On critical orbits and sectional hyperbolicity of the nonwandering set for flows (Q631885)

This note provides an extension of the definition of a nonuniformly hyperbolic set, which was introduced in [\textit{A. Castro}, ``New criteria of generic hyperbolicity based on periodic points'', \url{arXiv:0906.2240v1}], see also [\textit{A. Castro, K. Oliveira} and \textit{V. Pinheiro}, Nonlinearity 20, No. 1, 75--85 (2007; Zbl 1125.37014)]. The authors define the notion of a nonuniformly sectional hyperbolic set and show how this property can be used for residual sets in order to have vector fields with sectional hyperbolic nonwandering sets. Indeed, using \textit{R. Mañé}'s ergodic closing lemma [Ann. Math. (2) 116, 503--540 (1982; Zbl 0511.58029)], the authors show that for a generic flow of class \(\mathcal{C}^1\), the convex hull of critical measures is dense in the set of invariant measures of the flow.