$\Gamma_0$ driving a perturbed cat map and centred on a fixed point (of period one). Solid curves are the averaged predictions (\[TrK\]) and (\[TrG\]) based on the classical autocorrelation function $A(n)$. Circles and triangles are respectively evaluations of $\Tr K^2$ and $\Tr \Gamma ^2$ for individual values of $N$: as $v$ is varied, $N$ is stepped from $N=200$ to $N=400$. Note that the averages (\[TrK\]) and (\[TrG\]) are independent of $N$: we let $N$ change in this figure simply to give a sense of the typical variation about the predicted mean for individual maps. This aspect is illustrated more explicitly in Fig. \[fig:Gfluc\] ](fig/logfluctuationvsvplain.pdf){width="0.9\linewidth"}

In this section, we use a perturbed cat map [@SCC94], in which a quantum cat map [@HB80; @Kea91; @AB95] is perturbed with a QKH map of the form used in Sec. \[sec:QKH\] (see \[app:B\] for more details). The perturbation is large enough to break any underlying time-reversal or spatial symmetries of the quantum version of the cat map, but small enough that the overall classical dynamics is still completely chaotic. A source term $\Gamma_0$ is used which corresponds to the nearly Gaussian density given in (\[WG0\]), with $\sigma_x
= \sigma_p = 1/2$. It is centred on 