Non-uniformly hyperbolic horseshoes unleashed by homoclinic bifurcations and density zero of attractors (Q2765874)

Let \(f\) be a \(C^{\infty}\)-diffeomorphism of a surface \(M^2\) such that the maximal invariant set in an open set \(V\subset M^2\) is the union of a horseshoe and a quadratic tangency between the stable and unstable foliations of this horseshoe. The dimension of the horseshoe is assumed to be larger than but close to one. The main result states that for most diffeomorphisms \(f\) close to \(f_0\), the maximal \(f\)-invariant set in \(V\) is a non-uniformly hyperbolic horseshoe with dynamics of the same type as met in Hénon attractors.