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arXiv · 2305.03910

On physical measures of multi-singular hyperbolic vector fields

Sylvain Crovisier·Xiaodong Wang·Dawei Yang·Jinhua Zhang
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Abstract

Bonatti and da Luz have introduced the class of \emph{multi-singular hyperbolic} vector fields to characterize systems whose periodic orbits and singularities do not bifurcate under perturbation (called star vector fields). In this paper, we study the Sina\"{\i}-Ruelle-Bowen measures for multi-singular hyperbolic vector fields: in a $C^1$ open and $C^1$ dense subset of multi-singular hyperbolic vector fields, each {$C^\infty$} one admits \emph{finitely} many physical measures whose basins cover a \emph{full} Lebesgue measure subset of the manifold. Similar results are also obtained for $C^1$ generic multi-singular hyperbolic vector fields.

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Sylvain Crovisier, Xiaodong Wang, Dawei Yang, Jinhua Zhang. 2023-05-06. On physical measures of multi-singular hyperbolic vector fields. https://arxiv.org/abs/2305.03910

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