unstable manifolds of the system and tracks, for example, the folds and switchbacks in them that arise from the small islands present. Some of these features are also seen in the propagated Wigner function but the details are lost.

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In Fig. \[fig:kr:k1:a\], we compare the classical (stationary) phase space distribution $\rho$ defined in (\[classprop\]) with the Wigner transforms of $\hat{K}$ and $\hat{\Gamma}$ for a damping value $v = 0.9$. The classical and wave results coincide remarkably well for $\rho$ and $W_{K}$ even without further averaging, see Fig. \[fig:kr:k1:0\], while only traces of the classical phase space structure are left for $W_{\Gamma}$ in this case. When performing a further parameter average in the Harper map, the quantum fluctuations are suppressed and the classical phase space structure comes out more clearly, both for $W_{K}$ and $W_{\Gamma}$, see Fig. \[fig:kr:k1:2\]. The fluctuations in $W_{\Gamma}$ are in both cases much larger than in $W_{K}$, with similar differences to be expected for the original objects $\hat{\Gamma}$ and $\hat{K}$. We can in fact quantify the increased fluctuations in $\hat{\Gamma}$ compared to $\hat{K}$ as a function of $v$, as will be described in the next section.

The results show that the classical phase space density gives 