Star flows: a characterization via Lyapunov functions

Abstract

We say that a differentiable flow or vector field $X$ is star on a compact invariant set $\Lambda$ of the Riemannian manifold M if there exist neighborhoods $\mathcal{U} \in \mathfrak{X}^1(M)$ of $X$ and $U \subset M$ of $\Lambda$ for which every closed orbit in $U$ of every vector field $Y$ in $\mathcal{U}$ is hyperbolic. In this work, it is presented a characterization of star condition for flows based on Lyapunov functions. It is obtained conditions to strong homogeneity for singular sets for a $C^1$ flow by using the notion of infinitesimal Lyapunov functions. As an application we obtain some results related to singular hyperbolic sets for flows.Comment: 22 pages, 1 figure. It were done many changes in the text and included more details into the proof of a main theorem. Proposition 4.3 stated to linear cocycles. arXiv admin note: text overlap with arXiv:1611.04072, arXiv:1201.255