American Institute of Mathematical Sciences

Advanced
  2018, 13: 1-42. doi: 10.3934/jmd.2018012

Rational ergodicity of step function skew products

Jon Aaronson 1, , Michael Bromberg 2, and  Nishant Chandgotia 1,

1. 

School of Mathematical Sciences, Tel Aviv University, 69978 Tel Aviv, Israel
2. 

School of Mathematics, Bristol University, Bristol BS8 1TW, UK

Dedicated to the memory of Roy Adler
JA: Partially supported by ISF grant No. 1599/13.
MB: Supported by ERC Grant Agreement n. 335989.
NC: Partially supported by ISF grant No. 1599/13 and ERC grant No. 678520.

Received  March 31, 2017 Revised  October 11, 2017 Published  December 2018

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.

Keywords: Infinite ergodic theory, skew product, step function, cylinder flow, renormalization, random affine transformation, affine random walk, stochastic matrix, perturbation, temporal central limit theorem, weak rough local limit theorem.
Mathematics Subject Classification: Primary: 37A40; Secondary: 11K38, 60F05.
Citation: Jon Aaronson, Michael Bromberg, Nishant Chandgotia. Rational ergodicity of step function skew products. Journal of Modern Dynamics, 2018, 13: 1-42. doi: 10.3934/jmd.2018012
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

J.-P. Conze, Equirépartition et ergodicité de transformations cylindriques, Séminaire de Probabilités, I (Univ. Rennes, Rennes), (1976), 1-21.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 3rd ed, Oxford, Clarendon Press, 1954.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

Y. Katznelson, Sigma-finite invariant measures for smooth mappings of the circle, J. Analyse Math., 31 (1977), 1-18.  doi: 10.1007/bf02813295.  Google Scholar

[12]

M. Keane, Irrational rotations and quasi-ergodic measures, Publications des Séminaires de Mathématiques (Univ. Rennes, Rennes), Fasc. 1: Probabilités, 1970, 17–26.  Google Scholar

[13]

A. Ya. Khintchine, Continued Fractions, translated by Peter Wynn, P. Noordhoff, Ltd., Groningen, 1963.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

K. Schmidt Cocycles on Ergodic Transformation Groups, Macmillan Lectures in Mathematics, Vol. 1, Macmillan, 1977.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

J.-P. Conze, Equirépartition et ergodicité de transformations cylindriques, Séminaire de Probabilités, I (Univ. Rennes, Rennes), (1976), 1-21.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 3rd ed, Oxford, Clarendon Press, 1954.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

Y. Katznelson, Sigma-finite invariant measures for smooth mappings of the circle, J. Analyse Math., 31 (1977), 1-18.  doi: 10.1007/bf02813295.  Google Scholar

[12]

M. Keane, Irrational rotations and quasi-ergodic measures, Publications des Séminaires de Mathématiques (Univ. Rennes, Rennes), Fasc. 1: Probabilités, 1970, 17–26.  Google Scholar

[13]

A. Ya. Khintchine, Continued Fractions, translated by Peter Wynn, P. Noordhoff, Ltd., Groningen, 1963.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

K. Schmidt Cocycles on Ergodic Transformation Groups, Macmillan Lectures in Mathematics, Vol. 1, Macmillan, 1977.  Google Scholar

[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.  Google Scholar

[1]

James Nolen. A central limit theorem for pulled fronts in a random medium. Networks & 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 & Continuous Dynamical Systems - A, 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 & Continuous Dynamical Systems - A, 2012, 32 (5) : 1597-1626. doi: 10.3934/dcds.2012.32.1597
[5]

Diogo Gomes, Levon Nurbekyan. An infinite-dimensional weak KAM theory via random variables. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6167-6185. doi: 10.3934/dcds.2016069
[6]

Weigu Li, Kening Lu. Takens theorem for random dynamical systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3191-3207. doi: 10.3934/dcdsb.2016093
[7]

Mathias Staudigl. A limit theorem for Markov decision processes. Journal of Dynamics & Games, 2014, 1 (4) : 639-659. doi: 10.3934/jdg.2014.1.639
[8]

Brendan Weickert. Infinite-dimensional complex dynamics: A quantum random walk. Discrete & Continuous Dynamical Systems - A, 2001, 7 (3) : 517-524. doi: 10.3934/dcds.2001.7.517
[9]

Krzysztof Barański. Hausdorff dimension of self-affine limit sets with an invariant direction. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 1015-1023. doi: 10.3934/dcds.2008.21.1015
[10]

Weigu Li, Kening Lu. A Siegel theorem for dynamical systems under random perturbations. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 635-642. doi: 10.3934/dcdsb.2008.9.635
[11]

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
[12]

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
[13]

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
[14]

Wafa Hamrouni, Ali Abdennadher. Random walk's models for fractional diffusion equation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2509-2530. doi: 10.3934/dcdsb.2016058
[15]

Edward Belbruno. Random walk in the three-body problem and applications. Discrete & Continuous Dynamical Systems - S, 2008, 1 (4) : 519-540. doi: 10.3934/dcdss.2008.1.519
[16]

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
[17]

Wei Wang, Yan Lv. Limit behavior of nonlinear stochastic wave equations with singular perturbation. Discrete & Continuous Dynamical Systems - B, 2010, 13 (1) : 175-193. doi: 10.3934/dcdsb.2010.13.175
[18]

Tomasz Komorowski, Łukasz Stȩpień. Kinetic limit for a harmonic chain with a conservative Ornstein-Uhlenbeck stochastic perturbation. Kinetic & Related Models, 2018, 11 (2) : 239-278. doi: 10.3934/krm.2018013
[19]

Núria Fagella, Àngel Jorba, Marc Jorba-Cuscó, Joan Carles Tatjer. Classification of linear skew-products of the complex plane and an affine route to fractalization. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3767-3787. doi: 10.3934/dcds.2019153
[20]

Julia Brettschneider. On uniform convergence in ergodic theorems for a class of skew product transformations. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 873-891. doi: 10.3934/dcds.2011.29.873

2018 Impact Factor: 0.295

Tools

Metrics

  • PDF downloads (71)
  • HTML views (554)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences