By McMullen C.

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This means f (t) = R eitx dµ(x). Corollary. Let U t be a unitary action of R on H with cyclic vector ξ. Then there is: 42 • a probability measure µ on R, and • an isomorphism H → L2 (R, µ), such that • ξ corresponds to the constant function 1, and • U t is sent to the action of multiplication by eixt . Proof. First let f (t) = U t ξ, ξ . Then f (ti − tj )ai aj = ai U ti ξ 2 ≥0 so f is positive definite. e. those N functions of the form g(x) = 1 ai eixti . ) Map A to H by sending g to N 1 ai U ξ.

2. First consider the representations that might occur in the regular representation on L2 (H). As is the case for R, the irreducible representations should correspond to eigenfunctions of the Laplacian (like exp(it) on R), since ∆ commutes with the action of G. Also we should not expect the eigenfunctions themselves to be in L2 . Finally the eigenvalues should be 63 real and positive since ∆ is a positive operator on L2 . ) To produce eigenfunctions of ∆, consider a conformal density µ = µ(x)|dx|s on S 1 = ∂H.

Using mixing of the geodesic flow, one can show the parallels Lt of γ at distance t are equidistributed on Y . ) Now consider the covering space Z → Y corresponding to π1 (γ) ⊂ π1 (Y ). Let E ⊂ Z be the projection of Γp to Z. Then it is not hard to see that N (R) is the same as |E ∩B(γ ′ , R)|, where γ ′ is the canonical lift of γ from Y to Z. Projecting Dirichlet regions based at Γp to Z gives the heuristic for the count; it is justified by equidistribution of parallels to γ. 5. M¨ ahler’s compactness criterion.