On the density of hyperbolicity and homoclinic bifurcations for 3D-diffeomorphisms in attracting regions
Hyperbolic systems constitute an ample and relevant class of dynamical systems (DS), but not a generic one: it is known that there are open sets in the space of DS which are nonhyperbolic. This was first shown by Abraham and Smale and by Shub in the early 70s by exhibiting open sets of diffeomorphisms in three dimensions with hyperbolic periodic points of different indices inside a transitive set. When periodic points of different indices are linked through the intersection (these are necessarily not all transversal) of their stable and unstable manifolds, one speaks of a heterodimensional cycle. Later on, Newhouse showed that hyperbolicity is also non dense within \(C^r\) (with \(r\geq 2\)) diffeomorphisms of compact surfaces. The underlying mechanism here is that of homoclinic tangency, i.e., nontransversal intersection of the stable and unstable manifolds of a periodic point. In the 80s, Palis conjectured that the mechanisms mentioned above are rather common in the complement of hyperbolic systems within DS. His conjectures are: (1) Every \(C^r\) diffeomorphism of a compact manifold \(M\) can be \(C^r\) approximated either by a hyperbolic one, or by one with a heterodimensional cycle, or by one exhibiting a homoclinic tangency. (2) When \(M\) is a two-dimensional compact manifold, every \(C^r\) diffeomorphism of \(M\) can be \(C^r\) approximated either by a hyperbolic one or by one exhibiting a homoclinic tangency. The present paper goes in the direction of proving the Palis conjecture in the \(C^1\) topology for an attracting homoclinic class of a three-dimensional \(C^2\) diffeomorphism. Given a periodic point \(p\), one defines the homoclinic class \(H_p\) associated to \(p\) as the closure of the transversal intersections of the stable and unstable manifolds of \(p\). Given \(H_p\), we say it is attracting if there exists an open set \(U\) such that \(H_p \subset U\) and \(H_p = \cap_{n>0} f^n (U)\). The present paper proves that given a maximally invariant attracting homoclinic class for a three-dimensional Kupka-Smale diffeomorphism, either the diffeomorphism is \(C^1\) approximated by one exhibiting a homoclinic tangency of a heterodimensional cycle, or the homoclinic class is topologically conjugate to a hyperbolic set. It also characterizes -- in arbitrary dimension -- the dynamics of a topologically hyperbolic homoclinic class and describes the continuation of this homoclinic class under a perturbation of the DS. It turns out that under certain topological conditions, discussed in the paper, the homoclinic class is contained in a two-dimensional manifold and is hyperbolic.
scientific article; zbMATH DE number 5121856