Physical measures for partially hyperbolic flows with mostly contracting centre

Abstract

In this paper, we study the physical measures for a class of partially hyperbolic flows with mostly contracting centre. Let X be a $C^2$ vector field on a compact Riemannian manifold M with partially hyperbolic splitting $TM=E^u\oplus E^c$. We prove that if the centre direction $E^c$ 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.

Keywords:

• Partially hyperbolic flows
• mostly contracting centre
• physical measures
• statistical stability

2010 Mathematics Subject Classifications:

• 37A35
• 37A60
• 37C40
• 37D25
• 37D30

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 $\Vert {\wedge }^{2}D{\phi }_t{|}_{E(x)}\Vert =\underset{P\in \mathcal{P}}{max}|\det D{\phi }_t{|}_P|$, where $\mathcal{P}$ is the set of two dimensional subspaces in $E(x)$.

2 It means that $T_x\gamma \oplus E^c(x)=T_{x}M$ for any $x\in \gamma$.

Additional information

Funding

Z. Mi would like to thank the support of National Natural Science Foundation of China (NSFC) (Grant Number 11801278).