We call a partially hyperbolic diffeomorphism partially volume expanding if the Jacobian restricted to any hyperplane that contains the unstable bundle \(E^u\) is larger than 1. This is a \(C^1\) open property. We show that any \(C^{1+}\) partially volume expanding diffeomorphisms admits finitely many physical measures, and the union of their basins has full volume.

中文翻译：

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部分体积扩展的亚同构

如果将雅可比行列式限制到包含不稳定束\（E ^ u \）的任何超平面大于1，则我们称部分双曲微分同质体部分扩展。这是\（C ^ 1 \）开放性质。我们证明，任何\（C ^ {1 +} \）部分体积扩展的微晶态都可以接受有限的物理量，并且它们盆地的并集具有全部体积。