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, 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, New York, 1982. doi: 10.1007/978-1-4615-6927-5.  Google Scholar

[2]

B. Fayad, Polynomial decay of correlations for a class of smooth flows on the two torus, Bull. Soc. Math. France, 129 (2001), 487-503.  Google Scholar

[3]

B. Fayad, Analytic mixing reparametrizations of irrational flows, Ergodic Theory Dynam. Systems, 22 (2002), 437-468. doi: 10.1017/S0143385702000214.  Google Scholar

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B. Fayad, Rank one and mixing differentiable flows, Invent. Math., 160 (2005), 305-340. doi: 10.1007/s00222-004-0408-x.  Google Scholar

[5]

B. Fayad, Smooth mixing flows with purely singular spectra, Duke Math. J., 132 (2006), 371-391. doi: 10.1215/S0012-7094-06-13225-8.  Google Scholar

[6]

K. Frączek and M. Lemańczyk, Ratner's property and mild mixing for special flows over two-dimensional rotations, J. Mod. Dyn., 4 (2010), 609-635. doi: 10.3934/jmd.2010.4.609.  Google Scholar

[7]

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.  Google Scholar

[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, Proc. Symp. Pure Math., 69, Amer. Math. Soc., Providence, RI, 2001, 107-173. doi: 10.1090/pspum/069.  Google Scholar

[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), 155-169. doi: 10.1007/BF01075628.  Google Scholar

[10]

A. Ya. Khinchin, Continued Fractions, Dover Publications, Inc., Mineola, NY, 1997.  Google Scholar

[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), 515-518.  Google Scholar

[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), 471-502, 504.  Google Scholar

[13]

A. V. Kochergin, A mixing special flow over a rotation of the circle with an almost Lipschitz function, Sb. Math., 193 (2002), 359-385. doi: 10.1070/SM2002v193n03ABEH000636.  Google Scholar

[14]

A. V. Kochergin, Nondegenerate fixed points and mixing in flows on a two-dimensional torus. II, Sb. Math., 195 (2004), 317-346. doi: 10.1070/SM2004v195n03ABEH000807.  Google Scholar

[15]

A. V. Kochergin, Causes of stretching of Birkhoff sums and mixing in flows on surfaces, in Dynamics, Ergodic Theory, and Geometry, Math. Sci. Res. Inst. Publ., 54, Cambridge Univ. Press, Cambridge, 2007, 129-144. doi: 10.1017/CBO9780511755187.006.  Google Scholar

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M. Ratner, Horocycle flows, joinings and rigidity of products, Ann. of Math. (2), 118 (1983), 277-313. doi: 10.2307/2007030.  Google Scholar

[17]

V. V. Ryzhikov, Around simple dynamical systems. Induced joinings and multiple mixing, J. Dynam. Control Systems, 3 (1997), 111-127. doi: 10.1007/BF02471764.  Google Scholar

[18]

V. V. Ryzhikov and J.-P. Thouvenot, Disjointness, divisibility, and quasi-simplicity of measure-preserving actions, Funct. Anal. Appl., 40 (2006), 237-240. doi: 10.1007/s10688-006-0038-8.  Google Scholar

[19]

J.-P. Thouvenot, Some properties and applicationsof joinings in ergodic theory, in Ergodic Theory and its Connections with Harmonic Analysis (Alexandria, 1993), London Math. Soc. Lecture Note Ser., 205, Cambridge Univ. Press, Cambridge, 1995, 207-235. doi: 10.1017/CBO9780511574818.004.  Google Scholar

[20]

J.-Ch. Yoccoz, Centralisateurs et conjugaison différentiable des difféomorphismes du cercle. Petits diviseurs en dimension $1$, Astérisque, 231 (1995), 89-242.  Google Scholar

show all references

References:
[1]

I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai, Ergodic Theory, Springer-Verlag, New York, 1982. doi: 10.1007/978-1-4615-6927-5.  Google Scholar

[2]

B. Fayad, Polynomial decay of correlations for a class of smooth flows on the two torus, Bull. Soc. Math. France, 129 (2001), 487-503.  Google Scholar

[3]

B. Fayad, Analytic mixing reparametrizations of irrational flows, Ergodic Theory Dynam. Systems, 22 (2002), 437-468. doi: 10.1017/S0143385702000214.  Google Scholar

[4]

B. Fayad, Rank one and mixing differentiable flows, Invent. Math., 160 (2005), 305-340. doi: 10.1007/s00222-004-0408-x.  Google Scholar

[5]

B. Fayad, Smooth mixing flows with purely singular spectra, Duke Math. J., 132 (2006), 371-391. doi: 10.1215/S0012-7094-06-13225-8.  Google Scholar

[6]

K. Frączek and M. Lemańczyk, Ratner's property and mild mixing for special flows over two-dimensional rotations, J. Mod. Dyn., 4 (2010), 609-635. doi: 10.3934/jmd.2010.4.609.  Google Scholar

[7]

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.  Google Scholar

[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, Proc. Symp. Pure Math., 69, Amer. Math. Soc., Providence, RI, 2001, 107-173. doi: 10.1090/pspum/069.  Google Scholar

[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), 155-169. doi: 10.1007/BF01075628.  Google Scholar

[10]

A. Ya. Khinchin, Continued Fractions, Dover Publications, Inc., Mineola, NY, 1997.  Google Scholar

[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), 515-518.  Google Scholar

[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), 471-502, 504.  Google Scholar

[13]

A. V. Kochergin, A mixing special flow over a rotation of the circle with an almost Lipschitz function, Sb. Math., 193 (2002), 359-385. doi: 10.1070/SM2002v193n03ABEH000636.  Google Scholar

[14]

A. V. Kochergin, Nondegenerate fixed points and mixing in flows on a two-dimensional torus. II, Sb. Math., 195 (2004), 317-346. doi: 10.1070/SM2004v195n03ABEH000807.  Google Scholar

[15]

A. V. Kochergin, Causes of stretching of Birkhoff sums and mixing in flows on surfaces, in Dynamics, Ergodic Theory, and Geometry, Math. Sci. Res. Inst. Publ., 54, Cambridge Univ. Press, Cambridge, 2007, 129-144. doi: 10.1017/CBO9780511755187.006.  Google Scholar

[16]

M. Ratner, Horocycle flows, joinings and rigidity of products, Ann. of Math. (2), 118 (1983), 277-313. doi: 10.2307/2007030.  Google Scholar

[17]

V. V. Ryzhikov, Around simple dynamical systems. Induced joinings and multiple mixing, J. Dynam. Control Systems, 3 (1997), 111-127. doi: 10.1007/BF02471764.  Google Scholar

[18]

V. V. Ryzhikov and J.-P. Thouvenot, Disjointness, divisibility, and quasi-simplicity of measure-preserving actions, Funct. Anal. Appl., 40 (2006), 237-240. doi: 10.1007/s10688-006-0038-8.  Google Scholar

[19]

J.-P. Thouvenot, Some properties and applicationsof joinings in ergodic theory, in Ergodic Theory and its Connections with Harmonic Analysis (Alexandria, 1993), London Math. Soc. Lecture Note Ser., 205, Cambridge Univ. Press, Cambridge, 1995, 207-235. doi: 10.1017/CBO9780511574818.004.  Google Scholar

[20]

J.-Ch. Yoccoz, Centralisateurs et conjugaison différentiable des difféomorphismes du cercle. Petits diviseurs en dimension $1$, Astérisque, 231 (1995), 89-242.  Google Scholar

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