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

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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

Krzysztof Frączek ^1, and Mariusz Lemańczyk ^2,

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

Abstract
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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.

Keywords: Ratner's property., Measure-preserving flows, mild mixing, special flows.

Mathematics Subject Classification: Primary: 37A10; Secondary: 37C4.

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.
[2]	V. I. Arnold, Topological and ergodic properties of closed 1-forms with incommensurable periods, (Russian) Funktsional. Anal. i Prilozhen., 25 (1991), 1-12, 96; translation in Funct. Anal. Appl., 25 (1991), 81-90.
[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], 245, Springer-Verlag, New York, 1982.
[4]	B. Fayad, Polynomial decay of correlations for a class of smooth flows on the two torus, Bull. Soc. Math. France, 129 (2001), 487-503.
[5]	B. Fayad, Analytic mixing reparametrizations of irrational flows, Ergodic Theory Dynam. Systems, 22 (2002), 437-468. doi: 10.1017/S0143385702000214.
[6]	B. Fayad, Smooth mixing flows with purely singular spectra, Duke Math. J., 132 (2006), 371-391. doi: 10.1215/S0012-7094-06-13225-8.
[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), 1083-1095. doi: 10.1017/S0143385704000112.
[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), 719-738. doi: 10.1017/S0143385706000046.
[9]	K. Frączek, M. Lemańczyk and E. Lesigne, Mild mixing property for special flows under piecewise constant functions, Discrete Contin. Dynam. Syst., 19 (2007), 691-710. doi: 10.3934/dcds.2007.19.691.
[10]	K. Frączek and M. Lemańczyk, On the self-similarity problem for ergodic flows, Proc. London Math. Soc., 99 (2009), 658-696. doi: 10.1112/plms/pdp013.
[11]	K. Frączek and M. Lemańczyk, A class of mixing special flows over two-dimensional rotations,, submitted., ().
[12]	K. Frączek and M. Lemańczyk, Ratner's property and mixing for special flows over two-dimensional rotations,, \arXiv{1002.2734}., ().
[13]	H. Furstenberg and B. Weiss, The finite multipliers of infinite ergodic transformations. The structure of attractors in dynamical systems, (Proc. Conf., North Dakota State Univ., Fargo, N.D., 1977), 127-132, Lecture Notes in Math., 668, Springer, Berlin, 1978.
[14]	B. Host, Mixing of all orders and pairwise independent joinings of systems with singular spectrum, Israel J. Math., 76 (1991), 289-298. doi: 10.1007/BF02773866.
[15]	A. Iwanik, M. Lemańczyk and C. Mauduit, Piecewise absolutely continuous cocycles over irrational rotations, J. London Math. Soc. (2), 59 (1999), 171-187. doi: 10.1112/S0024610799006961.
[16]	A. Katok, Cocycles, cohomology and combinatorial constructions in ergodic theory, In collaboration with E. A. Robinson, Jr. Proc. Sympos. Pure Math., 69, Smooth ergodic theory and its applications (Seattle, WA, 1999), 107-173, Amer. Math. Soc., Providence, RI, 2001.
[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, 54. Cambridge University Press, Cambridge, 1995.
[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), 1-21; translation in Funct. Anal. Appl., 26 (1992), 155-169.
[19]	Y. Khinchin, "Continued Fractions," The University of Chicago Press, Chicago, Ill.-London 1964.
[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), 515-518.
[21]	A. V. Kochergin, Mixing in special flows over a rearrangement of segments and in smooth flows on surfaces, (Russian) Mat. Sb., 96 (1975), 471-502.
[22]	A. V. Kochergin, Non-degenerated saddles and absence of mixing, (Russian) Mat. Zametki, 19 (1976), 453-468.
[23]	A. V. Kochergin, A mixing special flow over a rotation of the circle with an almost Lipschitz function, (Russian) Mat. Sb., 193 (2002), 51-78; translation in Sb. Math., 193 (2002), 359-385.
[24]	A. V. Kochergin, Nondegenerate fixed points and mixing in flows on a two-dimensional torus. II, (Russian) Mat. Sb., 195 (2004), 15-46; translation in Sb. Math., 195 (2004), 317-346.
[25]	A. V. Kochergin, Causes of stretching of Birkhoff sums and mixing in flows on surfaces, Dynamics, ergodic theory, and geometry, 129-144, Math. Sci. Res. Inst. Publ., 54, Cambridge Univ. Press, Cambridge, 2007.
[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), part 1, 29-41.
[27]	J. von Neumann, Zur Operatorenmethode in der Klassichen Mechanik, (German), Ann. of Math., 33 (1932), 587-642. doi: 10.2307/1968537.
[28]	M. Ratner, Horocycle flows, joinings and rigidity of products, Ann. of Math. (2), 118 (1983), 277-313. doi: 10.2307/2007030.
[29]	V. V. Ryzhikov and J.-P. Thouvenot, Disjointness, divisibility, and quasi-simplicity of measure-preserving actions, (Russian) Funktsional. Anal. i Prilozhen., 40 (2006), 85-89; translation in Funct. Anal. Appl., 40 (2006), 237-240.
[30]	J.-P. Thouvenot, Some properties and applications of joinings in ergodic theory, Ergodic theory and its connections with harmonic analysis (Alexandria, 1993), 207-235, London Math. Soc. Lecture Note Ser., 205, Cambridge Univ. Press, Cambridge, 1995.
[31]	K. Schmidt, Dispersing cocycles and mixing flows under functions, Fund. Math., 173 (2002), 191-199. doi: 10.4064/fm173-2-6.
[32]	D. Witte, Rigidity of some translations on homogeneous spaces, Invent. Math., 81 (1985), 1-27. doi: 10.1007/BF01388769.
[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), 89-242.

