
-
Previous Article
Dynamics of consumer-resource systems with consumer's dispersal between patches
- DCDS-B Home
- This Issue
-
Next Article
Quasi-periodic travelling waves for beam equations with damping on 3-dimensional rectangular tori
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 |
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.
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. |
[2] |
Y. Benoist and J.-F. Quint, Random Walks on Reductive Groups, Springer, Cham, 2016.
doi: 10.1007/978-3-319-47721-3. |
[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. |
[4] |
R. Carmona and J. Lacroix, Spectral Theory of Random Schrödinger Operators, Birkhäuser, Basel, 1990.
doi: 10.1007/978-1-4612-4488-2. |
[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. |
[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. |
[7] |
F. Dorsch and H. Schulz-Baldes, Random perturbations of hyperbolic dynamics, Electronic J. Prob., 24 (2019), 23 pp.
doi: 10.1214/19-ejp340. |
[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. |
[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. |
[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. |
[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. |
[12] |
J. M. Luck, Systèmes Désordonnés Unidimensionnels, Aléa, Saclay, 1992. |
[13] |
L. Pastur and A. Figotin, Spectra of Random and Almost-Periodic Operators, Springer, Berlin, 1992. https://www.springer.com/gp/book/9783642743481 |
[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 |
[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. |
[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. |
[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. |
[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. |
[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. |
show all references
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. |
[2] |
Y. Benoist and J.-F. Quint, Random Walks on Reductive Groups, Springer, Cham, 2016.
doi: 10.1007/978-3-319-47721-3. |
[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. |
[4] |
R. Carmona and J. Lacroix, Spectral Theory of Random Schrödinger Operators, Birkhäuser, Basel, 1990.
doi: 10.1007/978-1-4612-4488-2. |
[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. |
[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. |
[7] |
F. Dorsch and H. Schulz-Baldes, Random perturbations of hyperbolic dynamics, Electronic J. Prob., 24 (2019), 23 pp.
doi: 10.1214/19-ejp340. |
[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. |
[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. |
[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. |
[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. |
[12] |
J. M. Luck, Systèmes Désordonnés Unidimensionnels, Aléa, Saclay, 1992. |
[13] |
L. Pastur and A. Figotin, Spectra of Random and Almost-Periodic Operators, Springer, Berlin, 1992. https://www.springer.com/gp/book/9783642743481 |
[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 |
[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. |
[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. |
[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. |
[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. |
[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. |






[1] |
Rajeshwari Majumdar, Phanuel Mariano, Hugo Panzo, Lowen Peng, Anthony Sisti. Lyapunov exponent and variance in the CLT for products of random matrices related to random Fibonacci sequences. Discrete and Continuous Dynamical Systems - B, 2020, 25 (12) : 4779-4799. doi: 10.3934/dcdsb.2020126 |
[2] |
Janusz Mierczyński. Averaging in random systems of nonnegative matrices. Conference Publications, 2015, 2015 (special) : 835-840. doi: 10.3934/proc.2015.0835 |
[3] |
Doan Thai Son. On analyticity for Lyapunov exponents of generic bounded linear random dynamical systems. Discrete and Continuous Dynamical Systems - B, 2017, 22 (8) : 3113-3126. doi: 10.3934/dcdsb.2017166 |
[4] |
Pedro Duarte, Silvius Klein, Manuel Santos. A random cocycle with non Hölder Lyapunov exponent. Discrete and Continuous Dynamical Systems, 2019, 39 (8) : 4841-4861. doi: 10.3934/dcds.2019197 |
[5] |
Lianfa He, Hongwen Zheng, Yujun Zhu. Shadowing in random dynamical systems. Discrete and Continuous Dynamical Systems, 2005, 12 (2) : 355-362. doi: 10.3934/dcds.2005.12.355 |
[6] |
Philippe Marie, Jérôme Rousseau. Recurrence for random dynamical systems. Discrete and Continuous Dynamical Systems, 2011, 30 (1) : 1-16. doi: 10.3934/dcds.2011.30.1 |
[7] |
Janusz Mierczyński, Sylvia Novo, Rafael Obaya. Lyapunov exponents and Oseledets decomposition in random dynamical systems generated by systems of delay differential equations. Communications on Pure and Applied Analysis, 2020, 19 (4) : 2235-2255. doi: 10.3934/cpaa.2020098 |
[8] |
Yujun Zhu. Preimage entropy for random dynamical systems. Discrete and Continuous Dynamical Systems, 2007, 18 (4) : 829-851. doi: 10.3934/dcds.2007.18.829 |
[9] |
Ji Li, Kening Lu, Peter W. Bates. Invariant foliations for random dynamical systems. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3639-3666. doi: 10.3934/dcds.2014.34.3639 |
[10] |
Weigu Li, Kening Lu. Takens theorem for random dynamical systems. Discrete and Continuous Dynamical Systems - B, 2016, 21 (9) : 3191-3207. doi: 10.3934/dcdsb.2016093 |
[11] |
Yong Chen, Hongjun Gao, María J. Garrido–Atienza, Björn Schmalfuss. Pathwise solutions of SPDEs driven by Hölder-continuous integrators with exponent larger than $1/2$ and random dynamical systems. Discrete and Continuous Dynamical Systems, 2014, 34 (1) : 79-98. doi: 10.3934/dcds.2014.34.79 |
[12] |
Björn Schmalfuss. Attractors for nonautonomous and random dynamical systems perturbed by impulses. Discrete and Continuous Dynamical Systems, 2003, 9 (3) : 727-744. doi: 10.3934/dcds.2003.9.727 |
[13] |
Ivan Werner. Equilibrium states and invariant measures for random dynamical systems. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 1285-1326. doi: 10.3934/dcds.2015.35.1285 |
[14] |
Weigu Li, Kening Lu. A Siegel theorem for dynamical systems under random perturbations. Discrete and Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 635-642. doi: 10.3934/dcdsb.2008.9.635 |
[15] |
Yuri Kifer. Computations in dynamical systems via random perturbations. Discrete and Continuous Dynamical Systems, 1997, 3 (4) : 457-476. doi: 10.3934/dcds.1997.3.457 |
[16] |
Thomas Bogenschütz, Achim Doebler. Large deviations in expanding random dynamical systems. Discrete and Continuous Dynamical Systems, 1999, 5 (4) : 805-812. doi: 10.3934/dcds.1999.5.805 |
[17] |
Fumihiko Nakamura, Yushi Nakano, Hisayoshi Toyokawa. Lyapunov exponents for random maps. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022058 |
[18] |
Petr Kůrka. Minimality in iterative systems of Möbius transformations. Conference Publications, 2011, 2011 (Special) : 903-912. doi: 10.3934/proc.2011.2011.903 |
[19] |
Petr Kůrka. Iterative systems of real Möbius transformations. Discrete and Continuous Dynamical Systems, 2009, 25 (2) : 567-574. doi: 10.3934/dcds.2009.25.567 |
[20] |
Shengtian Yang, Thomas Honold. Good random matrices over finite fields. Advances in Mathematics of Communications, 2012, 6 (2) : 203-227. doi: 10.3934/amc.2012.6.203 |
2021 Impact Factor: 1.497
Tools
Metrics
Other articles
by authors
[Back to Top]