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# Random Möbius dynamics on the unit disc and perturbation theory for Lyapunov exponents

• Randomly drawn $2\times 2$ matrices induce a random dynamics on the Riemann sphere via the Möbius transformation. Considering a situation where this dynamics is restricted to the unit disc and given by a random rotation perturbed by further random terms depending on two competing small parameters, the invariant (Furstenberg) measure of the random dynamical system is determined. The results have applications to the perturbation theory of Lyapunov exponents which are of relevance for one-dimensional discrete random Schrödinger operators.

Mathematics Subject Classification: Primary: 37H15, 37M25; Secondary: 37C40.

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• Figure 1.  Plot of an orbit $(z_n)_{n = 1,\ldots,N}$ in $\mathbb{D}$ with $N = 5\cdot 10^3$ iterations of the random model described in Section 5. The parameters are $\epsilon = 10^{-4}$ and $\delta = 10^{-3}$ so that $\epsilon = o(\delta)$, and the initial condition is $z_0 = 1$. The random variable $\eta_{\sigma}\equiv -2$ is a constant. The histogram shows the distribution of the radii $|z_n|$. The tail of the distribution is merely due to the thermalization and does not occur if $z_0 = 0$

Figure 2.  Same plot and histogram as in Figure 1, but with the values $\epsilon = 0.1$ and $\delta = 10^{-3}$. For the plot on the left, $10^5$ iterations were run; for the histogram on the right, $5\cdot 10^5$ iterations were run. As in Figure 1, the initial condition is $z_0 = 1$. If the system starts with $z_0 = 0$, the plot and the histogram look very similar

Figure 3.  Same plot and histogram as in Figure 1, but with the values $\epsilon = 0.1$ and $\delta = 10^{-5}$ so that $\delta = o(\epsilon^2)$. The number of iterations is $5\cdot 10^{4}$. The initial condition was $z_0 = 1$. If one chooses $z_0 = 0$ as initial condition, the orbit takes several hundreds of iterations to attain the boundary, but the histogram after a large number of iterations essentially looks the same

Figure 4.  Approximate radial density $\varrho_{\lambda}$ (blue) given by (15) and numerical histogram of values of $|z_n|^2$ obtained after $2\cdot 10^7$ iterations (yellow). The values are $(\epsilon,\delta) = (0.05,7.5\cdot 10^{-4})$ on the left, $(\epsilon,\delta) = (0.05,1.2\cdot 10^{-4})$ in the middle, $(\epsilon,\delta) = (0.05,2.5\cdot 10^{-5})$ on the right

Figure 5.  Numerical histograms of the distribution of $\frac{2}{\pi}\,\arctan(|z_n|^2)\in[0,1]$ after $2\cdot 10^7$ iterations (yellow) and suitably rescaled approximate radial density (blue) with $\mathcal{C}>0$ and $\mathcal{D}>0$ in violation of $\rm(v)$. The values are $(\epsilon,\delta) = (0.05,7.5\cdot 10^{-4})$ on the left and $(\epsilon,\delta) = (0.05,2.5\cdot 10^{-5})$ on the right. The model is described in detail in Section 5

Figure 6.  Numerical histograms of the distribution of $\frac{2}{\pi}\,\arctan(|z_n|^2)\in [0,1]$ after $2\cdot 10^7$ iterations for the same model as in Figure 5, but with $\mathcal{C} = 0$ and $\mathcal{D}>0$. The values are $(\epsilon,\delta) = (0.05,5\cdot 10^{-4})$ on the left and $(\epsilon,\delta) = (2\cdot 10^{-4},0.05)$ on the right, further details are again found in Section 5

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