---
abstract: 'In this work, it is presented a characterization of star condition for a $C^1$ vector field based on Lyapunov functions. It is obtained conditions to strong homogeneity for singular sets by using the notion of infinitesimal Lyapunov functions. As an application we obtain some results related to singular hyperbolic sets for flows.'
author:
- Luciana Salgado
title: 'Star Flows: a characterization via Lyapunov functions'
---

[^1]

Introduction and statement of results {#sec:int-stat}
=====================================

Star systems has been studied by many renowned researchers, among them R. Mañé and S. Liao, whom many years ago used it in order to prove the famous *stability conjecture* from Palis and Smale. For more details about star systems, see for instance [@Liao1980],[@Man82],[@Palis87],[@Man88],[@shi-gan-wen2014],[@GanWen2006],[@ArbMo2013].

\[def:star-flow\] A $C^1$ vector field $X$ (or its flow $X_t$) is said to be *star* if it cannot be $C^1$-approximated by ones exhibiting nonhyperbolic periodic orbits.

The definition for diffeomorphisms is analogous.

In the case of diffeomorphisms, $C^1$ structural stability is equivalent to star condition (and equivalent to Axiom A plus no cycle), see e.g. [@Man88].

However, in the case of flows, the situation is much more complex. In fact, in absence of singularities, Gan and Wen [@GanWen2006] proved that nonsingular 