Measures of chaos and a spectral decomposition of dynamical systems on the interval
Measures of chaos and a spectral decomposition of dynamical systems on the interval
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DOI：
10.1090/s0002-9947-1994-1227094-x
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发表时间：
1994-02
影响因子：
1.3
通讯作者：
B. Schweizer;J. Smítal
中科院分区：
文献类型：
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作者：
B. Schweizer;J. Smítal
设f：[0，1] -+ [0，1]是连续的。对于x，y E [0，1]，上和下（距离）分布函数Fx*y和Fxy对于任何t > 0被定义为在前n次迭代期间x和y的轨迹之间的距离Ifi（x）fi（y）I小于t的平均次数的lim sup和lim inf。f的谱是系统吗？(f)的低分布函数的特征在于以下性质：（1）元素？(f)是相互不可比的;（2）对于任何F E？(f)（3）如果S是f的一个加扰集，则存在F，G在？(f)以及分解S = SF U SG（SG可以是空的），使得如果u，v E SF，则FUV > F，并且如果u，v E SG，则FUV > G。我们的主要结果是：（1）若f有正拓扑熵，则？(f)是非空的有限的，任何F E？(f)在区间[0，e]上为零，其中e > 0（因此任何PF都是Li和Yorke意义上的加扰集）。(2)如果f的拓扑熵为零，那么？(f)= {F}，其中F 1。因此，f的谱提供了f的混沌程度的度量。此外，一个有用的数值措施是最大的数字fo（1 F（t））dt，其中F E？（f）.
Let f: [0, 1] -+ [0, 1] be continuous. For x, y E [0, 1], the upper and lower (distance) distribution functions, Fx*y and Fxy, are defined for any t > 0 as the lim sup and lim inf as n -+ oc of the average number of times that the distance Ifi (x) fi(y)I between the trajectories of x and y is less than t during the first n iterations. The spectrum of f is the system ?(f) of lower distribution functions which is characterized by the following properties: (1) The elements of ?(f) are mutually incomparable; (2) for any F E ?(f), there is a perfect set PF #8 0 such that FUV = F and FUV 1 for any distinct u, v E PF; (3) if S is a scrambled set for f, then there are F, G in ?(f) and a decomposition S = SF U SG (SG may be empty) such that FUV > F if u, v E SF and FUV > G if u, v E SG . Our principal results are: (1) If f has positive topological entropy, then ?(f) is nonempty and finite, and any F E ?(f) is zero on an interval [0, e], where e > 0 (and hence any PF is a scrambled set in the sense of Li and Yorke). (2) If f has zero topological entropy, then ?(f) = {F} where F 1. It follows that the spectrum of f provides a measure of the degree of chaos of f . In addition, a useful numerical measure is the largest of the numbers fo(1 F(t))dt, where F E ?(f).