Non-uniformly hyperbolic horseshoes arising from bifurcations of Poincaré heteroclinic cycles
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In the present paper, we advance considerably the current knowledge on the topic of bifurcations of heteroclinic cycles for smooth, meaning C∞, parametrized families {gt∣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 g0=gt=0 is K∪o(q), where o(q) denotes the orbit of q for g0. 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, gt is a non-uniformly hyperbolic horseshoe in W, and so gt has no attractors in W. Most t, and thus most gt, here means that t is taken in a set of parameter values with Lebesgue density one at t=0.
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