On the Lebesgue Measure of Li-Yorke Pairs for Interval Maps
On the Lebesgue Measure of Li-Yorke Pairs for Interval Maps
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DOI：
10.1007/s00220-010-1085-9
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发表时间：
2009-11
影响因子：
2.4
通讯作者：
H. Bruin;V. Jiménez López
中科院分区：
文献类型：
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作者：
H. Bruin;V. Jiménez López
我们研究了具有非平坦临界点的C2和C3多模态映射f的Li-Yorke对的普遍性。我们证明了每个可测扰集都有零勒贝格测度，所有强游荡集都有零勒贝格测度，渐进（但不是渐进周期）点对的集合也是如此。如果是拓扑混合的，并且没有康托吸引子，则典型（w.r.t.二维Lebesgue测度）对是Li-Yorke的;如果f另外允许一个绝对连续的不变概率测度（acip），则典型的对有f ×f的稠密轨道。这些结果利用了一般多峰映射临界集的所谓好邻域，从而证明了一致扩展的Markov诱导映射的存在性，并对其中有Cantor吸引子的情形，给出了Li-Yorke对和远点对的集合有正二维Lebesgue测度的一个剖分.
We investigate the prevalence of Li-Yorke pairs forC2andC3multimodal mapsfwith non-flat critical points. We show that every measurable scrambled set has zero Lebesgue measure and that all strongly wandering sets have zero Lebesgue measure, as does the set of pairs of asymptotic (but not asymptotically periodic) points.Iffis topologically mixing and has no Cantor attractor, then typical (w.r.t. two-dimensional Lebesgue measure) pairs are Li-Yorke; if additionallyfadmits an absolutely continuous invariant probability measure (acip), then typical pairs have a dense orbit forf×f. These results make use of so-called nice neighborhoods of the critical set of general multimodal maps, and hence uniformly expanding Markov induced maps, the existence of either is proved in this paper as well.For the setting wherefhas a Cantor attractor, we present a trichotomy explaining when the set of Li-Yorke pairs and distal pairs have positive two-dimensional Lebesgue measure.