Your cart is empty.

11M36 Number theory -- Zeta and L-functions: analytic theory -- Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. Explicit formulas

22E46 Topological groups, Lie groups -- Lie groups -- Semisimple Lie groups and their representations

11F72 Number theory -- Discontinuous groups and automorphic forms -- Spectral theory; Selberg trace formula

58Jxx Global analysis, analysis on manifolds -- Partial differential equations on manifolds; differential operators En-ligne: Springerlink | MSN | zbMath

No physical items for this record

Publisher’s description: 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 Ruelles 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 functi lso 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 not only 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.

There are no comments for this item.