In the 1960s Manin, Mazur and Mumford noted that from the viewpoint of étale topology there was an intriguing analogy between prime numbers embedded into the spectrum of the integers and knots in 3-space. Later Kapranov, Reznikov, Morishita and other authors discovered further analogies between number rings and the topology of 3-manifolds. For example, the Iwasawa zeta function corresponds to the Alexander polynomial of a knot. The search for a cohomology theory related to the Riemann zeta function led to the discovery of analogies between number rings and a class of 3-dimensional dynamical systems, where the primes would correspond to the periodic orbits. For example, Riemann’s explicit formulas in analytic number theory correspond to a transversal index theorem in the dynamical context, proved by Álvarez-López, Kordyukov and Leichtnam. The dynamical systems analogy refines the previous analogy because a periodic orbit gives a knot. The analogies extend to higher dimensional arithmetic schemes and for example Lichtenbaum's conjecture on the special value at zero of the Hasse Weil zeta function in terms of Weil étale cohomology has a word by word dynamical analog which can be proved using the Cheeger Mueller theorem. We have constructed foliated dynamical systems for number rings and even for all arithmetic schemes that have some but not yet all the expected properties.
