Limit theorems for skew translations

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

Limit theorems for skew translations

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

Received September 2013 Published November 2014

Abstract
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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. doi: 10.4007/annals.2014.179.2.2.
[2]	A. Bufetov and G. Forni, Limit theorems for horocycle flows,, , ().
[3]	A. Bufetov and B. Solomyak, Limit theorems for self-similar tilings,, Comm. Math. Phys., 319 (2013), 761. doi: 10.1007/s00220-012-1624-7.
[4]	D. Dolgopyat and B. Fayad, Deviations of ergodic sums for toral translations I. Convex bodies,, Geom. Funct. Anal., 24 (2014), 85. doi: 10.1007/s00039-014-0254-y.
[5]	D. Dolgopyat and B. Fayad, Deviations of ergodic sums for toral translations II. Boxes,, , ().
[6]	F. Cellarosi, Limiting curlicue measures for theta sums,, Ann. Inst. Henri Poincaré Probab. Stat., 47 (2011), 466. doi: 10.1214/10-AIHP361.
[7]	A. Fedotov and F. Klopp, An exact renormalization formula for Gaussian exponential sums and applications,, Amer. J. Math., 134 (2012), 711. doi: 10.1353/ajm.2012.0016.
[8]	L. Flaminio and G. Forni, Equidistribution of nilflows and applications to theta sums,, Ergodic Theory Dynam. Systems, 26 (2006), 409. doi: 10.1017/S014338570500060X.
[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. doi: 10.1215/S0012-7094-81-04848-1.
[10]	W. B. Jurkat and J. W. Van Horne, On the central limit theorem for theta series,, Michigan Math. J., 29 (1982), 65. doi: 10.1307/mmj/1029002615.
[11]	W. B. Jurkat and J. W. Van Horne, The uniform central limit theorem for theta sums,, Duke Math. J., 50 (1983), 649. doi: 10.1215/S0012-7094-83-05030-5.
[12]	H. Kesten, Uniform distribution mod 1,, Ann. of Math. (2), 71 (1960), 445. doi: 10.2307/1969938.
[13]	H. Kesten, Uniform distribution mod 1. II,, Acta Arith., 7 (): 355.
[14]	J. Marklof, Limit theorems for theta sums,, Duke Math. J., 97 (1999), 127. doi: 10.1215/S0012-7094-99-09706-5.
[15]	J. Marklof, Almost modular functions and the distribution of $n^2 x$ modulo one,, Int. Math. Res. Not., 39 (2003), 2131. doi: 10.1155/S1073792803130292.
[16]	J. Marklof, Pair correlation densities of inhomogeneous quadratic forms,, Ann. of Math. (2), 158 (2003), 419. doi: 10.4007/annals.2003.158.419.
[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, (2008), 317. doi: 10.1090/conm/469/09174.

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References:

[1]	A. Bufetov, Limit theorems for translation flows,, Ann. of Math. (2), 179 (2014), 431. doi: 10.4007/annals.2014.179.2.2.
[2]	A. Bufetov and G. Forni, Limit theorems for horocycle flows,, , ().
[3]	A. Bufetov and B. Solomyak, Limit theorems for self-similar tilings,, Comm. Math. Phys., 319 (2013), 761. doi: 10.1007/s00220-012-1624-7.
[4]	D. Dolgopyat and B. Fayad, Deviations of ergodic sums for toral translations I. Convex bodies,, Geom. Funct. Anal., 24 (2014), 85. doi: 10.1007/s00039-014-0254-y.
[5]	D. Dolgopyat and B. Fayad, Deviations of ergodic sums for toral translations II. Boxes,, , ().
[6]	F. Cellarosi, Limiting curlicue measures for theta sums,, Ann. Inst. Henri Poincaré Probab. Stat., 47 (2011), 466. doi: 10.1214/10-AIHP361.
[7]	A. Fedotov and F. Klopp, An exact renormalization formula for Gaussian exponential sums and applications,, Amer. J. Math., 134 (2012), 711. doi: 10.1353/ajm.2012.0016.
[8]	L. Flaminio and G. Forni, Equidistribution of nilflows and applications to theta sums,, Ergodic Theory Dynam. Systems, 26 (2006), 409. doi: 10.1017/S014338570500060X.
[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. doi: 10.1215/S0012-7094-81-04848-1.
[10]	W. B. Jurkat and J. W. Van Horne, On the central limit theorem for theta series,, Michigan Math. J., 29 (1982), 65. doi: 10.1307/mmj/1029002615.
[11]	W. B. Jurkat and J. W. Van Horne, The uniform central limit theorem for theta sums,, Duke Math. J., 50 (1983), 649. doi: 10.1215/S0012-7094-83-05030-5.
[12]	H. Kesten, Uniform distribution mod 1,, Ann. of Math. (2), 71 (1960), 445. doi: 10.2307/1969938.
[13]	H. Kesten, Uniform distribution mod 1. II,, Acta Arith., 7 (): 355.
[14]	J. Marklof, Limit theorems for theta sums,, Duke Math. J., 97 (1999), 127. doi: 10.1215/S0012-7094-99-09706-5.
[15]	J. Marklof, Almost modular functions and the distribution of $n^2 x$ modulo one,, Int. Math. Res. Not., 39 (2003), 2131. doi: 10.1155/S1073792803130292.
[16]	J. Marklof, Pair correlation densities of inhomogeneous quadratic forms,, Ann. of Math. (2), 158 (2003), 419. doi: 10.4007/annals.2003.158.419.
[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, (2008), 317. doi: 10.1090/conm/469/09174.

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