# American Institute of Mathematical Sciences

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

 Department Mathematik, Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstraße 11, 91058 Erlangen, Germany

Received  August 2020 Revised  January 2021 Published  April 2021

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.

Citation: Florian Dorsch, Hermann Schulz-Baldes. Random Möbius dynamics on the unit disc and perturbation theory for Lyapunov exponents. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021076
##### References:
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##### References:
 [1] P. Bougerol and J. Lacroix, Products of Random Matrices with Applications to Schrödinger Operators, Birkhäuser, Boston, 1985. doi: 10.1007/978-1-4684-9172-2.  Google Scholar [2] Y. Benoist and J.-F. Quint, Random Walks on Reductive Groups, Springer, Cham, 2016. doi: 10.1007/978-3-319-47721-3.  Google Scholar [3] A. Bovier and A. Klein, Weak disorder expansion of the invariant measure for the one-dimensional Anderson model, J. Statist. Phys., 51 (1988), 501-517.  doi: 10.1007/BF01028469.  Google Scholar [4] R. Carmona and J. Lacroix, Spectral Theory of Random Schrödinger Operators, Birkhäuser, Basel, 1990. doi: 10.1007/978-1-4612-4488-2.  Google Scholar [5] B. Derrida and E. J. Gardner, Lyapunov exponent of the one dimensional Anderson model: Weak disorder expansion, J. Physique, 45 (1984), 1283-1295.  doi: 10.1051/jphys:019840045080128300.  Google Scholar [6] T.-C. Dinh, L. Kaufmann and H. Wu, Products of random matrices: A dynamical point of view, Pure and Applied Mathematics Quarterly, 2020, https://www.intlpress.com/site/pub/pages/journals/items/pamq/_home/acceptedpapers/index.php. Google Scholar [7] F. Dorsch and H. Schulz-Baldes, Random perturbations of hyperbolic dynamics, Electronic J. Prob., 24 (2019), 23 pp. doi: 10.1214/19-ejp340.  Google Scholar [8] F. Dorsch and H. Schulz-Baldes, Pseudo-gaps for random hopping models, J. Phys. A, 53 (2020), 185201, 21 pp. doi: 10.1088/1751-8121/ab5e8c.  Google Scholar [9] M. Drabkin and H. Schulz-Baldes, Gaussian fluctuations of products of random matrices distributed close to the identity, J. Difference Equ. Appl., 21 (2015), 467-485.  doi: 10.1080/10236198.2015.1024672.  Google Scholar [10] S. Jitomirskaya, H. Schulz-Baldes and G. Stolz, Delocalization in random polymer models, Commun. Math. Phys., 233 (2003), 27-48.  doi: 10.1007/s00220-002-0757-5.  Google Scholar [11] M. Kappus and F. Wegner, Anomaly in the band centre of the one-dimensional Anderson model, Z. Phys., B 45 (1981), 15-21.  doi: 10.1007/BF01294272.  Google Scholar [12] J. M. Luck, Systèmes Désordonnés Unidimensionnels, Aléa, Saclay, 1992. Google Scholar [13] L. Pastur and A. Figotin, Spectra of Random and Almost-Periodic Operators, Springer, Berlin, 1992. https://www.springer.com/gp/book/9783642743481  Google Scholar [14] R. Römer and H. Schulz-Baldes, Weak disorder expansion for localization lengths of quasi-1D systems, Euro. Phys. Lett., 68 (2004), 247–253. https://iopscience.iop.org/article/10.1209/epl/i2004-10190-9 Google Scholar [15] C. Sadel and H. Schulz-Baldes, Scaling diagram for the localization length at a band edge, Ann. Henri Poincaré, 8 (2007), 1595-1621.  doi: 10.1007/s00023-007-0347-3.  Google Scholar [16] C. Sadel and B. Virág, A central limit theorem for products of random matrices and GOE statistics for the Anderson model on long boxes, Commun. Math. Phys., 343 (2016), 881-919.  doi: 10.1007/s00220-016-2600-4.  Google Scholar [17] R. Schrader, H. Schulz-Baldes and A. Sedrakyan, Perturbative test of single parameter scaling for $1D$ random media, Ann. Henri Poincare, 5 (2004), 1159-1180.  doi: 10.1007/s00023-004-0195-3.  Google Scholar [18] H. Schulz-Baldes, Perturbation theory for Lyapunov exponents of an Anderson model on a strip, Geom. Funct. Anal., 14 (2004), 1089-1117.  doi: 10.1007/s00039-004-0484-5.  Google Scholar [19] H. Schulz-Baldes, Lyapunov Exponents at Anomalies of $\rm SL (2, \mathbb{R})$-actions, Operator Theory, Analysis and Mathematical Physics, 174, 159–172. Birkhäuser, Basel, 2007. doi: 10.1007/978-3-7643-8135-6_10.  Google Scholar
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 $z_0 = 0$
, 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 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
, 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 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
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
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
, 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">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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