Some analogies between number theory and dynamical systems on foliated spaces.

By C. Deninger. 2002

124 citations recorded by the source corpus.

We discuss analogies between number theory and the theory of dynamical systems on spaces with a one-codimensional foliation. The emphasis is on comparing the "explicit formulas" of analytic number theory with certain dynamical Lefschetz trace formulas. We also point out a possible relation between an Arakelov-Euler characteristic and an Euler characteristic in the sense of Connes. Finally the role of generalized solenoids as phase spaces in our picture is explained.

Subfields: Geometry and Topology, Mathematical Physics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/2819f958bd8b2cb9ba66c3c5dc0b29f4505cb86e

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

In Mathematical Physics

In Nuclear and High Energy Physics

Citations in the corpus

Noncommutative geometry, dynamics, and ∞-adic Arakelov geometry (2004, background): “Deninger’s theory of “arithmetic cohomology”, for which the importance of solenoids was stressed in the remarks of §5.8 of [ 21 ].”
The notion of dimension in geometry and algebra (2005, background): “In particular, [ De3 ], [De4] postulate existence of dynamical systems underlying this geometry, whereas [ConsMar] introduces non-commutative spaces responsible for Γ–factors for curves, and Connes in [Co5] outlines an approach to the Riemann hypothesis by way of non-commutative geometry.”
Euler's constant: Euler's work and modern developments (2013, methodology): “ing (possibly) associated to a dynamical entropy, one approach to the Riemann hypothesis suggests that it may be related to an (as yet unknown) arithmetical dynamical system, cf. Deninger [65], [66], [67], [68]. There is a statistical mechanics interpretation of the Riemann zeta function as a partit”