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October  2010, 4(4): 609-635. doi: 10.3934/jmd.2010.4.609

## Ratner's property and mild mixing for special flows over two-dimensional rotations

 1 Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12/18, 87-100 Toruń 2 Faculty of Mathematics and Computer Science, N. Copernicus University, ul. Chopina 12/18, 87-100 Toruń, Poland

Received  February 2010 Revised  December 2010 Published  January 2011

We consider special flows over two-dimensional rotations by $(\alpha,\beta)$ on $\T^2$ and under piecewise $C^2$ roof functions $f$ satisfying von Neumann's condition $\int_{\T^2}f_x(x,y)dxdy\ne 0$ or $\int_{\T^2}f_y(x,y)dxdy\ne 0$. Such flows are shown to be always weakly mixing and never partially rigid. It is proved that while specifying to a subclass of roof functions and to ergodic rotations for which $\alpha$ and $\beta$ are of bounded partial quotients the corresponding special flows enjoy the so-called weak Ratner property. As a consequence, such flows turn out to be mildly mixing.
Citation: Krzysztof Frączek, Mariusz Lemańczyk. Ratner's property and mild mixing for special flows over two-dimensional rotations. Journal of Modern Dynamics, 2010, 4 (4) : 609-635. doi: 10.3934/jmd.2010.4.609
##### References:
 [1] J.-P. Allouche and J. Shallit, "Automatic Sequences. Theory, Applications, Generalizations,", Cambridge Univ. Press, (2003).  doi: 10.1017/CBO9780511546563.  Google Scholar [2] V. I. Arnold, Topological and ergodic properties of closed 1-forms with incommensurable periods,, (Russian) Funktsional. Anal. i Prilozhen., 25 (1991), 1.   Google Scholar [3] I. P. Cornfeld, S. V. Fomin and Y. G. Sinai, "Ergodic Theory,", Translated from the Russian by A. B. Sosinskii. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], (1982).   Google Scholar [4] B. Fayad, Polynomial decay of correlations for a class of smooth flows on the two torus,, Bull. Soc. Math. France, 129 (2001), 487.   Google Scholar [5] B. Fayad, Analytic mixing reparametrizations of irrational flows,, Ergodic Theory Dynam. Systems, 22 (2002), 437.  doi: 10.1017/S0143385702000214.  Google Scholar [6] B. Fayad, Smooth mixing flows with purely singular spectra,, Duke Math. J., 132 (2006), 371.  doi: 10.1215/S0012-7094-06-13225-8.  Google Scholar [7] K. Frączek and M. Lemańczyk, A class of special flows over irrational rotations which is disjoint from mixing flows,, Ergodic Theory Dynam. Systems, 24 (2004), 1083.  doi: 10.1017/S0143385704000112.  Google Scholar [8] K. Frączek and M. Lemańczyk, On mild mixing of special flows over irrational rotations under piecewise smooth functions,, Ergodic Theory Dynam. Systems, 26 (2006), 719.  doi: 10.1017/S0143385706000046.  Google Scholar [9] K. Frączek, M. Lemańczyk and E. Lesigne, Mild mixing property for special flows under piecewise constant functions,, Discrete Contin. Dynam. Syst., 19 (2007), 691.  doi: 10.3934/dcds.2007.19.691.  Google Scholar [10] K. Frączek and M. Lemańczyk, On the self-similarity problem for ergodic flows,, Proc. London Math. Soc., 99 (2009), 658.  doi: 10.1112/plms/pdp013.  Google Scholar [11] K. Frączek and M. Lemańczyk, A class of mixing special flows over two-dimensional rotations,, submitted., ().   Google Scholar [12] K. Frączek and M. Lemańczyk, Ratner's property and mixing for special flows over two-dimensional rotations,, \arXiv{1002.2734}., ().   Google Scholar [13] H. Furstenberg and B. Weiss, The finite multipliers of infinite ergodic transformations. The structure of attractors in dynamical systems,, (Proc. Conf., (1977), 127.   