Abstract: We show that any diffeomorphism with a co-index one heterocimensional cycle is on the C^r (r=2,..,\infty,\omega) boundary of a C^1 open set where diffeomorphisms having robust heterodimensional cycles are dense. We also given conditions on which a heterodimensional cycle can be C^r stablized (i.e. the cycle can be made robust by an arbitrarily small C^r perturbation and the new cycle is associated to the continuations of the saddles of the original one). This is a joint work with Dmitry Turaev.
