American Institute of Mathematical Sciences

Advanced
January  2007, 1(1): 147-153. doi: 10.3934/jmd.2007.1.147

The Hopf argument

Yves Coudène 1,

1. 

IRMAR, Université Rennes 1, Campus Beaulieu, bat.23, 35042 Rennes cedex, France

Received  July 2006 Published  October 2006

Let $T$ be a measure-preserving transformation of a metric space $X$. Assume $T$ is conservative and $X$ can be covered by a countable family of open sets, each of finite measure. Then any eigenfunction is invariant with respect to the stable foliation of $T$.
Keywords: ergodicity., Hopf argument.
Mathematics Subject Classification: 37D10, 37A40, 37A2.
Citation: Yves Coudène. The Hopf argument. Journal of Modern Dynamics, 2007, 1 (1) : 147-153. doi: 10.3934/jmd.2007.1.147
[1]

Charles Pugh, Michael Shub, Alexander Starkov. Unique ergodicity, stable ergodicity, and the Mautner phenomenon for diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2006, 14 (4) : 845-855. doi: 10.3934/dcds.2006.14.845
[2]

Bi Ping, Maoan Han. Oscillation of second order difference equations with advanced argument. Conference Publications, 2003, 2003 (Special) : 108-112. doi: 10.3934/proc.2003.2003.108
[3]

Marian Gidea, Clark Robinson. Obstruction argument for transition chains of tori interspersed with gaps. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 393-416. doi: 10.3934/dcdss.2009.2.393
[4]

Gabriel Rivière. Remarks on quantum ergodicity. Journal of Modern Dynamics, 2013, 7 (1) : 119-133. doi: 10.3934/jmd.2013.7.119
[5]

Jon Chaika, Rodrigo Treviño. Logarithmic laws and unique ergodicity. Journal of Modern Dynamics, 2017, 11: 563-588. doi: 10.3934/jmd.2017022
[6]

Marat Akhmet. Quasilinear retarded differential equations with functional dependence on piecewise constant argument. Communications on Pure & Applied Analysis, 2014, 13 (2) : 929-947. doi: 10.3934/cpaa.2014.13.929
[7]

Irena Lasiecka, Roberto Triggiani. Global exact controllability of semilinear wave equations by a double compactness/uniqueness argument. Conference Publications, 2005, 2005 (Special) : 556-565. doi: 10.3934/proc.2005.2005.556
[8]

Marat Akhmet, Duygu Aruğaslan. Lyapunov-Razumikhin method for differential equations with piecewise constant argument. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 457-466. doi: 10.3934/dcds.2009.25.457
[9]

Jianqing Chen. A variational argument to finding global solutions of a quasilinear Schrödinger equation. Communications on Pure & Applied Analysis, 2008, 7 (1) : 83-88. doi: 10.3934/cpaa.2008.7.83
[10]

Karl Grill, Christian Tutschka. Ergodicity of two particles with attractive interaction. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4831-4838. doi: 10.3934/dcds.2015.35.4831
[11]

Henk Bruin, Gregory Clack. Inducing and unique ergodicity of double rotations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4133-4147. doi: 10.3934/dcds.2012.32.4133
[12]

Dubi Kelmer. Quantum ergodicity for products of hyperbolic planes. Journal of Modern Dynamics, 2008, 2 (2) : 287-313. doi: 10.3934/jmd.2008.2.287
[13]

Federico Rodriguez Hertz, María Alejandra Rodriguez Hertz, Raúl Ures. Partial hyperbolicity and ergodicity in dimension three. Journal of Modern Dynamics, 2008, 2 (2) : 187-208. doi: 10.3934/jmd.2008.2.187
[14]

Ryan T. Botts, Ale Jan Homburg, Todd R. Young. The Hopf bifurcation with bounded noise. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2997-3007. doi: 10.3934/dcds.2012.32.2997
[15]

Matteo Franca, Russell Johnson, Victor Muñoz-Villarragut. On the nonautonomous Hopf bifurcation problem. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1119-1148. doi: 10.3934/dcdss.2016045
[16]

John Guckenheimer, Hinke M. Osinga. The singular limit of a Hopf bifurcation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2805-2823. doi: 10.3934/dcds.2012.32.2805
[17]

Jan Philipp Schröder. Ergodicity and topological entropy of geodesic flows on surfaces. Journal of Modern Dynamics, 2015, 9: 147-167. doi: 10.3934/jmd.2015.9.147
[18]

Marianne Akian, Stéphane Gaubert, Antoine Hochart. Ergodicity conditions for zero-sum games. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 3901-3931. doi: 10.3934/dcds.2015.35.3901
[19]

Dmitry Dolgopyat. The work of Federico Rodriguez Hertz on ergodicity of dynamical systems. Journal of Modern Dynamics, 2016, 10: 175-189. doi: 10.3934/jmd.2016.10.175
[20]

David Ralston, Serge Troubetzkoy. Ergodicity of certain cocycles over certain interval exchanges. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2523-2529. doi: 10.3934/dcds.2013.33.2523

2018 Impact Factor: 0.295

Tools

Metrics

  • PDF downloads (17)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences