By Don Blasius, Jonathan Rogawski (auth.), Alexander Reznikov, Norbert Schappacher (eds.)

This publication is an outgrowth of the Workshop on "Regulators in research, Geom etry and quantity thought" held on the Edmund Landau heart for study in Mathematical research of The Hebrew collage of Jerusalem in 1996. throughout the guidance and the preserving of the workshop we have been tremendously helped by means of the director of the Landau heart: Lior Tsafriri through the time of the making plans of the convention, and Hershel Farkas throughout the assembly itself. Organizing and working this workshop was once a real excitement, because of the specialist technical aid supplied through the Landau heart often, and through its secretary Simcha Kojman particularly. we wish to specific our hearty because of them all. even though, the articles assembled within the current quantity don't symbolize the lawsuits of this workshop; neither may perhaps all members to the ebook make it to the assembly, nor do the contributions herein inevitably mirror talks given in Jerusalem. within the creation, we define our view of the speculation to which this quantity intends to give a contribution. The an important goal of the current quantity is to assemble strategies, equipment, and effects from research, differential in addition to algebraic geometry, and quantity concept with a view to paintings in the direction of a deeper and extra accomplished figuring out of regulators and secondary invariants. Our thank you visit the entire contributors of the workshop and authors of this quantity. may perhaps the readers of this booklet take pleasure in and take advantage of the combo of mathematical rules right here documented.

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**Example text**

Thus we see that a good theory of motives will be useful to produce many sheaves F = Fn(M), but also that at least the problems one faces with respect to motives are no obstacle to building up the dynamical theory we are hoping for. 9. We close this section by giving another heuristic argument why the condition T U EB rs integrable should be essential in our context. Consider a smooth Anosov flow (X, qi) on a compact manifold X and let h be the topological entropy of the flow. For any unitary character X : HI (X, IE) -7 U (1), set L(X,s) = n(l- X([yDe-sl(y»-1 y where y runs over the closed orbits of the flow.

2) The restriction di : A 07i* ~ A o+I7i* agrees with dTj for i = 0, 1. Q in general. * ® AP7o* ~ A Q+ 171* ® A P7o* equals the extension of the flat TI-connection on A P7o* induced by the flat TI-connection /\ -I 0 d 1 110* : TQ* ~ 71* ® 70*· PROOF. *. We have to show that dT,ApQ C AP+1,Q EB AP,Q+I. The assertion for dT' A 10 for example follows from the formula: and the integrability of TI which implies that dT'w has no component in A 02 if w E AIO. The general case follows similarly using equation (1).

If(X, C) agree with the cohomologies HV(specWq/Lq) of [DI, §4]. Note that the assignment specWq t-+ (X = lRjlogqZ, ¢t) is functorial! , to its only closed orbit. 8. 6 has been extended to a functor from smooth quasiprojective varieties Xo/ k to dynamical systems (X, ¢t) with a decomposition T X = ToX €a TO X as above. We also write "Xo" = X. For such a variety 1l'0 : Xo -+ speck, let 1l' = "1l'0" : "Xo" -+ "speck" be the associated morphism of dynamical systems. The functor Xo I~ R i 1l'*(Rx) defines a cohomology theory with values in the abelian category of R"speck"modules with action opposite to the F -flow on "spec k".