Persistence of the Feigenbaum attractor in one-parameter families (Q1586929)

The paper deals with the question whether, far away from the fixed map, a whole open set of transformations exhibiting Feigenbaum attractors can exist and whether or not these attractors can always be destroyed by arbitrary small perturbations. Indeed consider a map \(\psi_0\) of class \(C^r\) for large \(r\) of a manifold of dimension \(n\) greater than or equal to 2 having a Feigenbaum attractor. The authors prove that any such \(\psi_0\) is a point of a local codimension-one manifold of \(C^r\) transformations also exhibiting Feigenbaum attractors. As a result, the attractor persists when perturbing a one-parameter family transversal to that manifold at \(\psi_0\). Moreover the authors, using their construction, prove a conjecture by J. Palis stating that a map exhibiting a Feigenbaum attractor can be perturbed to obtain homoclinic tangencies.