Physical measures for partially hyperbolic flows with mostly contracting centre. Issue 3 (3rd July 2021)

• Record Type: Journal Article
• Authors: Mi, Zeya
• Is Part Of: Dynamical systems. Volume 36:Issue 3(2021)
• Page Start: 427
• Page End: 444
• Publication Date: 2021-07-03
• Subjects: Partially hyperbolic flows -- mostly contracting centre -- physical measures -- statistical stability; 37A35 -- 37A60 -- 37C40 -- 37D25 -- 37D30; Differentiable dynamical systems -- Periodicals; 515.35205
• DOI: 10.1080/14689367.2021.1927988
• Languages: English
• ISSNs: 1468-9367

Abstract

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 T M = E u ⊕ 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.