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Zbl 1003.11029
Deninger, Christopher
Number theory and dynamical systems on foliated spaces. (English)
[J] Jahresber. Dtsch. Math.-Ver. 103, No.3, 79-100 (2001). ISSN 0012-0456

This is a report on some ongoing developments in the search for a new understanding of number theoretical zeta functions in terms of dynamical systems and foliations, resp. their cohomology. Central to this theory is a conjectural Lefschetz formula connecting closed orbits of a dynamical system to its action on a foliation cohomology. \par The paper deals with three topics. Firstly, it reports on progress on the dynamical Lefschetz formula for foliations of codimension one which is mainly due to Álvarez López and Kordyukov. Next the explicit formulas of number theory are compared with conjectural Lefschetz formulas. Motivated by this comparison, the author finishes with an extension of the conjectural Lefschetz formulas to the case of laminated spaces or solenoids, i.e., spaces which locally look like $\bbfR^n$ times a totally disconnected space. The author considers solenoids which are foliated by solenoids of lower dimension and formulates a Lefschetz formula in this setting. He gives examples where the formula can be proved.
[Anton Deitmar (Exeter)]
MSC 2000:
*11G40 L-functions of varieties over global fields
11M36 Selberg zeta functions and regularized determinants
14G10 Zeta-functions and related questions
37C27 Periodic orbits of vector fields and flows

Keywords: zeta functions; dynamical systems; foliations; cohomology; conjectural Lefschetz formula; dynamical Lefschetz formula; foliations of codimension one; solenoids

Cited in: Zbl 1138.14028 Zbl 1131.14024

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Highlights
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