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By Andreas Juhl

Dynamical zeta services are linked to dynamical platforms with a countable set of periodic orbits. The dynamical zeta services of the geodesic move of lo­ cally symmetric areas of rank one are recognized additionally because the generalized Selberg zeta services. the current e-book is anxious with those zeta services from a cohomological viewpoint. initially, the Selberg zeta functionality seemed within the spectral idea of automorphic types and have been advised through an analogy among Weil's specific formulation for the Riemann zeta functionality and Selberg's hint formulation ([261]). the aim of the cohomological idea is to appreciate the analytical homes of the zeta services at the foundation of compatible analogs of the Lefschetz mounted aspect formulation within which periodic orbits of the geodesic move take where of mounted issues. This strategy is parallel to Weil's thought to research the zeta features of professional­ jective algebraic forms over finite fields at the foundation of appropriate models of the Lefschetz mounted element formulation. The Lefschetz formulation formalism indicates that the divisors of the rational Hassc-Wcil zeta services are made up our minds by way of the spectra of Frobenius operators on l-adic cohomology.

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In particular, the Riemannian structure on stable and unstable leaves in SY projects to a Riemannian structure on stable and unstable leaves in SX = r\SY. We recall also that for Y = IHIn the induced metric on the horospheres is Euclidean. Hence by the preceding construction the stable and unstable leaves in SX become isometric images of lR n - 1 with scalar multiples of the usual Euclidean metric. We choose orientations on the horospheres which are preserved by G and fix orientations on the stable (unstable) leaves in SY by transport of the orientations on the corresponding horospheres.

In fact, suitable constant multiples of the map w f---t 0;- 1\ w provide solution operators CCi~~~(SY) ~ ker6; -7 cci~~~~ol1)(SY). Here 0;- = d- He; E n~20) (SY), where He; E n~20) (SY) is defined by He; (kan-) = Ilogn-1 2 , no· I . , the complex {w E n~n_-;~:~I)(SY) 16-w = O} '-7 cct~~~021)(SY) ~ cct~~~022)(SY) E .... ~=... cciO,O) (SY) -70 is a resolution of the spherical principal series representation on {w E nt-;~:21)(SY) 16-w = O}. The latter argument proving the triviality of the cohomology of the canonical complexes on SY, however, does not apply to the canonical complexes of currents on the compact quotient SX = r\SY since 0;- does not drop to a form on SX.

The currents which are responsible for the divisor are constant along P-. In dimension n :::: 4, however, the analogous description of D(Zs) requires us to consider currents which are polynomials of arbitrary large order on the leaves of P- . Next we consider the canonical complexes and the Hodge decompositions. The canonical complexes are the finite-dimensional complexes A 0--+ CC(1,O) (SX) A d---+ A CC(O,O) (SX) A --+ 0 of currents, where CC ,O) (SX) = {w E nZ1 •0 ) (SX) I rd-w = O} nzo. 18-w = 0 and Hence and CH~,o)(SX) = {w E O) (SX) O} /d- {w E nz 1 ,0) (SX) 18-d-w = The space CH(l,O)(SX) is the space of invariant transverse currents = o}.

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