the treatment of losses). Here, $X=(x,p)$ denotes the collective phase space coordinates on the SOS (with $p$ the momentum conjugate to $x$) and $\varphi:X'\to X$ is the classical map describing the flow of trajectories from the SOS back to itself after a single reflection. We use that $\varphi$ is Hamiltonian, and, in particular, phase space volume preserving. The operator $\cal L$ is also referred to as the Frobenius-Perron (FP) operator [@book]. The integral representation in (\[FP\]) is useful for considering effects like absorption and mode conversion as well as uncertainty, see [@CT14]. The ray-dynamical, or classical, analogues of equations (\[K\]) are now provided by $$\label{classprop}
\rho = {\cal L}\rho + \rho_0
\quad\Rightarrow\quad \rho=\frac{1}{1-{\cal L}}\, \rho_0=
\sum_{n=0}^\infty {\cal L}^n\, \rho_0
= \sum_{n=0}^\infty \rho_n.$$ From the relation derived in \[app:A\], we can now write $$\label{Wrho}
\langle W_{K_n}(X)\rangle \approx 
W_{\Gamma_0}\left(\varphi^{-n}(X)\right)
= {\cal L}^n \left[ \rho_0 \right](X) = \rho_n(X),$$ and we obtain for the Wigner function of the full $\hat{K}$ operator to leading order $$\label{FPK}
\langle W_K \rangle = \sum_{n=0}^\infty \langle W_{K_n} \rangle\approx \sum^{\infty}_{n=0} {\cal L}^n[\rho_0] = \frac{1}{1-{\cal L}}\,  \rho_0 = \rho\, .$$ The averaging $\langle \cdot \rangle$ is understood as defined in (\[Gx1x2\]) and can be performed in terms of an average over 