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  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

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