Patterson-Sullivan distributions and quantum ergodicity

Abstract

We relate two types of phase space distributions associated to eigenfunctions ${\varphi }_{i{r}_j}$ of the Laplacian on a compact hyperbolic surface $X_{\Gamma }$: (1) Wigner distributions $\int_{S^*\X} a dW_{ir_j}=< Op(a)ϕ_{ir_j}, ϕ_{ir_j}>_{L^2(\X)}$, which arise in quantum chaos. They are invariant under the wave group. (2) Patterson-Sullivan distributions $P{S}_{i{r}_j}$, which are the residues of the dynamical zeta-functions $\lcal(s; a): = \sum_γ\frac{e^{-sL_γ}}{1-e^{-L_γ}} \int_{γ_0} a$ (where the sum runs over closed geodesics) at the poles $s=1/2+i{r}_j$. They are invariant under the geodesic flow. We prove that these distributions (when suitably normalized) are asymptotically equal as $r_j\to \infty$. 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.

54 pages, no figures. Added some references

References in corpus (1)

• Microlocal lifts of eigenfunctions on hyperbolic surfaces and trilinear invariant functionals

Cited by in corpus (3)

• Local and global analysis of eigenfunctions
• Eigenfunctions of the Laplacian and associated Ruelle operator
• Patterson--Sullivan distributions for rank one symmetric spaces of the noncompact type