An example derived from the Lorenz attractor

Abstract 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.