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 and 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.

[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.

[3]

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

[4]

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

[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.

[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.

[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.

[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.

[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.

[10]

A. Ya. Khinchin, Continued Fractions, Dover Publications, Inc., Mineola, NY, 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), 515-518.

[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.

[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.

[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.

[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.

[16]

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

[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.

[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.

[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.

[20]

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

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.

[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.

[3]

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

[4]

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

[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.

[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.

[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.

[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.

[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.

[10]

A. Ya. Khinchin, Continued Fractions, Dover Publications, Inc., Mineola, NY, 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), 515-518.

[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.

[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.

[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.

[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.

[16]

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

[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.

[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.

[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.

[20]

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

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