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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
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References:
 [1] Cambridge Univ. Press, 2003. doi: 10.1017/CBO9780511546563.  Google Scholar [2] (Russian) Funktsional. Anal. i Prilozhen., 25 (1991), 1-12, 96; translation in Funct. Anal. Appl., 25 (1991), 81-90.  Google Scholar [3] Translated from the Russian by A. B. Sosinskii. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 245, Springer-Verlag, New York, 1982.  Google Scholar [4] Bull. Soc. Math. France, 129 (2001), 487-503.  Google Scholar [5] Ergodic Theory Dynam. Systems, 22 (2002), 437-468. doi: 10.1017/S0143385702000214.  Google Scholar [6] Duke Math. J., 132 (2006), 371-391. doi: 10.1215/S0012-7094-06-13225-8.  Google Scholar [7] Ergodic Theory Dynam. Systems, 24 (2004), 1083-1095. doi: 10.1017/S0143385704000112.  Google Scholar [8] Ergodic Theory Dynam. Systems, 26 (2006), 719-738. doi: 10.1017/S0143385706000046.  Google Scholar [9] Discrete Contin. Dynam. Syst., 19 (2007), 691-710. doi: 10.3934/dcds.2007.19.691.  Google Scholar [10] Proc. London Math. Soc., 99 (2009), 658-696. 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] (Proc. Conf., North Dakota State Univ., Fargo, N.D., 1977), 127-132, Lecture Notes in Math., 668, Springer, Berlin, 1978.  Google Scholar [14] Israel J. Math., 76 (1991), 289-298. doi: 10.1007/BF02773866.  Google Scholar [15] J. London Math. Soc. (2), 59 (1999), 171-187. doi: 10.1112/S0024610799006961.  Google Scholar [16] 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.  Google Scholar [17] With a supplementary chapter by Katok and Leonardo Mendoza. Encyclopedia of Mathematics and its Applications, 54. Cambridge University Press, Cambridge, 1995.  Google Scholar [18] (Russian) Funktsional. Anal. i Prilozhen., 26 (1992), 1-21; translation in Funct. Anal. Appl., 26 (1992), 155-169.  Google Scholar [19] The University of Chicago Press, Chicago, Ill.-London 1964.  Google Scholar [20] (Russian) Dokl. Akad. Nauk SSSR, 205 (1972), 515-518.  Google Scholar [21] (Russian) Mat. Sb., 96 (1975), 471-502.  Google Scholar [22] (Russian) Mat. Zametki, 19 (1976), 453-468.  Google Scholar [23] (Russian) Mat. Sb., 193 (2002), 51-78; translation in Sb. Math., 193 (2002), 359-385.  Google Scholar [24] (Russian) Mat. Sb., 195 (2004), 15-46; translation in Sb. Math., 195 (2004), 317-346.  Google Scholar [25] Dynamics, ergodic theory, and geometry, 129-144, Math. Sci. Res. Inst. Publ., 54, Cambridge Univ. Press, Cambridge, 2007.  Google Scholar [26] (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.  Google Scholar [27] (German), Ann. of Math., 33 (1932), 587-642. doi: 10.2307/1968537.  Google Scholar [28] Ann. of Math. (2), 118 (1983), 277-313. doi: 10.2307/2007030.  Google Scholar [29] (Russian) Funktsional. Anal. i Prilozhen., 40 (2006), 85-89; translation in Funct. Anal. Appl., 40 (2006), 237-240.  Google Scholar [30] Ergodic theory and its connections with harmonic analysis (Alexandria, 1993), 207-235, London Math. Soc. Lecture Note Ser., 205, Cambridge Univ. Press, Cambridge, 1995.  Google Scholar [31] Fund. Math., 173 (2002), 191-199. doi: 10.4064/fm173-2-6.  Google Scholar [32] Invent. Math., 81 (1985), 1-27. doi: 10.1007/BF01388769.  Google Scholar [33] (French) [Centralizers and differentiable conjugacy of diffeomorphisms of the circle] Petits diviseurs en dimension $1$. Astérisque No. 231, (1995), 89-242.  Google Scholar
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