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Partially hyperbolic and transitive dynamics generated by heteroclinic cycles. (English) Zbl 0972.37018
The authors study $$C^k$$ diffeomorphisms, $$k \geq 1$$, $$f :M \rightarrow M$$, where $$M$$ is a compact and smooth boundaryless manifold, exhibiting heterodimensional cycles (i.e. cycles containing periodic points of different stable indices). They prove that if $$f$$ can not be $$C^k$$ approximated by diffeomorphisms with homoclinic tangencies, then $$f$$ is in the closure of an open set $${\mathcal U} \subset \text{Diff}^k(M)$$ consisting of diffeomorphisms $$g$$ with a non-hyperbolic transitive set $$\Lambda_g$$ which is locally maximal and strongly partially hyperbolic. As a consequence, in the case of 3-dimensional manifolds, they give new examples of open sets of $$C^1$$ diffeomorphisms for which residually infinitely sinks and sources coexist. They also prove that the occurence of non-hyperbolic dynamics has persistent character in the unfolding of heterodimensional cycles.

##### MSC:
 37D30 Partially hyperbolic systems and dominated splittings 37C29 Homoclinic and heteroclinic orbits for dynamical systems
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