American Institute of Mathematical Sciences

Advanced
February  2006, 15(1): 121-142. doi: 10.3934/dcds.2006.15.121

An introduction to joinings in ergodic theory

Thierry de la Rue 1,

1. 

Laboratoire de Mathématiques Raphaël Salem, Université de Rouen, CNRS, Avenue de l'Université, Avenue de l'Université, 76801 Saint Étienne du Rouvray, France

Received  December 2004 Revised  July 2005 Published  February 2006

Since their introduction by Furstenberg [3], joinings have proved a very powerful tool in ergodic theory. We present here some aspects of the use of joinings in the study of measurable dynamical systems, emphasizing
  • the links between the existence of a non trivial common factor and the existence of a joining which is not the product measure,
  • how joinings can be employed to provide elegant proofs of classical results,
  • how joinings are involved in important questions of ergodic theory, such as pointwise convergence or Rohlin's multiple mixing problem.
Keywords: minimal self-joinings, disjointness, Joinings, weak disjointness, 2-fold and 3-fold mixing..
Mathematics Subject Classification: Primary: 37A05, 37A35; Secondary: 60G1.
Citation: Thierry de la Rue. An introduction to joinings in ergodic theory. Discrete & Continuous Dynamical Systems - A, 2006, 15 (1) : 121-142. doi: 10.3934/dcds.2006.15.121
[1]

Younghwan Son. Substitutions, tiling dynamical systems and minimal self-joinings. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4855-4874. doi: 10.3934/dcds.2014.34.4855
[2]

Piotr Oprocha. Double minimality, entropy and disjointness with all minimal systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 263-275. doi: 10.3934/dcds.2019011
[3]

Peter W. Bates, Ji Li, Mingji Zhang. Singular fold with real noise. Discrete & Continuous Dynamical Systems - B, 2016, 21 (7) : 2091-2107. doi: 10.3934/dcdsb.2016038
[4]

Jie Li, Kesong Yan, Xiangdong Ye. Recurrence properties and disjointness on the induced spaces. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1059-1073. doi: 10.3934/dcds.2015.35.1059
[5]

Gheorghe Tigan. Degenerate with respect to parameters fold-Hopf bifurcations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 2115-2140. doi: 10.3934/dcds.2017091
[6]

Philipp Kunde. Smooth diffeomorphisms with homogeneous spectrum and disjointness of convolutions. Journal of Modern Dynamics, 2016, 10: 439-481. doi: 10.3934/jmd.2016.10.439
[7]

Jacek Brzykcy, Krzysztof Frączek. Disjointness of interval exchange transformations from systems of probabilistic origin. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 53-73. doi: 10.3934/dcds.2010.27.53
[8]

Karim Boulabiar, Gerard Buskes and Gleb Sirotkin. A strongly diagonal power of algebraic order bounded disjointness preserving operators. Electronic Research Announcements, 2003, 9: 94-98.

[9]

Jon Chaika, Alex Eskin. Möbius disjointness for interval exchange transformations on three intervals. Journal of Modern Dynamics, 2019, 14: 55-86. doi: 10.3934/jmd.2019003
[10]

Wen Huang, Zhiren Wang, Guohua Zhang. Möbius disjointness for topological models of ergodic systems with discrete spectrum. Journal of Modern Dynamics, 2019, 14: 277-290. doi: 10.3934/jmd.2019010
[11]

Carles Bonet-Revés, Tere M-Seara. Regularization of sliding global bifurcations derived from the local fold singularity of Filippov systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3545-3601. doi: 10.3934/dcds.2016.36.3545
[12]

Matúš Dirbák. Minimal skew products with hypertransitive or mixing properties. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1657-1674. doi: 10.3934/dcds.2012.32.1657
[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]

Bernard Helffer, Thomas Hoffmann-Ostenhof, Susanna Terracini. Nodal minimal partitions in dimension $3$. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 617-635. doi: 10.3934/dcds.2010.28.617
[16]

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
[17]

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
[18]

Lidong Wang, Hui Wang, Guifeng Huang. Minimal sets and $\omega$-chaos in expansive systems with weak specification property. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1231-1238. doi: 10.3934/dcds.2015.35.1231
[19]

Hiromichi Nakayama, Takeo Noda. Minimal sets and chain recurrent sets of projective flows induced from minimal flows on $3$-manifolds. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 629-638. doi: 10.3934/dcds.2005.12.629
[20]

Bernard Ducomet, Eduard Feireisl, Hana Petzeltová, Ivan Straškraba. Global in time weak solutions for compressible barotropic self-gravitating fluids. Discrete & Continuous Dynamical Systems - A, 2004, 11 (1) : 113-130. doi: 10.3934/dcds.2004.11.113

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (31)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences