• Previous Article
    Structure of attractors for $(a,b)$-continued fraction transformations
  • JMD Home
  • This Issue
  • Next Article
    New cases of differentiable rigidity for partially hyperbolic actions: Symplectic groups and resonance directions
October  2010, 4(4): 609-635. doi: 10.3934/jmd.2010.4.609

Ratner's property and mild mixing for 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, Poland

Received  February 2010 Revised  December 2010 Published  January 2011

We consider special flows over two-dimensional rotations by $(\alpha,\beta)$ on $\T^2$ and under piecewise $C^2$ roof functions $f$ satisfying von Neumann's condition $\int_{\T^2}f_x(x,y)dxdy\ne 0$ or $\int_{\T^2}f_y(x,y)dxdy\ne 0 $. Such flows are shown to be always weakly mixing and never partially rigid. It is proved that while specifying to a subclass of roof functions and to ergodic rotations for which $\alpha$ and $\beta$ are of bounded partial quotients the corresponding special flows enjoy the so-called weak Ratner property. As a consequence, such flows turn out to be mildly mixing.
Citation: Krzysztof Frączek, Mariusz Lemańczyk. Ratner's property and mild mixing for special flows over two-dimensional rotations. Journal of Modern Dynamics, 2010, 4 (4) : 609-635. doi: 10.3934/jmd.2010.4.609
References:
[1]

Cambridge Univ. Press, 2003. doi: 10.1017/CBO9780511546563.  Google Scholar

[2]

(Russian) Funktsional. Anal. i Prilozhen., 25 (1991), 1-12, 96; translation in Funct. Anal. Appl., 25 (1991), 81-90.  Google Scholar

[3]

Translated from the Russian by A. B. Sosinskii. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 245, Springer-Verlag, New York, 1982.  Google Scholar

[4]

Bull. Soc. Math. France, 129 (2001), 487-503.  Google Scholar

[5]

Ergodic Theory Dynam. Systems, 22 (2002), 437-468. doi: 10.1017/S0143385702000214.  Google Scholar

[6]

Duke Math. J., 132 (2006), 371-391. doi: 10.1215/S0012-7094-06-13225-8.  Google Scholar

[7]

Ergodic Theory Dynam. Systems, 24 (2004), 1083-1095. doi: 10.1017/S0143385704000112.  Google Scholar

[8]

Ergodic Theory Dynam. Systems, 26 (2006), 719-738. doi: 10.1017/S0143385706000046.  Google Scholar

[9]

Discrete Contin. Dynam. Syst., 19 (2007), 691-710. doi: 10.3934/dcds.2007.19.691.  Google Scholar

[10]

Proc. London Math. Soc., 99 (2009), 658-696. doi: 10.1112/plms/pdp013.  Google Scholar

[11]

K. Frączek and M. Lemańczyk, A class of mixing special flows over two-dimensional rotations,, submitted., ().   Google Scholar

[12]

K. Frączek and M. Lemańczyk, Ratner's property and mixing for special flows over two-dimensional rotations,, \arXiv{1002.2734}., ().   Google Scholar

[13]

(Proc. Conf., North Dakota State Univ., Fargo, N.D., 1977), 127-132, Lecture Notes in Math., 668, Springer, Berlin, 1978.  Google Scholar

[14]

Israel J. Math., 76 (1991), 289-298. doi: 10.1007/BF02773866.  Google Scholar

[15]

J. London Math. Soc. (2), 59 (1999), 171-187. doi: 10.1112/S0024610799006961.  Google Scholar

[16]

In collaboration with E. A. Robinson, Jr. Proc. Sympos. Pure Math., 69, Smooth ergodic theory and its applications (Seattle, WA, 1999), 107-173, Amer. Math. Soc., Providence, RI, 2001.  Google Scholar

[17]

With a supplementary chapter by Katok and Leonardo Mendoza. Encyclopedia of Mathematics and its Applications, 54. Cambridge University Press, Cambridge, 1995.  Google Scholar

[18]

(Russian) Funktsional. Anal. i Prilozhen., 26 (1992), 1-21; translation in Funct. Anal. Appl., 26 (1992), 155-169.  Google Scholar

[19]

The University of Chicago Press, Chicago, Ill.-London 1964.  Google Scholar

[20]

(Russian) Dokl. Akad. Nauk SSSR, 205 (1972), 515-518.  Google Scholar

[21]

(Russian) Mat. Sb., 96 (1975), 471-502.  Google Scholar

[22]

(Russian) Mat. Zametki, 19 (1976), 453-468.  Google Scholar

[23]

(Russian) Mat. Sb., 193 (2002), 51-78; translation in Sb. Math., 193 (2002), 359-385.  Google Scholar

[24]

(Russian) Mat. Sb., 195 (2004), 15-46; translation in Sb. Math., 195 (2004), 317-346.  Google Scholar

[25]

Dynamics, ergodic theory, and geometry, 129-144, Math. Sci. Res. Inst. Publ., 54, Cambridge Univ. Press, Cambridge, 2007.  Google Scholar

