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April  2014, 8(2): 177-189. doi: 10.3934/jmd.2014.8.177

Limit theorems for skew translations

Jory Griffin 1, and  Jens Marklof 1,

1. 

School of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom

Received  September 2013 Published  November 2014

Bufetov, Bufetov-Forni and Bufetov-Solomyak have recently proved limit theorems for translation flows, horocycle flows and tiling flows, respectively. We present here analogous results for skew translations of a torus.
Keywords: theta series., skew translation, Central limit theorem.
Mathematics Subject Classification: Primary: 37A50; Secondary: 11F27, 60F05, 37D40, 28D0.
Citation: 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
References:
[1]

A. Bufetov, Limit theorems for translation flows, Ann. of Math. (2), 179 (2014), 431-499. doi: 10.4007/annals.2014.179.2.2.  Google Scholar

[2]

A. Bufetov and G. Forni, Limit theorems for horocycle flows,, , ().   Google Scholar

[3]

A. Bufetov and B. Solomyak, Limit theorems for self-similar tilings, Comm. Math. Phys., 319 (2013), 761-789. doi: 10.1007/s00220-012-1624-7.  Google Scholar

[4]

D. Dolgopyat and B. Fayad, Deviations of ergodic sums for toral translations I. Convex bodies, Geom. Funct. Anal., 24 (2014), 85-115. doi: 10.1007/s00039-014-0254-y.  Google Scholar

[5]

D. Dolgopyat and B. Fayad, Deviations of ergodic sums for toral translations II. Boxes,, , ().   Google Scholar

[6]

F. Cellarosi, Limiting curlicue measures for theta sums, Ann. Inst. Henri Poincaré Probab. Stat., 47 (2011), 466-497. doi: 10.1214/10-AIHP361.  Google Scholar

[7]

A. Fedotov and F. Klopp, An exact renormalization formula for Gaussian exponential sums and applications, Amer. J. Math., 134 (2012), 711-748. doi: 10.1353/ajm.2012.0016.  Google Scholar

[8]

L. Flaminio and G. Forni, Equidistribution of nilflows and applications to theta sums, Ergodic Theory Dynam. Systems, 26 (2006), 409-433. doi: 10.1017/S014338570500060X.  Google Scholar

[9]

W. B. Jurkat and J. W. Van Horne, The proof of the central limit theorem for theta sums, Duke Math. J., 48 (1981), 873-885. doi: 10.1215/S0012-7094-81-04848-1.  Google Scholar

[10]

W. B. Jurkat and J. W. Van Horne, On the central limit theorem for theta series, Michigan Math. J., 29 (1982), 65-77. doi: 10.1307/mmj/1029002615.  Google Scholar

[11]

W. B. Jurkat and J. W. Van Horne, The uniform central limit theorem for theta sums, Duke Math. J., 50 (1983), 649-666. doi: 10.1215/S0012-7094-83-05030-5.  Google Scholar

[12]

H. Kesten, Uniform distribution mod 1, Ann. of Math. (2), 71 (1960), 445-471. doi: 10.2307/1969938.  Google Scholar

[13]

H. Kesten, Uniform distribution mod 1. II,, Acta Arith., 7 (): 355.   Google Scholar

[14]

J. Marklof, Limit theorems for theta sums, Duke Math. J., 97 (1999), 127-153. doi: 10.1215/S0012-7094-99-09706-5.  Google Scholar

[15]

J. Marklof, Almost modular functions and the distribution of $n^2 x$ modulo one, Int. Math. Res. Not., 39 (2003), 2131-2151. doi: 10.1155/S1073792803130292.  Google Scholar

[16]

J. Marklof, Pair correlation densities of inhomogeneous quadratic forms, Ann. of Math. (2), 158 (2003), 419-471. doi: 10.4007/annals.2003.158.419.  Google Scholar

[17]

Y. G. Sinai and C. Ulcigrai, A limit theorem for Birkhoff sums of nonintegrable functions over rotations, in Geometric and Probabilistic Structures in Dynamics, Contemp. Math., 469, Amer. Math. Soc., Providence, RI, 2008, 317-340. doi: 10.1090/conm/469/09174.  Google Scholar

show all references

References:
[1]

A. Bufetov, Limit theorems for translation flows, Ann. of Math. (2), 179 (2014), 431-499. doi: 10.4007/annals.2014.179.2.2.  Google Scholar

[2]

A. Bufetov and G. Forni, Limit theorems for horocycle flows,, , ().   Google Scholar

[3]

A. Bufetov and B. Solomyak, Limit theorems for self-similar tilings, Comm. Math. Phys., 319 (2013), 761-789. doi: 10.1007/s00220-012-1624-7.  Google Scholar

[4]

D. Dolgopyat and B. Fayad, Deviations of ergodic sums for toral translations I. Convex bodies, Geom. Funct. Anal., 24 (2014), 85-115. doi: 10.1007/s00039-014-0254-y.  Google Scholar

[5]

D. Dolgopyat and B. Fayad, Deviations of ergodic sums for toral translations II. Boxes,, , ().   Google Scholar

[6]

F. Cellarosi, Limiting curlicue measures for theta sums, Ann. Inst. Henri Poincaré Probab. Stat., 47 (2011), 466-497. doi: 10.1214/10-AIHP361.  Google Scholar

[7]

A. Fedotov and F. Klopp, An exact renormalization formula for Gaussian exponential sums and applications, Amer. J. Math., 134 (2012), 711-748. doi: 10.1353/ajm.2012.0016.  Google Scholar

[8]

L. Flaminio and G. Forni, Equidistribution of nilflows and applications to theta sums, Ergodic Theory Dynam. Systems, 26 (2006), 409-433. doi: 10.1017/S014338570500060X.  Google Scholar

[9]

W. B. Jurkat and J. W. Van Horne, The proof of the central limit theorem for theta sums, Duke Math. J., 48 (1981), 873-885. doi: 10.1215/S0012-7094-81-04848-1.  Google Scholar

[10]

W. B. Jurkat and J. W. Van Horne, On the central limit theorem for theta series, Michigan Math. J., 29 (1982), 65-77. doi: 10.1307/mmj/1029002615.  Google Scholar

[11]

W. B. Jurkat and J. W. Van Horne, The uniform central limit theorem for theta sums, Duke Math. J., 50 (1983), 649-666. doi: 10.1215/S0012-7094-83-05030-5.  Google Scholar

[12]

H. Kesten, Uniform distribution mod 1, Ann. of Math. (2), 71 (1960), 445-471. doi: 10.2307/1969938.  Google Scholar

[13]

H. Kesten, Uniform distribution mod 1. II,, Acta Arith., 7 (): 355.   Google Scholar

[14]

J. Marklof, Limit theorems for theta sums, Duke Math. J., 97 (1999), 127-153. doi: 10.1215/S0012-7094-99-09706-5.  Google Scholar

[15]

J. Marklof, Almost modular functions and the distribution of $n^2 x$ modulo one, Int. Math. Res. Not., 39 (2003), 2131-2151. doi: 10.1155/S1073792803130292.  Google Scholar

[16]

J. Marklof, Pair correlation densities of inhomogeneous quadratic forms, Ann. of Math. (2), 158 (2003), 419-471. doi: 10.4007/annals.2003.158.419.  Google Scholar

[17]

Y. G. Sinai and C. Ulcigrai, A limit theorem for Birkhoff sums of nonintegrable functions over rotations, in Geometric and Probabilistic Structures in Dynamics, Contemp. Math., 469, Amer. Math. Soc., Providence, RI, 2008, 317-340. doi: 10.1090/conm/469/09174.  Google Scholar

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