Sufficient conditions for a partially hyperbolic attractor to be a homoclinic class

We prove that a partially hyperbolic attractor with two-dimensional central direction Λ is a homoclinic class if it exhibits a hyperbolic periodic orbit O and a Lorenz-like singularity σ with Wu(σ)∩Ws(O)≠∅ such that Ws(σ) is dense in Λ.