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.  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.   Google Scholar

[3]

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

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B. Fayad, Rank one and mixing differentiable flows,, Invent. Math., 160 (2005), 305.  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.  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.  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.  doi: 10.1007/BF02773866.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[10]

A. Ya. Khinchin, Continued Fractions,, Dover Publications, (1997).   Google Scholar

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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.   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.   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.  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.  doi: 10.1070/SM2004v195n03ABEH000807.  Google Scholar

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A. V. Kochergin, Causes of stretching of Birkhoff sums and mixing in flows on surfaces,, in Dynamics, (2007), 129.  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.  doi: 10.2307/2007030.  Google Scholar

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V. V. Ryzhikov, Around simple dynamical systems. Induced joinings and multiple mixing,, J. Dynam. Control Systems, 3 (1997), 111.  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.  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), 207.  doi: 10.1017/CBO9780511574818.004.  Google Scholar

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

show all references

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.  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.   Google Scholar

[3]

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

[4]

B. Fayad, Rank one and mixing differentiable flows,, Invent. Math., 160 (2005), 305.  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.  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.  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.  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, (2001), 107.  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.  doi: 10.1007/BF01075628.  Google Scholar

[10]

A. Ya. Khinchin, Continued Fractions,, Dover Publications, (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.   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.   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.  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.  doi: 10.1070/SM2004v195n03ABEH000807.  Google Scholar

[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.  Google Scholar

[16]

M. Ratner, Horocycle flows, joinings and rigidity of products,, Ann. of Math. (2), 118 (1983), 277.  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.  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.  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), 207.  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.   Google Scholar

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