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
We study statistical properties of a family of piecewise linear monotone circle maps related to the angle doubling map mod 1. In particular, we investigate whether for large , the deviations upon rescaling satisfies a -Gaussian distribution if and are both independently and uniformly distributed on the unit circle. This was motivated by the fact that if is the rotation by , then it was recently found that in this case the rescaled deviations are distributed as a -Gaussian with (a Cauchy distribution). This is the only case where a non-trivial (i.e. ) -Gaussian has been analytically established in a conservative dynamical system.
In this note, however, we prove that for the family considered here, converges to a random variable with a curious distribution which is clearly not a -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 与角度倍增图有关 mod 1. 特别是,我们调查是否对于大 ,偏差 重新缩放后满足 -高斯分布如果 和 既独立又均匀分布在单位圆上。这是因为如果 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 (柯西分布)。这是唯一一个非平凡的情况(即) -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, 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。