论文标题
分段线性地图家族的统计数据
Statistics of a Family of Piecewise Linear Maps
论文作者
论文摘要
我们研究截短的平点图的统计属性$ f_t（x）$。特别是，我们调查了对于$ n $，偏差$ \ sum_ {i = 0}^{n-1} \ left（f_t^i（x_0） - \ frac 12 \ right）$在重新恢复后满足$ q $ q $ -gaussian（如果$ x_0 $），并且在$ x_0 $和$ t $上是独立的，并且是独立的，则在单位上分布。这是由于以下事实，如果$ f_t $是$ t $的轮换，那么在这种情况下，将重新缩放的偏差分配为$ q $ -Gaussian，$ q = 2 $（cauchy分发）。这是唯一一个非平凡（即$ q \ neq 1 $）$ q $ -Gaussian在保守的动力系统中分析建立的情况。 但是，在本说明中，我们证明，对于此处考虑的家庭，$ \ lim_n s_n/n $收敛到带有好奇分布的随机变量，显然不是$ q $ - 高斯或任何其他标准平滑分布。
We study statistical properties of the truncated flat spot map $f_t(x)$. In particular, we investigate whether for large $n$, the deviations $\sum_{i=0}^{n-1} \left(f_t^i(x_0)-\frac 12\right)$ upon rescaling satisfy a $Q$-Gaussian distribution if $x_0$ and $t$ are both independently and uniformly distributed on the unit circle. This was motivated by the fact that if $f_t$ is the rotation by $t$, then it has been shown 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. $Q\neq 1$) $Q$-Gaussian has been analytically established in a conservative dynamical system. In this note, however, we prove that for the family considered here, $\lim_n S_n/n$ converges to a random variable with a curious distribution which is clearly not a $Q$-Gaussian or any other standard smooth distribution.