Google Scholar [14] B. Host, Mixing of all orders and pairwise independent joinings of systems with singular spectrum,, Israel J. Math., 76 (1991), 289.  doi: 10.1007/BF02773866.  Google Scholar [15] A. Iwanik, M. Lemańczyk and C. Mauduit, Piecewise absolutely continuous cocycles over irrational rotations,, J. London Math. Soc. (2), 59 (1999), 171.  doi: 10.1112/S0024610799006961.  Google Scholar [16] A. Katok, Cocycles, cohomology and combinatorial constructions in ergodic theory,, In collaboration with E. A. Robinson, (1999), 107.   Google Scholar [17] A. Katok and B. Hasselblatt, Introduction to the modern theory of dynamical systems,, With a supplementary chapter by Katok and Leonardo Mendoza. Encyclopedia of Mathematics and its Applications, (1995).   Google Scholar [18] K. M. Khanin and Y. G. Sinai, Mixing of some classes of special flows over rotations of the circle,, (Russian) Funktsional. Anal. i Prilozhen., 26 (1992), 1.   Google Scholar [19] Y. Khinchin, "Continued Fractions,", The University of Chicago Press, (1964).   Google Scholar [20] A. V. Kochergin, The absence of mixing in special flows over a rotation of the circle and in flows on a two-dimensional torus,, (Russian) Dokl. Akad. Nauk SSSR, 205 (1972), 515.   Google Scholar [21] A. V. Kochergin, Mixing in special flows over a rearrangement of segments and in smooth flows on surfaces,, (Russian) Mat. Sb., 96 (1975), 471.   Google Scholar [22] A. V. Kochergin, Non-degenerated saddles and absence of mixing,, (Russian) Mat. Zametki, 19 (1976), 453.   Google Scholar [23] A. V. Kochergin, A mixing special flow over a rotation of the circle with an almost Lipschitz function,, (Russian) Mat. Sb., 193 (2002), 51.   Google Scholar [24] A. V. Kochergin, Nondegenerate fixed points and mixing in flows on a two-dimensional torus. II,, (Russian) Mat. Sb., 195 (2004), 15.   Google Scholar [25] A. V. Kochergin, Causes of stretching of Birkhoff sums and mixing in flows on surfaces,, Dynamics, (2007), 129.   Google Scholar [26] M. Lemańczyk, Sur l'absence de mélange pour des flots spéciaux au dessus d'une rotation irrationnelle,, (French) [Absence of mixing for special flows over an irrational rotation] Dedicated to the memory of Anzelm Iwanik. Colloq. Math., 84/85 (2000), 29.   Google Scholar [27] J. von Neumann, Zur Operatorenmethode in der Klassichen Mechanik,, (German), 33 (1932), 587.  doi: 10.2307/1968537.  Google Scholar [28] M. Ratner, Horocycle flows, joinings and rigidity of products,, Ann. of Math. (2), 118 (1983), 277.  doi: 10.2307/2007030.  Google Scholar [29] V. V. Ryzhikov and J.-P. Thouvenot, Disjointness, divisibility, and quasi-simplicity of measure-preserving actions,, (Russian) Funktsional. Anal. i Prilozhen., 40 (2006), 85.   Google Scholar [30] J.-P. Thouvenot, Some properties and applications of joinings in ergodic theory,, Ergodic theory and its connections with harmonic analysis (Alexandria, (1993), 207.   Google Scholar [31] K. Schmidt, Dispersing cocycles and mixing flows under functions,, Fund. Math., 173 (2002), 191.  doi: 10.4064/fm173-2-6.  Google Scholar [32] D. Witte, Rigidity of some translations on homogeneous spaces,, Invent. Math., 81 (1985), 1.  doi: 10.1007/BF01388769.  Google Scholar [33] J.-Ch. Yoccoz, Centralisateurs et conjugaison différentiable des difféomorphismes du cercle,, (French) [Centralizers and differentiable conjugacy of diffeomorphisms of the circle] Petits diviseurs en dimension $1$. Astérisque No. 231, (1995), 89.   Google Scholar

