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

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.
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.

show all references

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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