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Statistics of a family of piecewise linear maps
Physica D: Nonlinear Phenomena ( IF 2.7 ) Pub Date : 2021-08-27 , DOI: 10.1016/j.physd.2021.133019
J.J.P. Veerman 1 , P.J. Oberly 2 , L.S. Fox 1
Affiliation  

We study statistical properties of a family of piecewise linear monotone circle maps ft(x) related to the angle doubling map x2x mod 1. In particular, we investigate whether for large n, the deviations i=0n1fti(x0)12 upon rescaling satisfies a Q-Gaussian distribution if x0 and t are both independently and uniformly distributed on the unit circle. This was motivated by the fact that if ft is the rotation by t, then it was recently found that in this case the rescaled deviations are distributed as a Q-Gaussian with Q=2 (a Cauchy distribution). This is the only case where a non-trivial (i.e. Q1) Q-Gaussian has been analytically established in a conservative dynamical system.

In this note, however, we prove that for the family considered here, limnSn/n converges to a random variable with a curious distribution which is clearly not a Q-Gaussian or any other standard smooth distribution.



中文翻译:

分段线性映射族的统计

我们研究了一系列分段线性单调圆图的统计特性 F(X) 与角度倍增图有关 X2X mod 1. 特别是,我们调查是否对于大 n,偏差 一世=0n-1F一世(X0)-12 重新缩放后满足 -高斯分布如果 X0既独立又均匀分布在单位圆上。这是因为如果F 是旋转 ,然后最近发现在这种情况下,重新调整的偏差分布为 -高斯与 =2(柯西分布)。这是唯一一个非平凡的情况(即1) -Gaussian 已在保守动力系统中解析建立。

然而,在本笔记中,我们证明对于这里考虑的家庭, nn/n 收敛到一个具有奇怪分布的随机变量,这显然不是 - 高斯或任何其他标准平滑分布。

更新日期:2021-09-08
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