Cohomological Theory of Dynamical Zeta Functions (Hardcover, 2001 ed.)

The periodic orbits of the geodesic flow of compact locally symmetric spaces of negative curvature give rise to meromorphic zeta functions (generalized Selberg zeta functions, Ruelle zeta functions). The book treats various aspects of the idea to understand the analytical properties of these zeta functions on the basis of appropriate analogs of the Lefschetz fixed point formula in which the periodic orbits of the flow take the place of the fixed points. According to geometric quantization the Anosov foliations of the sphere bundle provide a natural source for the definition of the cohomological data in the Lefschetz formula. The Lefschetz formula method can be considered as a link between the automorphic approach (Selberg trace formula) and Ruelle's approach (transfer operators). It yields a uniform cohomological characterization of the zeros and poles of the zeta functions and a new understanding of the functional equations from an index theoretical point of view. The divisors of the Selberg zeta functions also admit characterizations in terms of harmonic currents on the sphere bundle which represent the cohomology classes in the Lefschetz formulas in the sense of a Hodge theory. The concept of harmonic currents to be used for that purpose is introduced here for the first time. Harmonic currents for the geodesic flow of a noncompact hyperbolic space with a compact convex core generalize the Patterson-Sullivan measure on the limit set and are responsible for the zeros and poles of the corresponding zeta function. The book describes the present state of the research in a new field on the cutting edge of global analysis, harmonic analysis and dynamical systems. It should be appealing notonly to the specialists on zeta functions which will find their object of favorite interest connected in new ways with index theory, geometric quantization methods, foliation theory and representation theory. There are many unsolved problems and the book hopefully promotes further progress along the lines indicated here.

General

Imprint:	Birkhauser Verlag AG
Country of origin:	Switzerland
Series:	Progress in Mathematics, 194
Release date:	December 2000
First published:	2001
Authors:	Andreas Juhl
Dimensions:	235 x 155 x 38mm (L x W x T)
Format:	Hardcover
Pages:	709
Edition:	2001 ed.
ISBN-13:	978-3-7643-6405-2
Categories:	Books > Science & Mathematics > Mathematics > Calculus & mathematical analysis > Functional analysis
LSN:	3-7643-6405-X
Barcode:	9783764364052

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