The dynamics of vector fields with singularities
The article is a survey of recent results on dynamics of vector fields exhibiting singularities that are accumulated by recurrent regular points robustly. The author considers the notion of singular hyperbolicity as introduced in [\textit{C. A. Morales} et al., Ann. Math. (2) 160, No. 2, 375--432 (2004; Zbl 1071.37022)]. This notion is useful to characterize a rich family of vector fields with singularities, including the Lorenz attractor. The more general notion of multi-singular hyperbolicity introduced in [\textit{C. Bonatti} and \textit{A. Da Luz}, J. Eur. Math. Soc. (JEMS 23, No. 8, 2649--2705 (2021; Zbl 1487.37037)] is also considered. These notions are relevant to describe dense sets of vector fields. Several results and conjectures along this line of research are reviewed by the present author. This article also covers several results on ergodic theory of singular hyperbolic dynamics or multi-singular hyperbolic dynamics. The last section of the article contains an analysis of the extended linear Poincaré flow and the rescaled linear Poincaré flow showing their use for the study of vector fields with singularities.
scientific article; zbMATH DE number 7473247
scientific article