In this article we study physical measures for $\operatorname {C}^{1+\alpha }$ partially hyperbolic diffeomorphisms with a mostly expanding center. We show that every diffeomorphism with a mostly expanding center direction exhibits a geometrical-combinatorial structure, which we call skeleton, that determines the number, basins and supports of the physical measures. Furthermore, the skeleton allows us to describe how physical measures bifurcate as the diffeomorphism changes under $C^1$ topology.

Moreover, for each diffeomorphism with a mostly expanding center, there exists a $C^1$ neighbourhood, such that diffeomorphism among a $C^1$ residual subset of this neighbourhood admits finitely many physical measures, whose basins have full volume.

We also show that the physical measures for diffeomorphisms with a mostly expanding center satisfy exponential decay of correlation for any Hölder observes. In particular, we prove that every $C^2$ , partially hyperbolic, accessible diffeomorphism with 1-dimensional center and nonvanishing center exponent has exponential decay of correlations for Hölder functions.

在这篇文章中，我们研究了 $\operatorname {C}^{1+\alpha }$ 具有大部分扩展中心的部分双曲微分同胚的物理测量。我们表明，每个具有主要扩展中心方向的微分同胚都表现出一种几何组合结构，我们称之为骨架，它决定了物理措施的数量、盆地和支撑。此外，骨架允许我们描述物理测量如何随着微分同胚在 $C^1$ 拓扑下的变化而分叉。

此外，对于每个具有主要扩展中心的微分同胚，存在一个 $C^1$ 邻域，这样该邻域的 $C^1$ 剩余子集之间的微分同胚允许有限多个物理测量，其盆地具有完整体积。

我们还表明，具有主要扩展中心的微分同胚的物理测量满足任何 Hölder 观察到的相关指数衰减。特别是，我们证明每个 $C^2$ 、部分双曲、可访问的具有一维中心和非零中心指数的微分同胚都具有 Hölder 函数相关性的指数衰减。