American Institute of Mathematical Sciences

Advanced
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.  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.  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.  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.  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.  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.  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.  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.  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.  doi: 10.1215/S0012-7094-83-05030-5.  Google Scholar

[12]

H. Kesten, Uniform distribution mod 1,, Ann. of Math. (2), 71 (1960), 445.  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.  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.  doi: 10.1155/S1073792803130292.  Google Scholar

[16]

J. Marklof, Pair correlation densities of inhomogeneous quadratic forms,, Ann. of Math. (2), 158 (2003), 419.  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, (2008), 317.  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.  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.  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.  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.  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.  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.  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.  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.  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.  doi: 10.1215/S0012-7094-83-05030-5.  Google Scholar

[12]

H. Kesten, Uniform distribution mod 1,, Ann. of Math. (2), 71 (1960), 445.  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.  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.  doi: 10.1155/S1073792803130292.  Google Scholar

[16]

J. Marklof, Pair correlation densities of inhomogeneous quadratic forms,, Ann. of Math. (2), 158 (2003), 419.  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, (2008), 317.  doi: 10.1090/conm/469/09174.  Google Scholar

[1]

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
[2]

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
[3]

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
[4]

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
[5]

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
[6]

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
[7]

Kariane Calta, John Smillie. Algebraically periodic translation surfaces. Journal of Modern Dynamics, 2008, 2 (2) : 209-248. doi: 10.3934/jmd.2008.2.209
[8]

José A. Conejero, Alfredo Peris. Chaotic translation semigroups. Conference Publications, 2007, 2007 (Special) : 269-276. doi: 10.3934/proc.2007.2007.269
[9]

Martin Swaczyna, Petr Volný. Uniform motions in central fields. Journal of Geometric Mechanics, 2017, 9 (1) : 91-130. doi: 10.3934/jgm.2017004
[10]

Sergey Kryzhevich, Sergey Tikhomirov. Partial hyperbolicity and central shadowing. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2901-2909. doi: 10.3934/dcds.2013.33.2901
[11]

Benjamin Dozier. Equidistribution of saddle connections on translation surfaces. Journal of Modern Dynamics, 2019, 14: 87-120. doi: 10.3934/jmd.2019004
[12]

Armengol Gasull, Francesc Mañosas. Subseries and signed series. Communications on Pure & Applied Analysis, 2019, 18 (1) : 479-492. doi: 10.3934/cpaa.2019024
[13]

Roy Adler, Bruce Kitchens, Michael Shub. Stably ergodic skew products. Discrete & Continuous Dynamical Systems - A, 1996, 2 (3) : 349-350. doi: 10.3934/dcds.1996.2.349
[14]

Pascal Hubert, Gabriela Schmithüsen. Infinite translation surfaces with infinitely generated Veech groups. Journal of Modern Dynamics, 2010, 4 (4) : 715-732. doi: 10.3934/jmd.2010.4.715
[15]

Wolf-Jürgen Beyn, Raphael Kruse. Two-sided error estimates for the stochastic theta method. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 389-407. doi: 10.3934/dcdsb.2010.14.389
[16]

Weidong Zhao, Jinlei Wang, Shige Peng. Error estimates of the $\theta$-scheme for backward stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2009, 12 (4) : 905-924. doi: 10.3934/dcdsb.2009.12.905
[17]

Bram van Asch, Frans Martens. Lee weight enumerators of self-dual codes and theta functions. Advances in Mathematics of Communications, 2008, 2 (4) : 393-402. doi: 10.3934/amc.2008.2.393
[18]

Weidong Zhao, Yang Li, Guannan Zhang. A generalized $\theta$-scheme for solving backward stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1585-1603. doi: 10.3934/dcdsb.2012.17.1585
[19]

Roy Adler, Bruce Kitchens, Michael Shub. Errata to "Stably ergodic skew products". Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 456-456. doi: 10.3934/dcds.1999.5.456
[20]

Alexis Eduardo Almendras Valdebenito, Andrea Luigi Tironi. On the dual codes of skew constacyclic codes. Advances in Mathematics of Communications, 2018, 12 (4) : 659-679. doi: 10.3934/amc.2018039

2018 Impact Factor: 0.295

Tools

Metrics

  • PDF downloads (15)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences