Topological entropy of saturated sets for Lorenz attractors

Abstract

We study the topological entropy of saturated sets for continuous flows, with emphasis on singular hyperbolic dynamics. For a $C^r$-generic ($r \geq 2$) vector field $X$ carrying a geometric Lorenz attractor $Λ$, we prove that for every nonempty compact connected set $\mathcal{K} \subseteq \mathcal{M}(Φ,Λ)$ and every open set $U \subseteq M$ with $U \cap Λ\neq \emptyset$, $h_{\mathrm{top}}(Φ,G_{\mathcal{K}}(Φ) \cap \operatorname{Trans}(Φ,Λ) \cap U)=\inf_{μ\in \mathcal{K}} h_μ(ϕ_1)$, where $Φ=(ϕ_t)_{t \in \mathbb{R}}$ is the flow induced by $X$ and $\operatorname{Trans}(Φ,Λ)$ denotes the set of points whose positive orbits are dense in $Λ$. The same formula holds $C^1$-generically for singular hyperbolic attractors. In contrast with the uniformly hyperbolic case, we also prove that, for vector fields in a dense subset of $\mathcal{X}^r(M^3)$, geometric Lorenz attractors admit a positive-entropy invariant measure with no generic points. An analogous conclusion holds for vector fields in a dense subset of $\mathcal{X}^1(M)$ whose non-trivial singular hyperbolic attractors contain singularities and satisfy $\dim E^{cu}=2$. These examples show that the discrete-time saturated entropy theory does not extend directly to singular flows.