We study statistical properties of a family of piecewise linear monotone circle maps $f_t\left(x\right)$ related to the angle doubling map $x\to 2x$ mod 1. In particular, we investigate whether for large $n$, the deviations ${\sum }_{i=0}^{n-1}\left(f_t^i\left(x_0\right)-\frac{1}{2}\right)$ upon rescaling satisfies 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 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. $Q\ne 1$) $Q$-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.

我们研究了一系列分段线性单调圆图的统计特性 $F_{吨}\left(X\right)$ 与角度倍增图有关 $X\to 2X$ mod 1. 特别是，我们调查是否对于大 $n$,偏差 ${\sum }_{\mathrm{一世}=0}^{n-1}\left(F_{吨}^{\mathrm{一世}}\left(X_0\right)-\frac{1}{2}\right)$ 重新缩放后满足 $问$-高斯分布如果 $X_0$ 和 $吨$既独立又均匀分布在单位圆上。这是因为如果$F_{吨}$ 是旋转 $吨$，然后最近发现在这种情况下，重新调整的偏差分布为 $问$-高斯与 $问=2$（柯西分布）。这是唯一一个非平凡的情况（即$问\ne 1$) $问$-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， ${林}_n{秒}_n/n$ 收敛到一个具有奇怪分布的随机变量，这显然不是 $问$- 高斯或任何其他标准平滑分布。