphi )$是不稳定熵之和和积分的上确界$\varphi $接受所有不变的措施，得到。研究了变分原理中达到最高的U-平衡态及其与吉布斯u-态的关系。还考虑了不稳定压力的可微性性质，例如切线泛函、Gateaux 可微性和 Fréchet 可微性以及它们与 u 平衡状态的关系。