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##### References:
 [1] J.-P. Allouche and J. Shallit, "Automatic Sequences. Theory, Applications, Generalizations,", Cambridge Univ. Press, (2003).  doi: 10.1017/CBO9780511546563.  Google Scholar [2] V. I. Arnold, Topological and ergodic properties of closed 1-forms with incommensurable periods,, (Russian) Funktsional. Anal. i Prilozhen., 25 (1991), 1.   Google Scholar [3] I. P. Cornfeld, S. V. Fomin and Y. G. Sinai, "Ergodic Theory,", Translated from the Russian by A. B. Sosinskii. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], (1982).   Google Scholar [4] B. Fayad, Polynomial decay of correlations for a class of smooth flows on the two torus,, Bull. Soc. Math. France, 129 (2001), 487.   Google Scholar [5] B. Fayad, Analytic mixing reparametrizations of irrational flows,, Ergodic Theory Dynam. Systems, 22 (2002), 437.  doi: 10.1017/S0143385702000214.  Google Scholar [6] B. Fayad, Smooth mixing flows with purely singular spectra,, Duke Math. J., 132 (2006), 371.  doi: 10.1215/S0012-7094-06-13225-8.  Google Scholar [7] K. Frączek and M. Lemańczyk, A class of special flows over irrational rotations which is disjoint from mixing flows,, Ergodic Theory Dynam. Systems, 24 (2004), 1083.  doi: 10.1017/S0143385704000112.  Google Scholar [8] K. Frączek and M. Lemańczyk, On mild mixing of special flows over irrational rotations under piecewise smooth functions,, Ergodic Theory Dynam. Systems, 26 (2006), 719.  doi: 10.1017/S0143385706000046.  Google Scholar [9] K. Frączek, M. Lemańczyk and E. Lesigne, Mild mixing property for special flows under piecewise constant functions,, Discrete Contin. Dynam. Syst., 19 (2007), 691.  doi: 10.3934/dcds.2007.19.691.  Google Scholar [10] K. Frączek and M. Lemańczyk, On the self-similarity problem for ergodic flows,, Proc. London Math. Soc., 99 (2009), 658.  doi: 10.1112/plms/pdp013.  Google Scholar [11] K. Frączek and M. Lemańczyk, A class of mixing special flows over two-dimensional rotations,, submitted., ().   Google Scholar [12] K. Frączek and M. Lemańczyk, Ratner's property and mixing for special flows over two-dimensional rotations,, \arXiv{1002.2734}., ().   Google Scholar [13] H. Furstenberg and B. Weiss, The finite multipliers of infinite ergodic transformations. The structure of attractors in dynamical systems,, (Proc. Conf., (1977), 127.   Google Scholar [14] B. Host, Mixing of all orders and pairwise independent joinings of systems with singular spectrum,, Israel J. Math., 76 (1991), 289.  doi: 10.1007/BF02773866.  Google Scholar [15] A. Iwanik, M. Lemańczyk and C. Mauduit, Piecewise absolutely continuous cocycles over irrational rotations,, J. London Math. Soc. (2), 59 (1999), 171.  doi: 10.1112/S0024610799006961.  Google Scholar [16] A. Katok, Cocycles, cohomology and combinatorial constructions in ergodic theory,, In collaboration with E. A. Robinson, (1999), 107.   Google Scholar [17] A. Katok and B. Hasselblatt, Introduction to the modern theory of dynamical systems,, With a supplementary chapter by Katok and Leonardo Mendoza. Encyclopedia of Mathematics and its Applications, (1995).   Google Scholar [18] K. M. Khanin and Y. G. Sinai, Mixing of some classes of special flows over rotations of the circle,, (Russian) Funktsional. Anal. i Prilozhen., 26 (1992), 1.   Google Scholar [19] Y. Khinchin, "Continued Fractions,", The University of Chicago Press, (1964).   Google Scholar [20] A. V. Kochergin, The absence of mixing in special flows over a rotation of the circle and in flows on a two-dimensional torus,, (Russian) Dokl. Akad. Nauk SSSR, 205 (1972), 515.   Google Scholar [21] A. V. Kochergin, Mixing in special flows over a rearrangement of segments and in smooth flows on surfaces,, (Russian) Mat. Sb., 96 (1975), 471.   Google Scholar [22] A. V. Kochergin, Non-degenerated saddles and absence of mixing,, (Russian) Mat. Zametki, 19 (1976), 453.   Google Scholar [23] A. V. Kochergin, A mixing special flow over a rotation of the circle with an almost Lipschitz function,, (Russian) Mat. Sb., 193 (2002), 51.   Google Scholar [24] A. V. Kochergin, Nondegenerate fixed points and mixing in flows on a two-dimensional torus. II,, (Russian) Mat. Sb., 195 (2004), 15.   Google Scholar [25] A. V. Kochergin, Causes of stretching of Birkhoff sums and mixing in flows on surfaces,, Dynamics, (2007), 129.   Google Scholar [26] M. Lemańczyk, Sur l'absence de mélange pour des flots spéciaux au dessus d'une rotation irrationnelle,, (French) [Absence of mixing for special flows over an irrational rotation] Dedicated to the memory of Anzelm Iwanik. Colloq. Math., 84/85 (2000), 29.   Google Scholar [27] J. von Neumann, Zur Operatorenmethode in der Klassichen Mechanik,, (German), 33 (1932), 587.  doi: 10.2307/1968537.  Google Scholar [28] M. Ratner, Horocycle flows, joinings and rigidity of products,, Ann. of Math. (2), 118 (1983), 277.  doi: 10.2307/2007030.  Google Scholar [29] V. V. Ryzhikov and J.-P. Thouvenot, Disjointness, divisibility, and quasi-simplicity of measure-preserving actions,, (Russian) Funktsional. Anal. i Prilozhen., 40 (2006), 85.   Google Scholar [30] J.-P. Thouvenot, Some properties and applications of joinings in ergodic theory,, Ergodic theory and its connections with harmonic analysis (Alexandria, (1993), 207.   Google Scholar [31] K. Schmidt, Dispersing cocycles and mixing flows under functions,, Fund. Math., 173 (2002), 191.  doi: 10.4064/fm173-2-6.  Google Scholar [32] D. Witte, Rigidity of some translations on homogeneous spaces,, Invent. Math., 81 (1985), 1.  doi: 10.1007/BF01388769.  Google Scholar [33] J.-Ch. Yoccoz, Centralisateurs et conjugaison différentiable des difféomorphismes du cercle,, (French) [Centralizers and differentiable conjugacy of diffeomorphisms of the circle] Petits diviseurs en dimension $1$. Astérisque No. 231, (1995), 89.   Google Scholar
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