American Institute of Mathematical Sciences

Advanced
April  2008, 2(2): 287-313. doi: 10.3934/jmd.2008.2.287

Quantum ergodicity for products of hyperbolic planes

Dubi Kelmer 1,

1. 

Raymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel

Received  August 2007 Revised  November 2007 Published  January 2008

For manifolds with geodesic flow that is ergodic on the unit tangent bundle, the Quantum Ergodicity Theorem implies that almost all Laplacian eigenfunctions become equidistributed as the eigenvalue goes to infinity. For a locally symmetric space with a universal cover that is a product of several upper half-planes, the geodesic flow has constants of motion so it cannot be ergodic. It is, however, ergodic when restricted to the submanifolds defined by these constants. Accordingly, we show that almost all eigenfunctions become equidistributed on these submanifolds.
Keywords: hyperbolic plane., quantum ergodicity.
Mathematics Subject Classification: Primary: 81Q50, Secondary: 43A8.
Citation: 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
[1]

Gabriel Rivière. Remarks on quantum ergodicity. Journal of Modern Dynamics, 2013, 7 (1) : 119-133. doi: 10.3934/jmd.2013.7.119
[2]

F. Rodriguez Hertz, M. A. Rodriguez Hertz, A. Tahzibi and R. Ures. A criterion for ergodicity for non-uniformly hyperbolic diffeomorphisms. Electronic Research Announcements, 2007, 14: 74-81. doi: 10.3934/era.2007.14.74
[3]

Keith Burns, Dmitry Dolgopyat, Yakov Pesin, Mark Pollicott. Stable ergodicity for partially hyperbolic attractors with negative central exponents. Journal of Modern Dynamics, 2008, 2 (1) : 63-81. doi: 10.3934/jmd.2008.2.63
[4]

C.P. Walkden. Stable ergodicity of skew products of one-dimensional hyperbolic flows. Discrete & Continuous Dynamical Systems - A, 1999, 5 (4) : 897-904. doi: 10.3934/dcds.1999.5.897
[5]

Carlos H. Vásquez. Stable ergodicity for partially hyperbolic attractors with positive central Lyapunov exponents. Journal of Modern Dynamics, 2009, 3 (2) : 233-251. doi: 10.3934/jmd.2009.3.233
[6]

Misha Bialy. Hopf rigidity for convex billiards on the hemisphere and hyperbolic plane. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 3903-3913. doi: 10.3934/dcds.2013.33.3903
[7]

M. Bauer, A. Lopes. A billiard in the hyperbolic plane with decay of correlation of type $n^{-2}$. Discrete & Continuous Dynamical Systems - A, 1997, 3 (1) : 107-116. doi: 10.3934/dcds.1997.3.107
[8]

David Brander. Results related to generalizations of Hilbert's non-immersibility theorem for the hyperbolic plane. Electronic Research Announcements, 2008, 15: 8-16. doi: 10.3934/era.2008.15.8
[9]

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
[10]

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

Helmut Kröger. From quantum action to quantum chaos. Conference Publications, 2003, 2003 (Special) : 492-500. doi: 10.3934/proc.2003.2003.492
[12]

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
[13]

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
[14]

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
[15]

Paolo Antonelli, Pierangelo Marcati. Quantum hydrodynamics with nonlinear interactions. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 1-13. doi: 10.3934/dcdss.2016.9.1
[16]

Sergei Avdonin, Pavel Kurasov. Inverse problems for quantum trees. Inverse Problems & Imaging, 2008, 2 (1) : 1-21. doi: 10.3934/ipi.2008.2.1
[17]

Dmitry Jakobson. On quantum limits on flat tori. Electronic Research Announcements, 1995, 1: 80-86.

[18]

Luiza H. F. Andrade, Rui F. Vigelis, Charles C. Cavalcante. A generalized quantum relative entropy. Advances in Mathematics of Communications, 2019, 0 (0) : 0-0. doi: 10.3934/amc.2020063
[19]

Alessandro Fonda, Rafael Ortega. Positively homogeneous equations in the plane. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 475-482. doi: 10.3934/dcds.2000.6.475
[20]

Jorge Groisman. Expansive homeomorphisms of the plane. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 213-239. doi: 10.3934/dcds.2011.29.213

2018 Impact Factor: 0.295

Tools

Metrics

  • PDF downloads (13)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences