-
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] |
Pablo D. Carrasco, Túlio Vales. A symmetric Random Walk defined by the time-one map of a geodesic flow. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020390 |
[2] |
Isabeau Birindelli, Françoise Demengel, Fabiana Leoni. Boundary asymptotics of the ergodic functions associated with fully nonlinear operators through a Liouville type theorem. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020395 |
[3] |
Peng Luo. Comparison theorem for diagonally quadratic BSDEs. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020374 |
[4] |
Bixiang Wang. Mean-square random invariant manifolds for stochastic differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1449-1468. doi: 10.3934/dcds.2020324 |
[5] |
Van Duong Dinh. Random data theory for the cubic fourth-order nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020284 |
[6] |
Timothy Chumley, Renato Feres. Entropy production in random billiards. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1319-1346. doi: 10.3934/dcds.2020319 |
[7] |
Yangrong Li, Shuang Yang, Qiangheng Zhang. Odd random attractors for stochastic non-autonomous Kuramoto-Sivashinsky equations without dissipation. Electronic Research Archive, 2020, 28 (4) : 1529-1544. doi: 10.3934/era.2020080 |
[8] |
Leanne Dong. Random attractors for stochastic Navier-Stokes equation on a 2D rotating sphere with stable Lévy noise. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020352 |
[9] |
Wenlong Sun, Jiaqi Cheng, Xiaoying Han. Random attractors for 2D stochastic micropolar fluid flows on unbounded domains. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 693-716. doi: 10.3934/dcdsb.2020189 |
[10] |
Xinfu Chen, Huiqiang Jiang, Guoqing Liu. Boundary spike of the singular limit of an energy minimizing problem. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3253-3290. doi: 10.3934/dcds.2020124 |
[11] |
Hai-Liang Li, Tong Yang, Mingying Zhong. Diffusion limit of the Vlasov-Poisson-Boltzmann system. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021003 |
[12] |
Hideki Murakawa. Fast reaction limit of reaction-diffusion systems. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1047-1062. doi: 10.3934/dcdss.2020405 |
[13] |
Bing Sun, Liangyun Chen, Yan Cao. On the universal $ \alpha $-central extensions of the semi-direct product of Hom-preLie algebras. Electronic Research Archive, , () : -. doi: 10.3934/era.2021004 |
[14] |
Patrick W. Dondl, Martin Jesenko. Threshold phenomenon for homogenized fronts in random elastic media. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 353-372. doi: 10.3934/dcdss.2020329 |
[15] |
Shuxing Chen, Jianzhong Min, Yongqian Zhang. Weak shock solution in supersonic flow past a wedge. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 115-132. doi: 10.3934/dcds.2009.23.115 |
[16] |
Dmitry Dolgopyat. The work of Sébastien Gouëzel on limit theorems and on weighted Banach spaces. Journal of Modern Dynamics, 2020, 16: 351-371. doi: 10.3934/jmd.2020014 |
[17] |
Meilan Cai, Maoan Han. Limit cycle bifurcations in a class of piecewise smooth cubic systems with multiple parameters. Communications on Pure & Applied Analysis, 2021, 20 (1) : 55-75. doi: 10.3934/cpaa.2020257 |
[18] |
Zhiting Ma. Navier-Stokes limit of globally hyperbolic moment equations. Kinetic & Related Models, 2021, 14 (1) : 175-197. doi: 10.3934/krm.2021001 |
[19] |
Shiqi Ma. On recent progress of single-realization recoveries of random Schrödinger systems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020121 |
[20] |
Josselin Garnier, Knut Sølna. Enhanced Backscattering of a partially coherent field from an anisotropic random lossy medium. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1171-1195. doi: 10.3934/dcdsb.2020158 |
2019 Impact Factor: 0.465
Tools
Metrics
Other articles
by authors
[Back to Top]