An example derived from the Lorenz attractor

We consider a DA-type surgery of the famous Lorenz attractor in dimension 4. This kind of surgery was first used by Smale [Differentiable dynamical systems. Bull. Amer. Math. Soc. (N.S.) 73 (6) (1967), 747–817] and Mañé [Contributions to the stability conjecture. Topology 17 (4) (1978), 383–396] to give important examples in the study of partially hyperbolic systems. Our construction gives the first example of a singular chain recurrence class which is Lyapunov stable, away from homoclinic tangencies, and exhibits robustly heterodimensional cycles. Moreover, the chain recurrence class has the following interesting property: there exists robustly a two-dimensional sectionally expanding subbundle (containing the flow direction) of the tangent bundle such that it is properly included in a subbundle of the finest dominated splitting for the tangent flow.