Number theory and dynamical systems on foliated spaces

TL;DR

This paper explores deep analogies between number theory and dynamical systems on foliated spaces, highlighting connections through trace formulas, Euler characteristics, and the role of solenoids as phase spaces.

Contribution

It introduces a novel perspective linking number theory and dynamical systems via trace formulas and geometric invariants, proposing new interpretations of classical concepts.

Findings

Comparison of explicit formulas with Lefschetz trace formulas

Potential relation between Arakelov-Euler and Connes' Euler characteristic

Role of generalized solenoids as phase spaces

Abstract

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.