# American Institute of Mathematical Sciences

October  2015, 35(10): 4823-4829. doi: 10.3934/dcds.2015.35.4823

## A class of mixing 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

Received  October 2014 Revised  February 2015 Published  April 2015

We consider special flows over two-dimensional rotations by $(\alpha,\beta)$ on $\mathbb{T}^2$ and under piecewise $C^2$ roof functions $f$ satisfying von Neumann's condition $\int_{\mathbb{T}^2}f_x(x,y)\,dx\,dy\neq 0\quad\text{ and }\quad \int_{\mathbb{T}^2}f_y(x,y)\,dx \,dy\neq 0.$ For an uncountable set of $(\alpha,\beta)$ with both $\alpha$ and $\beta$ of unbounded partial quotients the mixing property is proved to hold.
Citation: Krzysztof Frączek, Mariusz Lemańczyk. A class of mixing special flows over two--dimensional rotations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4823-4829. doi: 10.3934/dcds.2015.35.4823
##### References:
 [1] I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai, Ergodic Theory,, Springer-Verlag, (1982). doi: 10.1007/978-1-4615-6927-5. [2] B. Fayad, Polynomial decay of correlations for a class of smooth flows on the two torus,, Bull. Soc. Math. France, 129 (2001), 487. [3] B. Fayad, Analytic mixing reparametrizations of irrational flows,, Ergodic Theory Dynam. Systems, 22 (2002), 437. doi: 10.1017/S0143385702000214. [4] B. Fayad, Rank one and mixing differentiable flows,, Invent. Math., 160 (2005), 305. doi: 10.1007/s00222-004-0408-x. [5] B. Fayad, Smooth mixing flows with purely singular spectra,, Duke Math. J., 132 (2006), 371. doi: 10.1215/S0012-7094-06-13225-8. [6] K. Frączek and M. Lemańczyk, Ratner's property and mild mixing for special flows over two-dimensional rotations,, J. Mod. Dyn., 4 (2010), 609. doi: 10.3934/jmd.2010.4.609. [7] 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. [8] A. Katok, Cocycles, cohomology and combinatorial constructions in ergodic theory (in collaboration with E. A. Robinson, Jr.,, in Smooth Ergodic Theory and its Applications, (2001), 107. doi: 10.1090/pspum/069. [9] K. M. Khanin and Ya. G. Sinai, Mixing of some classes of special flows over rotations of the circle,, Funct. Anal. Appl., 26 (1992), 155. doi: 10.1007/BF01075628. [10] A. Ya. Khinchin, Continued Fractions,, Dover Publications, (1997). [11] 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. [12] A. V. Kochergin, Mixing in special flows over a rearrangement of segments and in smooth flows on surfaces,, (Russian) Mat. Sb., 96(138) (1975), 471. [13] A. V. Kochergin, A mixing special flow over a rotation of the circle with an almost Lipschitz function,, Sb. Math., 193 (2002), 359. doi: 10.1070/SM2002v193n03ABEH000636. [14] A. V. Kochergin, Nondegenerate fixed points and mixing in flows on a two-dimensional torus. II,, Sb. Math., 195 (2004), 317. doi: 10.1070/SM2004v195n03ABEH000807. [15] A. V. Kochergin, Causes of stretching of Birkhoff sums and mixing in flows on surfaces,, in Dynamics, (2007), 129. doi: 10.1017/CBO9780511755187.006. [16] M. Ratner, Horocycle flows, joinings and rigidity of products,, Ann. of Math. (2), 118 (1983), 277. doi: 10.2307/2007030. [17] V. V. Ryzhikov, Around simple dynamical systems. Induced joinings and multiple mixing,, J. Dynam. Control Systems, 3 (1997), 111. doi: 10.1007/BF02471764. [18] V. V. Ryzhikov and J.-P. Thouvenot, Disjointness, divisibility, and quasi-simplicity of measure-preserving actions,, Funct. Anal. Appl., 40 (2006), 237. doi: 10.1007/s10688-006-0038-8. [19] J.-P. Thouvenot, Some properties and applicationsof joinings in ergodic theory,, in Ergodic Theory and its Connections with Harmonic Analysis (Alexandria, (1993), 207. doi: 10.1017/CBO9780511574818.004. [20] J.-Ch. Yoccoz, Centralisateurs et conjugaison différentiable des difféomorphismes du cercle. Petits diviseurs en dimension $1$,, Astérisque, 231 (1995), 89.

