Partially hyperbolic and transitive dynamics generated by heteroclinic cycles

By Lorenzo J. Díaz, J. Rocha. Ergodic Theory and Dynamical Systems, 2001

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Periodic points and homoclinic classes (2006, background): “But a homoclinic class may fail to be uniformly hyperbolic (for instance, it may contain in a robust way hyperbolic saddles having different indices as p, see the constructions in [D´ i1 ,D ´i2, DR ] and the examples of non-hyperbolic robustly transitive diffeomorphisms in [BD1, BV]) and locally ma”
On maximal transitive sets of generic diffeomorphisms (2003, background): “This cycle is far from tangencies if every diffeomorphism C1-close to f has no homoclinic tangencies associated to P or Q . For this result see [ DR ].”
MINIMALITY OF STRONG STABLE AND UNSTABLE FOLIATIONS FOR PARTIALLY HYPERBOLIC DIFFEOMORPHISMS (2002, background): “Later, in [12] was introduced the generalized blenders (hyperbolic sets with skew Markov partitions) and proved that the previous result always holds in the context of bifurcations of heterodimensional cycles in parametrized families of diffeomorphisms (i.”
Nonsingular star flows satisfy Axiom A and the no-cycle condition (2006, background): “Note that for as ystem X to exhibit a heterodimensional cycle, there is a restriction to the dimension of M, i.e., dim M ≥ 4. The dynamics of systems with heterodimensional cycle are extensively studied by Diaz and other authors, see [ 3 ] and references listed there for more details.”