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 ($r\ge 2$) vector field $X$ carrying a geometric Lorenz attractor $\Lambda$, we prove that for every nonempty compact connected set $\mathcal{K}\subseteq \mathcal{M}(\Phi ,\Lambda )$ and every open set $U\subseteq M$ with $U\cap \Lambda \ne \varnothing$, $h_{\mathrm{top}}(\Phi ,G_{\mathcal{K}}(\Phi )\cap \mathrm{Trans}(\Phi ,\Lambda )\cap U)={\inf }_{\mu \in \mathcal{K}}{h}_{\mu }({\varphi }_1)$, where $\Phi =({\varphi }_t{)}_{t\in \mathbb{R}}$ is the flow induced by $X$ and $\mathrm{Trans}(\Phi ,\Lambda )$ denotes the set of points whose positive orbits are dense in $\Lambda$. 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.

arXiv comment34 pages