The Rovella attractor is a compact invariant set for a vector eld X 0 that mimics the construction of the geometric Lorenz attractor, but considering a central contractive singularity instead of a central expansive one. In this paper we will prove that the Rovella attractor is asymptotically sectional-hyperbolic. Furthermore, it is proved that for a generic two-parameter family of vector elds containing X 0 , asymptotically sectional-hyperbolicity is an almost 2-persistent property.

<p style='text-indent:20px;'>In this article we define a notion of metric entropy for an invariant measure associated to a set-valued map <inline-formula><tex-math id="M1">\begin{document}$ F $\end{document}</tex-math></inline-formula> on a compact metric space <inline-formula><tex-math id="M2">\begin{document}$ X $\end{document}</tex-math></inline-formula>. Besides, we describe its main properties and prove the <i>Half Variational Principle</i>, which relates the metric entropy with the notion of topological entropy given in [<xref ref-type="bibr" rid="b13">13</xref>] for this class of maps.</p>

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The notion of asymptotically sectional-hyperbolic set was recently introduced. The main feature is that any point outside of the stable manifolds of its singularities has arbitrarily large hyperbolic times. In this paper we prove the existence, on any three-dimensional Riemannian manifold, of attractors with Rovella-like singularities satisfying this kind of hyperbolicity. Furthermore, we prove that asymptotically sectional-hyperbolic Lyapunov-stable sets, under certain conditions, have positive topological entropy.