Cohomological Theory of Dynamical Zeta Functions

Juhl, Andreas

doi:10.1007/978-3-0348-8340-5

Cited by 23 publications

(16 citation statements)

References 95 publications

(177 reference statements)

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“…The trace on chains gave the logarithmic derivative of the (Ruelle) zeta function, while the trace on homology gave the spectral side of the Selberg trace formula. For later developments in this direction (by C. Denninger, A. Deitmar, U. Bunke, M. Olbrich and others) we refer to [17]. This paper develops the close relation on the level of eigenfunctions and invariant distributions rather than just eigenvalues and lengths of closed geodesics.…”

Section: More Generally If σ Is An Eigenfunction Of Casimir Parametementioning

confidence: 93%

Patterson–Sullivan Distributions and Quantum Ergodicity

Anantharaman

Zelditch

2007

Ann. Henri Poincaré

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Abstract. This article gives relations between two types of phase space distributions associated to eigenfunctions φir j of the Laplacian on a compact hyperbolic surface XΓ:, which arise in quantum chaos. They are invariant under the wave group.• Patterson-Sullivan distributions P Sir j , which are the residues of the We prove that these distributions (when suitably normalized) are asymptotically equal as rj → ∞. 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.

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Section: More Generally If σ Is An Eigenfunction Of Casimir Parametementioning

confidence: 93%

Patterson–Sullivan Distributions and Quantum Ergodicity

Anantharaman

Zelditch

2007

Ann. Henri Poincaré

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“…They also consider general locally symmetric spaces and address the question of what happens at the exceptional points (which in our case are contained in − n 2 − 1 2 N 0 ), relating the behavior of the zeta functions at these points to topological invariants. It is possible that the results [Ju,BuOl95,BuOl96,BuOl99,BuOl01] together with an appropriate representation theoretic calculation recover our description of resonances, even though no explicit description featuring the spectrum of the Laplacian on trace-free divergence-free symmetric tensors as in (1.3), (1.4) seems to be available in the literature. The direct spectral approach used in this paper, unlike the zeta function techniques, gives an explicit relation between resonant states and eigenstates of the Laplacian (see the remarks following (1.7)) and is a step towards a more quantitative understanding of decay of correlations.…”

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confidence: 92%

“…However, the Ruelle zeta function does not recover all resonances on functions, due to cancellations with singularities coming from differential forms of different orders. For example, [Ju,Theorem 3.7] describes the spectral singularities of the Ruelle zeta function for n = 3 in terms of the spectrum of the Laplacian on functions and 1-forms, which is much smaller than the set obtained in Theorem 2.…”

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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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“…Algebraic group averaging methods have been used to prove trace formulae in [Guillopé and Zworski 1999;Guillarmou and Naud 2006;Perry 2003] on manifolds with constant negative curvature and infinite volume. Further results include [Arthur 1989;Borthwick et al 2005;Patterson and Perry 2001;Juhl 2001;Müller 1983]. For manifolds with infinite volume, a renormalized wave trace replaces the standard wave trace in the dynamical formula.…”

Section: Introductionmentioning

confidence: 99%

Dynamics of asymptotically hyperbolic manifolds

Rowlett¹

2009

Pacific J. Math.

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We prove a dynamical wave trace formula for asymptotically hyperbolic (n + 1)-dimensional manifolds with negative (but not necessarily constant) sectional curvatures; the formula equates the renormalized wave trace to the lengths of closed geodesics. This result generalizes the classical theorem of Duistermaat and Guillemin for compact manifolds and the results of Guillopé and Zworski, Perry, and Guillarmou and Naud for hyperbolic manifolds with infinite volume. A corollary of this dynamical trace formula is a dynamical resonance-wave trace formula for compact perturbations of convex cocompact hyperbolic manifolds. We define a dynamical zeta function and prove its analyticity in a half plane. In our main result, we produce a prime orbit theorem for the geodesic flow. This is the first such result for manifolds that have neither constant curvature nor finite volume. As a corollary to the prime orbit theorem, using our dynamical resonance-wave trace formula, we show that the existence of pure point spectrum for the Laplacian on negatively curved compact perturbations of convex cocompact hyperbolic manifolds is related to the dynamics of the geodesic flow.

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Cohomological Theory of Dynamical Zeta Functions

Cited by 23 publications

References 95 publications

Patterson–Sullivan Distributions and Quantum Ergodicity

Patterson–Sullivan Distributions and Quantum Ergodicity

Power spectrum of the geodesic flow on hyperbolic manifolds

Dynamics of asymptotically hyperbolic manifolds

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