Non-uniformly hyperbolic horseshoes arising from bifurcations of Poincaré heteroclinic cycles
Non-uniformly hyperbolic horseshoes arising from bifurcations of Poincaré heteroclinic cycles
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庞加莱异宿循环分岔产生的非均匀双曲马蹄铁
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发表时间：
2006
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通讯作者：
J. Yoccoz
中科院分区：
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作者：
J. Palis;J. Yoccoz
在本文中，我们极大地推进了关于光滑的异斜环分岔的现有知识，即曲面微分同态的C∞，参数化族{gt∣t∈λ}。我们假设形成二次接触q t = 0之间的稳定和不稳定两个周期点,不属于同一轨道,(一致双曲)的马蹄K(看到一个例子介绍),等线交叉互相积极的相对速度参数的演化,从t = 0和q点。我们也认为,在一些社区的K和W的轨道相切o (q)的最大不变集g0 = gt = 0 K∪o (q),其中o(q)表示q对g0的轨道。然后，我们证明，当Hausdorff维数HD(K)大于1，但不太大时（精确表述见1.2节（H.4）），那么对于大多数t， |到|小，gt在W中是一个非均匀双曲马蹄形，因此gt在W中没有吸引子。大多数t，因此大多数gt，这里意味着t在t=0时被取为一组勒贝格密度为1的参数值。
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.