-
Previous Article
Symbolic dynamics for non-uniformly hyperbolic diffeomorphisms of compact smooth manifolds
- JMD Home
- This Volume
-
Next Article
Roy Adler and the lasting impact of his work
Rational ergodicity of step function skew products
1. | School of Mathematical Sciences, Tel Aviv University, 69978 Tel Aviv, Israel |
2. | School of Mathematics, Bristol University, Bristol BS8 1TW, UK |
We study rational step function skew products over certain rotations of the circle proving ergodicity and bounded rational ergodicity when the rotation number is a quadratic irrational. The latter arises from a consideration of the asymptotic temporal statistics of an orbit as modelled by an associated affine random walk.
References:
[1] |
J. Aaronson, M. Bromberg and H. Nakada,
Discrepancy skew products and affine random walks, Israel J. Math., 221 (2017), no. 2,973-1010.
doi: 10.1007/s11856-017-1560-5. |
[2] |
J. Aaronson and M. Keane,
The visits to zero of some deterministic random walks, Proc. London Math. Soc.(3), 44 (1982), no. 3,535-553.
doi: 10.1112/plms/s3-44.3.535. |
[3] |
J. Beck,
Probabilistic Diophantine Approximation. Randomness in Lattice Point Counting, Springer Monographs in Mathematics, Springer, 2014.
doi: 10.1007/978-3-319-10741-7. |
[4] |
M. Bromberg and C. Ulcigrai,
A temporal central limit theorem for real-valued cocycles over
rotations, Ann. Inst. Henri Poincaré Probab. Stat., 54 (2018), no. 4, 2304-2334.
doi: 10.1214/17-AIHP872. |
[5] |
J.-P. Conze,
Equirépartition et ergodicité de transformations cylindriques, Séminaire de
Probabilités, I (Univ. Rennes, Rennes), (1976), 1-21.
|
[6] |
J.-P. Conze and A. Piȩkniewska,
On multiple ergodicity of affine cocycles over irrational rotations, Israel J. Math., 201 (2014), no. 2,543-584.
doi: 10.1007/s11856-014-0033-3. |
[7] |
D. Dolgopyat and O. Sarig,
Temporal distributional limit theorems for dynamical systems, J. Stat. Phys., 166 (2017), no. 3-4,680-713.
doi: 10.1007/s10955-016-1689-3. |
[8] |
G. H. Hardy and E. M. Wright,
An Introduction to the Theory of Numbers, 3rd ed, Oxford, Clarendon Press, 1954. |
[9] |
H. Hennion and L. Hervé,
Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-Compactness, Lecture Notes in Mathematics, 1766, Springer-Verlag, Berlin, 2001.
doi: 10.1007/b87874. |
[10] |
M.-R. Herman,
Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Inst. Hautes Études Sci. Publ. Math., 49 (1979), 5-233.
doi: 10.1007/bf02684798. |
[11] |
Y. Katznelson,
Sigma-finite invariant measures for smooth mappings of the circle, J. Analyse Math., 31 (1977), 1-18.
doi: 10.1007/bf02813295. |
[12] |
M. Keane, Irrational rotations and quasi-ergodic measures, Publications des Séminaires de
Mathématiques (Univ. Rennes, Rennes), Fasc. 1: Probabilités, 1970, 17–26. |
[13] |
A. Ya. Khintchine,
Continued Fractions, translated by Peter Wynn, P. Noordhoff, Ltd., Groningen, 1963. |
[14] |
C. Kraaikamp and H. Nakada,
On normal numbers for continued fractions, Ergodic Theory Dynam. Systems, 20 (2000), no. 5, 1405-1421.
doi: 10.1017/S0143385700000766. |
[15] |
L. Kronecker,
Zwei Sätze über Gleichungen mit ganzzahligen Coefficienten, J. Reine Angew. Math., 53 (1857), 173-175.
doi: 10.1515/crll.1857.53.173. |
[16] |
I. Oren,
Ergodicity of cylinder flows arising from irregularities of distribution, Israel J. Math, 44 (1993), no. 2,127-138.