show all references

References:

[1]	J.-P. Allouche and J. Shallit, "Automatic Sequences. Theory, Applications, Generalizations," Cambridge Univ. Press, 2003. doi: 10.1017/CBO9780511546563.
[2]	V. I. Arnold, Topological and ergodic properties of closed 1-forms with incommensurable periods, (Russian) Funktsional. Anal. i Prilozhen., 25 (1991), 1-12, 96; translation in Funct. Anal. Appl., 25 (1991), 81-90.
[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], 245, Springer-Verlag, New York, 1982.
[4]	B. Fayad, Polynomial decay of correlations for a class of smooth flows on the two torus, Bull. Soc. Math. France, 129 (2001), 487-503.
[5]	B. Fayad, Analytic mixing reparametrizations of irrational flows, Ergodic Theory Dynam. Systems, 22 (2002), 437-468. doi: 10.1017/S0143385702000214.
[6]	B. Fayad, Smooth mixing flows with purely singular spectra, Duke Math. J., 132 (2006), 371-391. doi: 10.1215/S0012-7094-06-13225-8.
[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), 1083-1095. doi: 10.1017/S0143385704000112.
[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), 719-738. doi: 10.1017/S0143385706000046.
[9]	K. Frączek, M. Lemańczyk and E. Lesigne, Mild mixing property for special flows under piecewise constant functions, Discrete Contin. Dynam. Syst., 19 (2007), 691-710. doi: 10.3934/dcds.2007.19.691.
[10]	K. Frączek and M. Lemańczyk, On the self-similarity problem for ergodic flows, Proc. London Math. Soc., 99 (2009), 658-696. doi: 10.1112/plms/pdp013.
[11]	K. Frączek and M. Lemańczyk, A class of mixing special flows over two-dimensional rotations,, submitted., ().
[12]	K. Frączek and M. Lemańczyk, Ratner's property and mixing for special flows over two-dimensional rotations,, \arXiv{1002.2734}., ().
[13]	H. Furstenberg and B. Weiss, The finite multipliers of infinite ergodic transformations. The structure of attractors in dynamical systems, (Proc. Conf., North Dakota State Univ., Fargo, N.D., 1977), 127-132, Lecture Notes in Math., 668, Springer, Berlin, 1978.
[14]	B. Host, Mixing of all orders and pairwise independent joinings of systems with singular spectrum, Israel J. Math., 76 (1991), 289-298. doi: 10.1007/BF02773866.
[15]	A. Iwanik, M. Lemańczyk and C. Mauduit, Piecewise absolutely continuous cocycles over irrational rotations, J. London Math. Soc. (2), 59 (1999), 171-187. doi: 10.1112/S0024610799006961.
[16]	A. Katok, Cocycles, cohomology and combinatorial constructions in ergodic theory, In collaboration with E. A. Robinson, Jr. Proc. Sympos. Pure Math., 69, Smooth ergodic theory and its applications (Seattle, WA, 1999), 107-173, Amer. Math. Soc., Providence, RI, 2001.
[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, 54. Cambridge University Press, Cambridge, 1995.
[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), 1-21; translation in Funct. Anal. Appl., 26 (1992), 155-169.
[19]	Y. Khinchin, "Continued Fractions," The University of Chicago Press, Chicago, Ill.-London 1964.
[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), 515-518.
[21]	A. V. Kochergin, Mixing in special flows over a rearrangement of segments and in smooth flows on surfaces, (Russian) Mat. Sb., 96 (1975), 471-502.
[22]	A. V. Kochergin, Non-degenerated saddles and absence of mixing, (Russian) Mat. Zametki, 19 (1976), 453-468.
[23]	A. V. Kochergin, A mixing special flow over a rotation of the circle with an almost Lipschitz function, (Russian) Mat. Sb., 193 (2002), 51-78; translation in Sb. Math., 193 (2002), 359-385.
[24]	A. V. Kochergin, Nondegenerate fixed points and mixing in flows on a two-dimensional torus. II, (Russian) Mat. Sb., 195 (2004), 15-46; translation in Sb. Math., 195 (2004), 317-346.
[25]	A. V. Kochergin, Causes of stretching of Birkhoff sums and mixing in flows on surfaces, Dynamics, ergodic theory, and geometry, 129-144, Math. Sci. Res. Inst. Publ., 54, Cambridge Univ. Press, Cambridge, 2007.
[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), part 1, 29-41.
[27]	J. von Neumann, Zur Operatorenmethode in der Klassichen Mechanik, (German), Ann. of Math., 33 (1932), 587-642. doi: 10.2307/1968537.
[28]	M. Ratner, Horocycle flows, joinings and rigidity of products, Ann. of Math. (2), 118 (1983), 277-313. doi: 10.2307/2007030.
[29]	V. V. Ryzhikov and J.-P. Thouvenot, Disjointness, divisibility, and quasi-simplicity of measure-preserving actions, (Russian) Funktsional. Anal. i Prilozhen., 40 (2006), 85-89; translation in Funct. Anal. Appl., 40 (2006), 237-240.
[30]	J.-P. Thouvenot, Some properties and applications of joinings in ergodic theory, Ergodic theory and its connections with harmonic analysis (Alexandria, 1993), 207-235, London Math. Soc. Lecture Note Ser., 205, Cambridge Univ. Press, Cambridge, 1995.
[31]	K. Schmidt, Dispersing cocycles and mixing flows under functions, Fund. Math., 173 (2002), 191-199. doi: 10.4064/fm173-2-6.
[32]	D. Witte, Rigidity of some translations on homogeneous spaces, Invent. Math., 81 (1985), 1-27. doi: 10.1007/BF01388769.
[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), 89-242.

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