Dynamical Systems ( IF 0.5 ) Pub Date : 2021-06-06 , DOI: 10.1080/14689367.2021.1927988 Zeya Mi 1
In this paper, we study the physical measures for a class of partially hyperbolic flows with mostly contracting centre. Let X be a vector field on a compact Riemannian manifold M with partially hyperbolic splitting . We prove that if the centre direction exhibits the asmptotically sectionally contracting behaviour with respect to Gibbs u-states, then X admits finitely many physical measures, and their basins cover Lebesgue almost all points of the ambient manifold. Moreover, when the unstable manifolds are dense, we prove that X admits only one physical measure whose basin covers a full Lebesgue measure subset. By tracing the physical measures via typical hyperbolic periodic orbits, we study the statistical stability of this kind of partially hyperbolic flows.
中文翻译:
大部分收缩中心的部分双曲线流的物理测量
在本文中,我们研究了一类具有大部分收缩中心的部分双曲线流的物理测度。设X为具有部分双曲分裂的紧黎曼流形M 上的向量场. 我们证明如果中心方向表现出关于 Gibbs u 态的渐近截面收缩行为,那么X 承认有限的许多物理测度,并且它们的盆地几乎覆盖了环境流形的所有点。此外,当不稳定流形密集时,我们证明X只承认一个物理测度,其盆地覆盖了一个完整的 Lebesgue 测度子集。通过典型的双曲周期轨道跟踪物理量度,我们研究了这种部分双曲流的统计稳定性。