[26]

(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, 29-41.  Google Scholar

[27]

(German), Ann. of Math., 33 (1932), 587-642. doi: 10.2307/1968537.  Google Scholar

[28]

Ann. of Math. (2), 118 (1983), 277-313. doi: 10.2307/2007030.  Google Scholar

[29]

(Russian) Funktsional. Anal. i Prilozhen., 40 (2006), 85-89; translation in Funct. Anal. Appl., 40 (2006), 237-240.  Google Scholar

[30]

Ergodic theory and its connections with harmonic analysis (Alexandria, 1993), 207-235, London Math. Soc. Lecture Note Ser., 205, Cambridge Univ. Press, Cambridge, 1995.  Google Scholar

[31]

Fund. Math., 173 (2002), 191-199. doi: 10.4064/fm173-2-6.  Google Scholar

[32]

Invent. Math., 81 (1985), 1-27. doi: 10.1007/BF01388769.  Google Scholar

[33]

(French) [Centralizers and differentiable conjugacy of diffeomorphisms of the circle] Petits diviseurs en dimension $1$. Astérisque No. 231, (1995), 89-242.  Google Scholar

show all references

References:
[1]

Cambridge Univ. Press, 2003. doi: 10.1017/CBO9780511546563.  Google Scholar

[2]

(Russian) Funktsional. Anal. i Prilozhen., 25 (1991), 1-12, 96; translation in Funct. Anal. Appl., 25 (1991), 81-90.  Google Scholar

[3]

Translated from the Russian by A. B. Sosinskii. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 245, Springer-Verlag, New York, 1982.  Google Scholar

[4]

Bull. Soc. Math. France, 129 (2001), 487-503.  Google Scholar

[5]

Ergodic Theory Dynam. Systems, 22 (2002), 437-468. doi: 10.1017/S0143385702000214.  Google Scholar

[6]

Duke Math. J., 132 (2006), 371-391. doi: 10.1215/S0012-7094-06-13225-8.  Google Scholar

[7]

Ergodic Theory Dynam. Systems, 24 (2004), 1083-1095. doi: 10.1017/S0143385704000112.  Google Scholar

[8]

Ergodic Theory Dynam. Systems, 26 (2006), 719-738. doi: 10.1017/S0143385706000046.  Google Scholar

[9]

Discrete Contin. Dynam. Syst., 19 (2007), 691-710. doi: 10.3934/dcds.2007.19.691.  Google Scholar

[10]

Proc. London Math. Soc., 99 (2009), 658-696. doi: 10.1112/plms/pdp013.  Google Scholar

[11]

K. Frączek and M. Lemańczyk, A class of mixing special flows over two-dimensional rotations,, submitted., ().   Google Scholar

[12]

K. Frączek and M. Lemańczyk, Ratner's property and mixing for special flows over two-dimensional rotations,, \arXiv{1002.2734}., ().   Google Scholar

[13]

(Proc. Conf., North Dakota State Univ., Fargo, N.D., 1977), 127-132, Lecture Notes in Math., 668, Springer, Berlin, 1978.  Google Scholar

[14]

Israel J. Math., 76 (1991), 289-298. doi: 10.1007/BF02773866.  Google Scholar

[15]

J. London Math. Soc. (2), 59 (1999), 171-187. doi: 10.1112/S0024610799006961.  Google Scholar

[16]

In collaboration with E. A. Robinson, Jr. Proc. Sympos. Pure Math., 69, Smooth ergodic theory and its applications (Seattle, WA, 1999), 107-173, Amer. Math. Soc., Providence, RI, 2001.  Google Scholar

[17]

With a supplementary chapter by Katok and Leonardo Mendoza. Encyclopedia of Mathematics and its Applications, 54. Cambridge University Press, Cambridge, 1995.  Google Scholar

[18]

(Russian) Funktsional. Anal. i Prilozhen., 26 (1992), 1-21; translation in Funct. Anal. Appl., 26 (1992), 155-169.  Google Scholar

[19]

The University of Chicago Press, Chicago, Ill.-London 1964.  Google Scholar

[20]

(Russian) Dokl. Akad. Nauk SSSR, 205 (1972), 515-518.  Google Scholar

[21]

(Russian) Mat. Sb., 96 (1975), 471-502.  Google Scholar

[22]

(Russian) Mat. Zametki, 19 (1976), 453-468.  Google Scholar

[23]

(Russian) Mat. Sb., 193 (2002), 51-78; translation in Sb. Math., 193 (2002), 359-385.  Google Scholar

[24]

(Russian) Mat. Sb., 195 (2004), 15-46; translation in Sb. Math., 195 (2004), 317-346.  Google Scholar

[25]

Dynamics, ergodic theory, and geometry, 129-144, Math. Sci. Res. Inst. Publ., 54, Cambridge Univ. Press, Cambridge, 2007.  Google Scholar

[26]

(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, 29-41.  Google Scholar

[27]

(German), Ann. of Math., 33 (1932), 587-642. doi: 10.2307/1968537.  Google Scholar

