Abundance of Observable Lyapunov Irregular Sets
Abundance of Observable Lyapunov Irregular Sets
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DOI：
10.1007/s00220-022-04337-6
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发表时间：
2021-10
影响因子：
2.4
通讯作者：
Shin Kiriki;Xiaolong Li;Yushi Nakano;Teruhiko Soma
中科院分区：
文献类型：
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作者：
Shin Kiriki;Xiaolong Li;Yushi Nakano;Teruhiko Soma
李亚普诺夫指数在自然科学中广泛用于寻找混沌信号，但其存在性却很少被讨论。在本文中，我们考虑李亚普诺夫指数不存在的点集（称为李亚普诺夫不规则集）是否具有正勒贝格测度的问题。唯一已知的李亚普诺夫不规则集正勒贝格测度的例子是 Ott 和 Yorke 的工作的 8 字形吸引子（Phys Rev 78, 056203, 2008），其关键机制（同宿环）很容易被小扰动破坏。在本文中，我们证明了由 Colli 和 Vargas 给出的具有鲁棒同宿正切的表面微分同胚（Ergod Theory Dyn Syst 21, 1657–1681, 2001），以及其他几种已知的非双曲动力学，具有正勒贝格测度的李亚普诺夫不规则集。当时间平均值存在和不存在时，我们可以构造这样的正勒贝格测度集。
Lyapunov exponent is widely used in natural science to find chaotic signal, but its existence is seldom discussed. In the present paper, we consider the problem of whether the set of points at which Lyapunov exponent fails to exist, called the Lyapunov irregular set, has positive Lebesgue measure. The only known example with the Lyapunov irregular set of positive Lebesgue measure is a figure-8 attractor by the work of Ott and Yorke (Phys Rev 78, 056203, 2008), whose key mechanism (homoclinic loop) is easy to be broken by small perturbations. In this paper, we show that surface diffeomorphisms with a robust homoclinic tangency given by Colli and Vargas (Ergod Theory Dyn Syst 21, 1657–1681, 2001), as well as other several known nonhyperbolic dynamics, have the Lyapunov irregular set of positive Lebesgue measure. We can construct such positive Lebesgue measure sets both as the time averages exist and do not exist on it.