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

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