Ratner's property and mild mixing for special flows over twodimensional rotations
Pages: 609  635,
Issue 4,
October
2010
doi:10.3934/jmd.2010.4.609 Abstract
References
Full text (324.7K)
Related Articles
Krzysztof Frączek  Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12/18, 87100 Toruń, Poland (email)
Mariusz Lemańczyk  Faculty of Mathematics and Computer Science, N. Copernicus University, ul. Chopina 12/18, 87100 Toruń, Poland (email)
1 
J.P. Allouche and J. Shallit, "Automatic Sequences. Theory, Applications, Generalizations," Cambridge Univ. Press, 2003. 

2 
V. I. Arnold, Topological and ergodic properties of closed 1forms with incommensurable periods, (Russian) Funktsional. Anal. i Prilozhen., 25 (1991), 112, 96; translation in Funct. Anal. Appl., 25 (1991), 8190. 

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, SpringerVerlag, 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), 487503. 

5 
B. Fayad, Analytic mixing reparametrizations of irrational flows, Ergodic Theory Dynam. Systems, 22 (2002), 437468. 

6 
B. Fayad, Smooth mixing flows with purely singular spectra, Duke Math. J., 132 (2006), 371391. 

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), 10831095. 

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), 719738. 

9 
K. Frączek, M. Lemańczyk and E. Lesigne, Mild mixing property for special flows under piecewise constant functions, Discrete Contin. Dynam. Syst., 19 (2007), 691710. 

10 
K. Frączek and M. Lemańczyk, On the selfsimilarity problem for ergodic flows, Proc. London Math. Soc., 99 (2009), 658696. 

11 
K. Frączek and M. Lemańczyk, A class of mixing special flows over twodimensional rotations, submitted. 

12 
K. Frączek and M. Lemańczyk, Ratner's property and mixing for special flows over twodimensional 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), 127132, 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), 289298. 

15 
A. Iwanik, M. Lemańczyk and C. Mauduit, Piecewise absolutely continuous cocycles over irrational rotations, J. London Math. Soc. (2), 59 (1999), 171187. 

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), 107173, 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), 121; translation in Funct. Anal. Appl., 26 (1992), 155169. 

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 twodimensional torus, (Russian) Dokl. Akad. Nauk SSSR, 205 (1972), 515518. 

21 
A. V. Kochergin, Mixing in special flows over a rearrangement of segments and in smooth flows on surfaces, (Russian) Mat. Sb., 96 (1975), 471502. 

22 
A. V. Kochergin, Nondegenerated saddles and absence of mixing, (Russian) Mat. Zametki, 19 (1976), 453468. 

23 
A. V. Kochergin, A mixing special flow over a rotation of the circle with an almost Lipschitz function, (Russian) Mat. Sb., 193 (2002), 5178; translation in Sb. Math., 193 (2002), 359385. 

24 
A. V. Kochergin, Nondegenerate fixed points and mixing in flows on a twodimensional torus. II, (Russian) Mat. Sb., 195 (2004), 1546; translation in Sb. Math., 195 (2004), 317346. 

25 
A. V. Kochergin, Causes of stretching of Birkhoff sums and mixing in flows on surfaces, Dynamics, ergodic theory, and geometry, 129144, 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, 2941. 

27 
J. von Neumann, Zur Operatorenmethode in der Klassichen Mechanik, (German), Ann. of Math., 33 (1932), 587642. 

28 
M. Ratner, Horocycle flows, joinings and rigidity of products, Ann. of Math. (2), 118 (1983), 277313. 

29 
V. V. Ryzhikov and J.P. Thouvenot, Disjointness, divisibility, and quasisimplicity of measurepreserving actions, (Russian) Funktsional. Anal. i Prilozhen., 40 (2006), 8589; translation in Funct. Anal. Appl., 40 (2006), 237240. 

30 
J.P. Thouvenot, Some properties and applications of joinings in ergodic theory, Ergodic theory and its connections with harmonic analysis (Alexandria, 1993), 207235, 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), 191199. 

32 
D. Witte, Rigidity of some translations on homogeneous spaces, Invent. Math., 81 (1985), 127. 

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), 89242. 

Go to top