[28]

Ann. of Math. (2), 118 (1983), 277-313. doi: 10.2307/2007030.  Google Scholar

[29]

(Russian) Funktsional. Anal. i Prilozhen., 40 (2006), 85-89; translation in Funct. Anal. Appl., 40 (2006), 237-240.  Google Scholar

[30]

Ergodic theory and its connections with harmonic analysis (Alexandria, 1993), 207-235, London Math. Soc. Lecture Note Ser., 205, Cambridge Univ. Press, Cambridge, 1995.  Google Scholar

[31]

Fund. Math., 173 (2002), 191-199. doi: 10.4064/fm173-2-6.  Google Scholar

[32]

Invent. Math., 81 (1985), 1-27. doi: 10.1007/BF01388769.  Google Scholar

[33]

(French) [Centralizers and differentiable conjugacy of diffeomorphisms of the circle] Petits diviseurs en dimension $1$. Astérisque No. 231, (1995), 89-242.  Google Scholar

[1]

Jihoon Lee, Ngocthach Nguyen. Flows with the weak two-sided limit shadowing property. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021040

[2]

A. Kochergin. Well-approximable angles and mixing for flows on T^2 with nonsingular fixed points. Electronic Research Announcements, 2004, 10: 113-121.

[3]

Fei Liu, Xiaokai Liu, Fang Wang. On the mixing and Bernoulli properties for geodesic flows on rank 1 manifolds without focal points. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021057

[4]

Yunjuan Jin, Aifang Qu, Hairong Yuan. Radon measure solutions for steady compressible hypersonic-limit Euler flows passing cylindrically symmetric conical bodies. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021048

[5]

Michael Schmidt, Emmanuel Trélat. Controllability of couette flows. Communications on Pure & Applied Analysis, 2006, 5 (1) : 201-211. doi: 10.3934/cpaa.2006.5.201

[6]

Ugo Bessi. Another point of view on Kusuoka's measure. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3241-3271. doi: 10.3934/dcds.2020404

[7]

Ronald E. Mickens. Positivity preserving discrete model for the coupled ODE's modeling glycolysis. Conference Publications, 2003, 2003 (Special) : 623-629. doi: 10.3934/proc.2003.2003.623

[8]

Charles Amorim, Miguel Loayza, Marko A. Rojas-Medar. The nonstationary flows of micropolar fluids with thermal convection: An iterative approach. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2509-2535. doi: 10.3934/dcdsb.2020193

[9]

Zhigang Pan, Chanh Kieu, Quan Wang. Hopf bifurcations and transitions of two-dimensional Quasi-Geostrophic flows. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021025

[10]

Annalisa Cesaroni, Valerio Pagliari. Convergence of nonlocal geometric flows to anisotropic mean curvature motion. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021065

[11]

Guanming Gai, Yuanyuan Nie, Chunpeng Wang. A degenerate elliptic problem from subsonic-sonic flows in convergent nozzles. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021070

[12]

Misha Bialy, Andrey E. Mironov. Rich quasi-linear system for integrable geodesic flows on 2-torus. Discrete & Continuous Dynamical Systems, 2011, 29 (1) : 81-90. doi: 10.3934/dcds.2011.29.81

[13]

Jan Březina, Eduard Feireisl, Antonín Novotný. On convergence to equilibria of flows of compressible viscous fluids under in/out–flux boundary conditions. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3615-3627. doi: 10.3934/dcds.2021009

[14]

Hirokazu Saito, Xin Zhang. Unique solvability of elliptic problems associated with two-phase incompressible flows in unbounded domains. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021051

[15]

Frank Sottile. The special Schubert calculus is real. Electronic Research Announcements, 1999, 5: 35-39.

[16]

Paul Deuring. Spatial asymptotics of mild solutions to the time-dependent Oseen system. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021044

[17]

Charlene Kalle, Niels Langeveld, Marta Maggioni, Sara Munday. Matching for a family of infinite measure continued fraction transformations. Discrete & Continuous Dynamical Systems, 2020, 40 (11) : 6309-6330. doi: 10.3934/dcds.2020281

[18]

Alina Chertock, Alexander Kurganov, Mária Lukáčová-Medvi${\rm{\check{d}}}$ová, Șeyma Nur Özcan. An asymptotic preserving scheme for kinetic chemotaxis models in two space dimensions. Kinetic & Related Models, 2019, 12 (1) : 195-216. doi: 10.3934/krm.2019009

[19]

Adrian Viorel, Cristian D. Alecsa, Titus O. Pinţa. Asymptotic analysis of a structure-preserving integrator for damped Hamiltonian systems. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3319-3341. doi: 10.3934/dcds.2020407

[20]

Chaudry Masood Khalique, Muhammad Usman, Maria Luz Gandarais. Special issue dedicated to Professor David Paul Mason. Discrete & Continuous Dynamical Systems - S, 2020, 13 (10) : iii-iv. doi: 10.3934/dcdss.2020416

2019 Impact Factor: 0.465

Metrics

  • PDF downloads (33)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]