doi: 10.1007/BF02760616. |
[17] |
K. Schmidt Cocycles on Ergodic Transformation Groups, Macmillan Lectures in Mathematics, Vol. 1, Macmillan, 1977. |
[18] |
O. Taussky, Eigenvalues of finite matrices: some topics concerning bounds for eigenvalues of
finite matrices, Survey of Numerical Analysis (ed. J. Todd), 1962, McGraw-Hill, New York,
279–297. |
show all references
References:
[1] |
J. Aaronson, M. Bromberg and H. Nakada,
Discrepancy skew products and affine random walks, Israel J. Math., 221 (2017), no. 2,973-1010.
doi: 10.1007/s11856-017-1560-5. |
[2] |
J. Aaronson and M. Keane,
The visits to zero of some deterministic random walks, Proc. London Math. Soc.(3), 44 (1982), no. 3,535-553.
doi: 10.1112/plms/s3-44.3.535. |
[3] |
J. Beck,
Probabilistic Diophantine Approximation. Randomness in Lattice Point Counting, Springer Monographs in Mathematics, Springer, 2014.
doi: 10.1007/978-3-319-10741-7. |
[4] |
M. Bromberg and C. Ulcigrai,
A temporal central limit theorem for real-valued cocycles over
rotations, Ann. Inst. Henri Poincaré Probab. Stat., 54 (2018), no. 4, 2304-2334.
doi: 10.1214/17-AIHP872. |
[5] |
J.-P. Conze,
Equirépartition et ergodicité de transformations cylindriques, Séminaire de
Probabilités, I (Univ. Rennes, Rennes), (1976), 1-21.
|
[6] |
J.-P. Conze and A. Piȩkniewska,
On multiple ergodicity of affine cocycles over irrational rotations, Israel J. Math., 201 (2014), no. 2,543-584.
doi: 10.1007/s11856-014-0033-3. |
[7] |
D. Dolgopyat and O. Sarig,
Temporal distributional limit theorems for dynamical systems, J. Stat. Phys., 166 (2017), no. 3-4,680-713.
doi: 10.1007/s10955-016-1689-3. |
[8] |
G. H. Hardy and E. M. Wright,
An Introduction to the Theory of Numbers, 3rd ed, Oxford, Clarendon Press, 1954. |
[9] |
H. Hennion and L. Hervé,
Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-Compactness, Lecture Notes in Mathematics, 1766, Springer-Verlag, Berlin, 2001.
doi: 10.1007/b87874. |
[10] |
M.-R. Herman,
Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Inst. Hautes Études Sci. Publ. Math., 49 (1979), 5-233.
doi: 10.1007/bf02684798. |
[11] |
Y. Katznelson,
Sigma-finite invariant measures for smooth mappings of the circle, J. Analyse Math., 31 (1977), 1-18.
doi: 10.1007/bf02813295. |
[12] |
M. Keane, Irrational rotations and quasi-ergodic measures, Publications des Séminaires de
Mathématiques (Univ. Rennes, Rennes), Fasc. 1: Probabilités, 1970, 17–26. |
[13] |
A. Ya. Khintchine,
Continued Fractions, translated by Peter Wynn, P. Noordhoff, Ltd., Groningen, 1963. |
[14] |
C. Kraaikamp and H. Nakada,
On normal numbers for continued fractions, Ergodic Theory Dynam. Systems, 20 (2000), no. 5, 1405-1421.
doi: 10.1017/S0143385700000766. |
[15] |
L. Kronecker,
Zwei Sätze über Gleichungen mit ganzzahligen Coefficienten, J. Reine Angew. Math., 53 (1857), 173-175.
doi: 10.1515/crll.1857.53.173. |
[16] |
I. Oren,
Ergodicity of cylinder flows arising from irregularities of distribution, Israel J. Math, 44 (1993), no. 2,127-138.
