2007, 1(1): 107-122. doi: 10.3934/jmd.2007.1.107

A dichotomy between discrete and continuous spectrum for a class of special flows over rotations

1. 

LAGA, CNRS UMR 7539, Université Paris 13, Villetaneuse 93430, France

2. 

University of Texas at Austin, 1 University Station C1200, Austin, TX 78712, United States

Received  April 2006 Published  October 2006

We provide sufficient conditions on a positive function so that the associated special flow over any irrational rotation is either weak mixing or $L^2$-conjugate to a suspension flow. This gives the first such complete classification within the class of Liouville dynamics. This rigidity coexists with a plethora of pathological behaviors.
Citation: 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
[1]

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

[2]

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

[3]

Benoît Saussol. Recurrence rate in rapidly mixing dynamical systems. Discrete & Continuous Dynamical Systems - A, 2006, 15 (1) : 259-267. doi: 10.3934/dcds.2006.15.259

[4]

Byungik Kahng, Miguel Mendes. The characterization of maximal invariant sets of non-linear discrete-time control dynamical systems. Conference Publications, 2013, 2013 (special) : 393-406. doi: 10.3934/proc.2013.2013.393

[5]

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

[6]

Sun-Sig Byun, Hongbin Chen, Mijoung Kim, Lihe Wang. Lp regularity theory for linear elliptic systems. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 121-134. doi: 10.3934/dcds.2007.18.121

[7]

Roman Šimon Hilscher. On general Sturmian theory for abnormal linear Hamiltonian systems. Conference Publications, 2011, 2011 (Special) : 684-691. doi: 10.3934/proc.2011.2011.684

[8]

Matthias Morzfeld, Daniel T. Kawano, Fai Ma. Characterization of damped linear dynamical systems in free motion. Numerical Algebra, Control & Optimization, 2013, 3 (1) : 49-62. doi: 10.3934/naco.2013.3.49

[9]

Wenlei Li, Shaoyun Shi. Weak-Painlevé property and integrability of general dynamical systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3667-3681. doi: 10.3934/dcds.2014.34.3667

[10]

Pedro J. Torres. Non-collision periodic solutions of forced dynamical systems with weak singularities. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 693-698. doi: 10.3934/dcds.2004.11.693

[11]

Oliver Díaz-Espinosa, Rafael de la Llave. Renormalization and central limit theorem for critical dynamical systems with weak external noise. Journal of Modern Dynamics, 2007, 1 (3) : 477-543. doi: 10.3934/jmd.2007.1.477

[12]

Tomás Caraballo, David Cheban. On the structure of the global attractor for non-autonomous dynamical systems with weak convergence. Communications on Pure & Applied Analysis, 2012, 11 (2) : 809-828. doi: 10.3934/cpaa.2012.11.809

[13]

Jean René Chazottes, F. Durand. Local rates of Poincaré recurrence for rotations and weak mixing. Discrete & Continuous Dynamical Systems - A, 2005, 12 (1) : 175-183. doi: 10.3934/dcds.2005.12.175

[14]

Oliver Knill. Singular continuous spectrum and quantitative rates of weak mixing. Discrete & Continuous Dynamical Systems - A, 1998, 4 (1) : 33-42. doi: 10.3934/dcds.1998.4.33

[15]

Piotr Oprocha. Chain recurrence in multidimensional time discrete dynamical systems. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 1039-1056. doi: 10.3934/dcds.2008.20.1039

[16]

Karl P. Hadeler. Quiescent phases and stability in discrete time dynamical systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 129-152. doi: 10.3934/dcdsb.2015.20.129

[17]

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

[18]

Michael Cranston, Benjamin Gess, Michael Scheutzow. Weak synchronization for isotropic flows. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3003-3014. doi: 10.3934/dcdsb.2016084

[19]

Fritz Colonius, Alexandre J. Santana. Topological conjugacy for affine-linear flows and control systems. Communications on Pure & Applied Analysis, 2011, 10 (3) : 847-857. doi: 10.3934/cpaa.2011.10.847

[20]

Xianpeng Hu, Hao Wu. Long-time behavior and weak-strong uniqueness for incompressible viscoelastic flows. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3437-3461. doi: 10.3934/dcds.2015.35.3437

2016 Impact Factor: 0.706

Metrics

  • PDF downloads (3)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]