American Institute of Mathematical Sciences

Advanced
December  2007, 19(4): 691-710. doi: 10.3934/dcds.2007.19.691

Mild mixing property for special flows under piecewise constant functions

Krzysztof Frączek 1, , M. Lemańczyk 2, and  E. Lesigne 3,

1. 

Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12/18, 87-100 Toruń, Poland
2. 

Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Ul. Chopina 12/18, 87-100 Toruń, Poland
3. 

Laboratoire de Mathématiques et Physique Théorique, Faculté des Sciences et Techniques, Université François Rabelais de Tours, parc de Grandmont, 37200 Tours, France

Received  November 2006 Revised  April 2007 Published  September 2007

We give a condition on a piecewise constant roof function and an irrational rotation by $\alpha$ on the circle to give rise to a special flow having the mild mixing property. Such flows will also satisfy Ratner's property. As a consequence we obtain a class of mildly mixing singular flows on the two-torus that arise from quasi-periodic Hamiltonians flows by velocity changes.
Keywords: Mild mixing property, measure-preserving flows, special flows..
Mathematics Subject Classification: Primary: 37A10, 37C40; Secondary: 37E3.
Citation: Krzysztof Frączek, M. Lemańczyk, E. Lesigne. Mild mixing property for special flows under piecewise constant functions. Discrete & Continuous Dynamical Systems - A, 2007, 19 (4) : 691-710. doi: 10.3934/dcds.2007.19.691
[1]

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
[2]

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
[3]

Rui Kuang, Xiangdong Ye. The return times set and mixing for measure preserving transformations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 817-827. doi: 10.3934/dcds.2007.18.817
[4]

Roland Gunesch, Anatole Katok. Construction of weakly mixing diffeomorphisms preserving measurable Riemannian metric and smooth measure. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 61-88. doi: 10.3934/dcds.2000.6.61
[5]

Dmitri Scheglov. Absence of mixing for smooth flows on genus two surfaces. Journal of Modern Dynamics, 2009, 3 (1) : 13-34. doi: 10.3934/jmd.2009.3.13
[6]

Przemysław Berk, Krzysztof Frączek. On special flows over IETs that are not isomorphic to their inverses. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 829-855. doi: 10.3934/dcds.2015.35.829
[7]

Anthony Quas, Terry Soo. Weak mixing suspension flows over shifts of finite type are universal. Journal of Modern Dynamics, 2012, 6 (4) : 427-449. doi: 10.3934/jmd.2012.6.427
[8]

Corinna Ulcigrai. Weak mixing for logarithmic flows over interval exchange transformations. Journal of Modern Dynamics, 2009, 3 (1) : 35-49. doi: 10.3934/jmd.2009.3.35
[9]

Luis Caffarelli, Fanghua Lin. Nonlocal heat flows preserving the L2 energy. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 49-64. doi: 10.3934/dcds.2009.23.49
[10]

Giovanni Forni. The cohomological equation for area-preserving flows on compact surfaces. Electronic Research Announcements, 1995, 1: 114-123.

[11]

V. Afraimovich, Jean-René Chazottes, Benoît Saussol. Pointwise dimensions for Poincaré recurrences associated with maps and special flows. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 263-280. doi: 10.3934/dcds.2003.9.263
[12]

Bassam Fayad, A. Windsor. A dichotomy between discrete and continuous spectrum for a class of special flows over rotations. Journal of Modern Dynamics, 2007, 1 (1) : 107-122. doi: 10.3934/jmd.2007.1.107
[13]

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

[14]

Jian Li. Localization of mixing property via Furstenberg families. Discrete & Continuous Dynamical Systems - A, 2015, 35 (2) : 725-740. doi: 10.3934/dcds.2015.35.725
[15]

Makoto Mori. Higher order mixing property of piecewise linear transformations. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 915-934. doi: 10.3934/dcds.2000.6.915
[16]

James Tanis. Exponential multiple mixing for some partially hyperbolic flows on products of $ {\rm{PSL}}(2, \mathbb{R})$. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 989-1006. doi: 10.3934/dcds.2018042
[17]

Roland Zweimüller. Asymptotic orbit complexity of infinite measure preserving transformations. Discrete & Continuous Dynamical Systems - A, 2006, 15 (1) : 353-366. doi: 10.3934/dcds.2006.15.353
[18]

S. Eigen, A. B. Hajian, V. S. Prasad. Universal skyscraper templates for infinite measure preserving transformations. Discrete & Continuous Dynamical Systems - A, 2006, 16 (2) : 343-360. doi: 10.3934/dcds.2006.16.343
[19]

Ian Melbourne, Dalia Terhesiu. Mixing properties for toral extensions of slowly mixing dynamical systems with finite and infinite measure. Journal of Modern Dynamics, 2018, 12: 285-313. doi: 10.3934/jmd.2018011
[20]

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

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences