Patterson-Sullivan distributions and quantum ergodicity

Nalini Anantharaman*, Steve Zelditch

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

17 Scopus citations


This article gives relations between two types of phase space distributions associated to eigenfunctions φirj of the Laplacian on a compact hyperbolic surface X Γ: Wigner distributions ∫S*XΓ a dWirj = Op(a)φirj,φ 〈 irj L2Γ, which arise in quantum chaos. They are invariant under the wave group. Patterson-Sullivan distributions PSirj, which are the residues of the dynamical zeta-functions Z(s; a) := ∑γ -sL_γ1-e-L_γ∫γ 0 a (where the sum runs over closed geodesics) at the poles s = 1/2 + ir j . They are invariant under the geodesic flow. We prove that these distributions (when suitably normalized) are asymptotically equal as r_j → ∞ . We also give exact relations between them. This correspondence gives a new relation between classical and quantum dynamics on a hyperbolic surface, and consequently a formulation of quantum ergodicity in terms of classical ergodic theory.

Original languageEnglish (US)
Pages (from-to)361-426
Number of pages66
JournalAnnales Henri Poincare
Issue number2
StatePublished - Apr 2007

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Nuclear and High Energy Physics
  • Mathematical Physics


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