doi: 10.1007/BF02760616. |
[17] |
K. Schmidt Cocycles on Ergodic Transformation Groups, Macmillan Lectures in Mathematics, Vol. 1, Macmillan, 1977. |
[18] |
O. Taussky, Eigenvalues of finite matrices: some topics concerning bounds for eigenvalues of
finite matrices, Survey of Numerical Analysis (ed. J. Todd), 1962, McGraw-Hill, New York,
279–297. |
[1] |
James Nolen. A central limit theorem for pulled fronts in a random medium. Networks and Heterogeneous Media, 2011, 6 (2) : 167-194. doi: 10.3934/nhm.2011.6.167 |
[2] |
Oliver Díaz-Espinosa, Rafael de la Llave. Renormalization and central limit theorem for critical dynamical systems with weak external noise. Journal of Modern Dynamics, 2007, 1 (3) : 477-543. doi: 10.3934/jmd.2007.1.477 |
[3] |
Yves Derriennic. Some aspects of recent works on limit theorems in ergodic theory with special emphasis on the "central limit theorem''. Discrete and Continuous Dynamical Systems, 2006, 15 (1) : 143-158. doi: 10.3934/dcds.2006.15.143 |
[4] |
Jean-Pierre Conze, Stéphane Le Borgne, Mikaël Roger. Central limit theorem for stationary products of toral automorphisms. Discrete and Continuous Dynamical Systems, 2012, 32 (5) : 1597-1626. doi: 10.3934/dcds.2012.32.1597 |
[5] |
Shige Peng. Law of large numbers and central limit theorem under nonlinear expectations. Probability, Uncertainty and Quantitative Risk, 2019, 4 (0) : 4-. doi: 10.1186/s41546-019-0038-2 |
[6] |
Diogo Gomes, Levon Nurbekyan. An infinite-dimensional weak KAM theory via random variables. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6167-6185. doi: 10.3934/dcds.2016069 |
[7] |
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 |
[8] |
Mathias Staudigl. A limit theorem for Markov decision processes. Journal of Dynamics and Games, 2014, 1 (4) : 639-659. doi: 10.3934/jdg.2014.1.639 |
[9] |
Brendan Weickert. Infinite-dimensional complex dynamics: A quantum random walk. Discrete and Continuous Dynamical Systems, 2001, 7 (3) : 517-524. doi: 10.3934/dcds.2001.7.517 |
[10] |
Pablo D. Carrasco, Túlio Vales. A symmetric Random Walk defined by the time-one map of a geodesic flow. Discrete and Continuous Dynamical Systems, 2021, 41 (6) : 2891-2905. doi: 10.3934/dcds.2020390 |
[11] |
Krzysztof Barański. Hausdorff dimension of self-affine limit sets with an invariant direction. Discrete and Continuous Dynamical Systems, 2008, 21 (4) : 1015-1023. doi: 10.3934/dcds.2008.21.1015 |
[12] |
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 |
[13] |
Qingshan Yang, Xuerong Mao. Stochastic dynamics of SIRS epidemic models with random perturbation. Mathematical Biosciences & Engineering, 2014, 11 (4) : 1003-1025. doi: 10.3934/mbe.2014.11.1003 |
[14] |
Wen Huang, Leiye Xu, Shengnan Xu. Ergodic measures of intermediate entropy for affine transformations of nilmanifolds. Electronic Research Archive, 2021, 29 (4) : 2819-2827. doi: 10.3934/era.2021015 |
[15] |
Jory Griffin, Jens Marklof. Limit theorems for skew translations. Journal of Modern Dynamics, 2014, 8 (2) : 177-189. doi: 10.3934/jmd.2014.8.177 |
[16] |
Florian Dorsch, Hermann Schulz-Baldes. Random Möbius dynamics on the unit disc and perturbation theory for Lyapunov exponents. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 945-976. doi: 10.3934/dcdsb.2021076 |
[17] |
Wafa Hamrouni, Ali Abdennadher. Random walk's models for fractional diffusion equation. Discrete and Continuous Dynamical Systems - B, 2016, 21 (8) : 2509-2530. doi: 10.3934/dcdsb.2016058 |
[18] |
Edward Belbruno. Random walk in the three-body problem and applications. Discrete and Continuous Dynamical Systems - S, 2008, 1 (4) : 519-540. doi: 10.3934/dcdss.2008.1.519 |
[19] |
N. D. Cong, T. S. Doan, S. Siegmund. A Bohl-Perron type theorem for random dynamical systems. Conference Publications, 2011, 2011 (Special) : 322-331. doi: 10.3934/proc.2011.2011.322 |
[20] |
Michael Björklund, Alexander Gorodnik. Central limit theorems in the geometry of numbers. Electronic Research Announcements, 2017, 24: 110-122. doi: 10.3934/era.2017.24.012 |
2020 Impact Factor: 0.848
Tools
Metrics
Other articles
by authors
[Back to Top]