Statistics of a Family of Piecewise Linear Maps

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.