Patterson–Sullivan Distributions and Quantum Ergodicity

“…This pairing formula is of independent interest as a step towards understanding the high frequency behavior of resonant states and attempting to prove quantum ergodicity of resonant states in the present setting. Anantharaman-Zelditch [AnZe07] obtained the pairing formula in dimension 2 and studied concentration of Patterson-Sullivan distributions, which are directly related to resonant states; see also [HHS].…”

mentioning

confidence: 99%

“…In dimension 2, the correspondence between the eigenfunctions of the Laplacian on the hyperbolic plane and distributions on the conformal boundary S 1 appeared in Pollicott [Po86b] and , it is also an important tool in the theory developed by to study Selberg zeta functions on convex co-compact hyperbolic manifolds (see also the book of Juhl [Ju] in the compact setting). These distributions on the conformal boundary S n , of Patterson-Sullivan type, are also the central object of the recent work of Anantharaman-Zelditch [AnZe07,AnZe12] studying quantum ergodicity on hyperbolic compact surfaces; a generalization to higher rank locally symmetric spaces was provided by Hansen-Hilgert-Schröder [HHS].…”

mentioning

confidence: 99%

Power spectrum of the geodesic flow on hyperbolic manifolds

2015

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Abstract. We describe the complex poles of the power spectrum of correlations for the geodesic flow on compact hyperbolic manifolds in terms of eigenvalues of the Laplacian acting on certain natural tensor bundles. These poles are a special case of Pollicott-Ruelle resonances, which can be defined for general Anosov flows. In our case, resonances are stratified into bands by decay rates. The proof also gives an explicit relation between resonant states and eigenstates of the Laplacian.In this paper, we consider the characteristic frequencies of correlations,for the geodesic flow ϕ t on a compact hyperbolic manifold M of dimension n + 1 (that is, M has constant sectional curvature −1). Here ϕ t acts on SM , the unit tangent bundle of M , and µ is the natural smooth probability measure. Such ϕ t are classical examples of Anosov flows; for this family of examples, we are able to prove much more precise results than in the general Anosov case.An important question, expanding on the notion of mixing, is the behavior of ρ f,g (t) as t → +∞. Following [Ru], we take the power spectrum, which in our convention is the Laplace transformρ f,g (λ) of ρ f,g restricted to t > 0. The long time behavior of ρ f,g (t) is related to the properties of the meromorphic extension ofρ f,g (λ) to the entire complex plane. The poles of this extension, called Pollicott-Ruelle resonances (see [Po86a,Ru,FaSj] and (1.7) below), are the complex characteristic frequencies of ρ f,g , describing its decay and oscillation and not depending on f, g.For the case of dimension n + 1 = 2, the following connection between resonances and the spectrum of the Laplacian was announced in [FaTs13a, Section 4] (see [FlFo] for a related result and the remarks below regarding the zeta function techniques). Theorem 1. Assume that M is a compact hyperbolic surface (n = 1) and the spectrum of the positive Laplacian on M is (see Figure 1)

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“…The choice of notation in (21) refers to the fact that g −1 −1 maps to D −1 , and all the other elements g For the case that χ is the trivial character, (24) coincides with ±L Mayer s , see (1). It is does not coincide with any other transfer operator existing for PSL 2 (Z).…”

Section: 1mentioning

confidence: 99%

“…Note that for α (1) s the set C * R is the largest domain that contains (0, 1) and on which all the cocycles in (36) (for all n ∈ N) are well-defined and holomorphic. For α (2) s , the slit plane C ′ is the largest domain with these properties.…”

Section: Letmentioning

confidence: 99%

A transfer-operator-based relation between Laplace eigenfunctions and zeros of Selberg zeta functions

Adam¹,

Pohl²

2018

Ergod. Th. Dynam. Sys.

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Over the last few years Pohl (partly jointly with coauthors) developed dual 'slow/fast' transfer operator approaches to automorphic functions, resonances, and Selberg zeta functions for a certain class of hyperbolic surfaces Γ\H with cusps and all finite-dimensional unitary representations χ of Γ.The eigenfunctions with eigenvalue 1 of the fast transfer operators determine the zeros of the Selberg zeta function for (Γ, χ). Further, if Γ is cofinite and χ is the trivial one-dimensional representation then highly regular eigenfunctions with eigenvalue 1 of the slow transfer operators characterize Maass cusp forms for Γ. Conjecturally, this characterization extends to more general automorphic functions as well as to residues at resonances.In this article we study, without relying on Selberg theory, the relation between the eigenspaces of these two types of transfer operators for any Hecke triangle surface Γ\H of finite or infinite area and any finite-dimensional unitary representation χ of the Hecke triangle group Γ. In particular we provide explicit isomorphisms between relevant subspaces. This solves a conjecture by Möller and Pohl, characterizes some of the zeros of the Selberg zeta functions independently of the Selberg trace formula, and supports the previously mentioned conjectures.

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Patterson–Sullivan Distributions and Quantum Ergodicity

Cited by 26 publications

References 42 publications

Building lattices and zeta functions

Building lattices and zeta functions

Power spectrum of the geodesic flow on hyperbolic manifolds

A transfer-operator-based relation between Laplace eigenfunctions and zeros of Selberg zeta functions

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