Non-uniformly hyperbolic horseshoes arising from bifurcations of Poincaré heteroclinic cycles

Abstract

In the present paper, we advance considerably the current knowledge on the topic of bifurcations of heteroclinic cycles for smooth, meaning C ∞, parametrized families {g t ∣t∈ℝ} of surface diffeomorphisms. We assume that a quadratic tangency q is formed at t=0 between the stable and unstable lines of two periodic points, not belonging to the same orbit, of a (uniformly hyperbolic) horseshoe K (see an example at the Introduction) and that such lines cross each other with positive relative speed as the parameter evolves, starting at t=0 and the point q. We also assume that, in some neighborhood W of K and of the orbit of tangency o(q), the maximal invariant set for g 0=g t=0 is K∪o(q), where o(q) denotes the orbit of q for g 0. We then prove that, when the Hausdorff dimension HD(K) is bigger than one, but not much bigger (see (H.4) in Section 1.2 for a precise statement), then for most t, |t| small, g t is a non-uniformly hyperbolic horseshoe in W, and so g t has no attractors in W. Most t, and thus most g t , here means that t is taken in a set of parameter values with Lebesgue density one at t=0.