The zeta functions of Ruelle and Selberg. I

Fried, David

doi:10.24033/asens.1515

Cited by 127 publications

(107 citation statements)

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“…Fried showed the same result using the same ideas as above, except that he worked in the framework of the associated geodesic flow [6]. The viewpoint above appears more satisfactory for commenting on the poles of Ç(s) in light of the previous sections.…”

Section: Zeta Functionsmentioning

confidence: 58%

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Kleinian Groups, Laplacian on Forms and Currents at Infinity

Pollicott¹

1990

Proceedings of the American Mathematical Society

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Abstract.In this note we consider the spectrum of the Laplacian acting on the space of (co-closed) differential forms on the quotient of /i-dimensional hyperbolic space by a co-compact Kleinian group. Using a result of P.-Y. Gaillard we relate these to currents on the sphere at infinity of hyperbolic space with distinctive transformation properties under the action of the group. We analyse these currents using zeta-functions and Ruelle's Transfer operator. This represents a partial extension of earlier work of the author related to Fuchsian groups. In an appendix we propose an alternative approach to related questions. IntroductionThe purpose of this note is to clarify the interaction of the objects in the title and to investigate their relationship with the "Transfer" operators (in the sense of Ruelle) which play an important role in symbolic dynamics and ergodic theory.We shall chiefly be concerned with co-compact Kleinian groups Y (which are discrete groups of isometries of zz-dimensional hyperbolic space). Work of Patterson [11] and Sullivan [15] has helped to understand the relationship between the first eigenvalue of the Laplacian on functions for V = Hn /Y and a distinguished measure zzz on the sphere at infinity S"~ (supported on the limit set A). In [13] we observed that, for Fuchsian groups, the other eigenvalues of the Laplacian could be understood in terms of certain distributions on the sphere at infinity (via a result of Helgason [10]), and then in terms of transfer operators. In this article we shall show that, after some necessary modifications, this correspondence continues for eigenvalues of the Laplacian on (co-closed) forms and currents on the sphere at infinity (i.e., linear functional on forms). In particular, these currents can be approached in terms of transfer operators on forms at infinity. We shall also consider the relationship with zeta functions.Besides purely aesthetic appeal, this analysis has the advantage of presenting an approach to Laplacian spectra which seems particularly amenable to explicit

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Section: Zeta Functionsmentioning

confidence: 58%

“…Recall the simple fact that a p-form is harmonic if and only if it is both closed and co-closed, cf. [6].…”

Section: Laplacian and Currentsmentioning

confidence: 99%

Kleinian Groups, Laplacian on Forms and Currents at Infinity

Pollicott¹

1990

Proceedings of the American Mathematical Society

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“…The latter zeta-functions coincide with Fredholmdeterminants det(l -L*(s)) (in the sense of Grothendieck) of holomorphic families L,(s) of nuclear transfer operators on certain spaces of differential forms. This representation implies that Zr has a meromorphic continuation to the complex plane (see [5], [14]). While these arguments establish the existence of a meromorphic continuation, it seems to be rather difficult to prove results on the positions and the multiplicities of the singularities of Zr by the same method.…”

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confidence: 99%

“…Now we use these periods to define the zeta-function The function Zr is well known as the Ruelle zeta-function of the geodesic flow [5]. The Euler product (1) defines a holomorphic function in the half-plane Re(s) > h. Now by symbolic dynamics the zeta-function Zr can be written for large Re(s) as an alternating product of zeta-functions associated to suspensions of subshifts of finite type.…”

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confidence: 99%

“…Generalized Selberg zeta-functions are also defined by Euler products similar to ( 1 ) but with more complicated local Euler factors containing monodromy contributions of the loops c in certain vector bundles on S(X) (see [3], [5], [7], [12], [17]). The generalized Selberg zetafunctions in turn can be investigated by using trace formula techniques which are at present the only known methods to uncover the deeper relations between periodic orbits and the geometry and topology of the underlying space.…”

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confidence: 99%

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Secondary invariants and the singularity of the Ruelle zeta-function in the central critical point

Juhl¹

1995

Bull. Amer. Math. Soc.

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Abstract.The Ruelle zeta-function of the geodesic flow on the sphere bundle S(X) of an even-dimensional compact locally symmetric space X of rank 1 is a meromorphic function in the complex plane that satisfies a functional equation relating its values in s and -s . The multiplicity of its singularity in the central critical point 5 = 0 only depends on the hyperbolic structure of the flow and can be calculated by integrating a secondary characteristic class canonically associated to the flow-invariant foliations of S(X) for which a representing differential form is given.Let Y = G/K be a rank one symmetric space of the non-compact type, i.e., Y is a real, complex, or quaternionic hyperbolic space or the ( 16-dimensional) hyperbolic Cayley-plane. Let T be a uniform lattice in the (connected simple) isometry group G of y without torsion. Y acts properly discontinuous on Y = G/K (K a maximal compact subgroup of G), and X = Y\G/K is a compact locally symmetric space. We consider I asa Riemannian manifold with respect to an arbitrary (constant) multiple g of the metric go induced by the Killing form on the Lie algebra of G. Then the Riemannian manifold (X, g) is a space of negative curvature.The negativity of the curvature of the metric g on X implies the existence of an infinite countable set of prime closed geodesies in X with a discrete set of prime periods accumulating at infinity.Let , be the geodesic flow on the unit sphere bundle S(X) of the space (X,g).The prime period of a periodic orbit of O, on S(X) coincides with the length of the closed geodesic in X obtained by projecting the periodic orbit into X. Now we use these periods to define the zeta-function

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Loop Spaces and the hypoelliptic Laplacian
Bismut
¹

2007
Comm Pure Appl Math
7
0
3
0
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The zeta functions of Ruelle and Selberg. I

Cited by 127 publications

References 2 publications

Kleinian Groups, Laplacian on Forms and Currents at Infinity

Kleinian Groups, Laplacian on Forms and Currents at Infinity

Secondary invariants and the singularity of the Ruelle zeta-function in the central critical point

Loop Spaces and the hypoelliptic Laplacian

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