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##### References:
 [1] I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai, Ergodic Theory,, Springer-Verlag, (1982). doi: 10.1007/978-1-4615-6927-5. [2] B. Fayad, Polynomial decay of correlations for a class of smooth flows on the two torus,, Bull. Soc. Math. France, 129 (2001), 487. [3] B. Fayad, Analytic mixing reparametrizations of irrational flows,, Ergodic Theory Dynam. Systems, 22 (2002), 437. doi: 10.1017/S0143385702000214. [4] B. Fayad, Rank one and mixing differentiable flows,, Invent. Math., 160 (2005), 305. doi: 10.1007/s00222-004-0408-x. [5] B. Fayad, Smooth mixing flows with purely singular spectra,, Duke Math. J., 132 (2006), 371. doi: 10.1215/S0012-7094-06-13225-8. [6] K. Frączek and M. Lemańczyk, Ratner's property and mild mixing for special flows over two-dimensional rotations,, J. Mod. Dyn., 4 (2010), 609. doi: 10.3934/jmd.2010.4.609. [7] 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. [8] A. Katok, Cocycles, cohomology and combinatorial constructions in ergodic theory (in collaboration with E. A. Robinson, Jr.,, in Smooth Ergodic Theory and its Applications, (2001), 107. doi: 10.1090/pspum/069. [9] K. M. Khanin and Ya. G. Sinai, Mixing of some classes of special flows over rotations of the circle,, Funct. Anal. Appl., 26 (1992), 155. doi: 10.1007/BF01075628. [10] A. Ya. Khinchin, Continued Fractions,, Dover Publications, (1997). [11] 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. [12] A. V. Kochergin, Mixing in special flows over a rearrangement of segments and in smooth flows on surfaces,, (Russian) Mat. Sb., 96(138) (1975), 471. [13] A. V. Kochergin, A mixing special flow over a rotation of the circle with an almost Lipschitz function,, Sb. Math., 193 (2002), 359. doi: 10.1070/SM2002v193n03ABEH000636. [14] A. V. Kochergin, Nondegenerate fixed points and mixing in flows on a two-dimensional torus. II,, Sb. Math., 195 (2004), 317. doi: 10.1070/SM2004v195n03ABEH000807. [15] A. V. Kochergin, Causes of stretching of Birkhoff sums and mixing in flows on surfaces,, in Dynamics, (2007), 129. doi: 10.1017/CBO9780511755187.006. [16] M. Ratner, Horocycle flows, joinings and rigidity of products,, Ann. of Math. (2), 118 (1983), 277. doi: 10.2307/2007030. [17] V. V. Ryzhikov, Around simple dynamical systems. Induced joinings and multiple mixing,, J. Dynam. Control Systems, 3 (1997), 111. doi: 10.1007/BF02471764. [18] V. V. Ryzhikov and J.-P. Thouvenot, Disjointness, divisibility, and quasi-simplicity of measure-preserving actions,, Funct. Anal. Appl., 40 (2006), 237. doi: 10.1007/s10688-006-0038-8. [19] J.-P. Thouvenot, Some properties and applicationsof joinings in ergodic theory,, in Ergodic Theory and its Connections with Harmonic Analysis (Alexandria, (1993), 207. doi: 10.1017/CBO9780511574818.004. [20] J.-Ch. Yoccoz, Centralisateurs et conjugaison différentiable des difféomorphismes du cercle. Petits diviseurs en dimension $1$,, Astérisque, 231 (1995), 89.
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