The basin equality conjecture for sectional-hyperbolic attracting sets

Let $G\in\EuScript{X}^1(M)$ have a trapping region $U$ whose attracting set $\Lambda=\cap_{t>0}\phi_t(U)$ is sectional-hyperbolic. Define the topological basin of attraction by

$${\mathcal U}=\{y\in M:d(\phi_t y,\Lambda)\xrightarrow[t\to+\infty]{}0\}.$$

Assume that there are finitely many hyperbolic periodic points $p_1,\dots,p_l$ of $\Lambda$ and $\varepsilon_0>0$, and let

$${\mathcal W}^{cs}=\{W^s_x:x\in W^u_{\EuScript{O}(p_i)}(\varepsilon_0);\ i=1,\dots,l\}.$$

Here $W^s_x$ is the stable manifold of $x$, and $W^u_{\EuScript{O}(p_i)}(\varepsilon_0)$ is the union of the local unstable manifolds of points on the periodic orbit $\EuScript{O}(p_i)$.

Basin equality conjecture. For any sectional-hyperbolic attracting set $\Lambda$, the topological basin of attraction coincides with the family of stable manifolds through the points of local unstable leaves of finitely many periodic orbits:

$${\mathcal U}={\mathcal W}^{cs}.$$

The preceding theorem proves that ${\mathcal W}^{cs}$ is open and dense in ${\mathcal U}$ and, for $C^{1+}$ vector fields, has full volume and contains the basin of every physical probability measure supported in $\Lambda$. The conjecture asks whether the remaining points of the topological basin are absent.