Some questions around quasi-periodic dynamics

TL;DR

This paper explores fundamental questions in quasi-periodic dynamics, emphasizing the impact of arithmetic properties like Diophantine and Liouville behavior on stability, rigidity, and phenomena across low-dimensional systems and their connections to number theory.

Contribution

It provides a comprehensive discussion of classical and new questions in quasi-periodic dynamics, highlighting the duality between Diophantine and Liouville cases and proposing unified approaches.

Findings

Diophantine dynamics exhibit rigidity and stability.

Liouville dynamics lack rigidity and stability.

Connections between dynamics and Diophantine approximation are discussed.

Abstract

We propose in these notes a list of some old and new questions related to quasi-periodic dynamics. A main aspect of quasi-periodic dynamics is the crucial influence of arithmetics on the dynamical features, with a strong duality in general between Diophantine and Liouville behavior. We will discuss rigidity and stability in Diophantine dynamics as well as their absence in Liouville ones. Beyond this classical dichotomy between the Diophantine and the Liouville worlds, we discuss some unified approaches and some phenomena that are valid in both worlds. Our focus is mainly on low dimensional dynamics such as circle diffeomorphisms, disc dynamics, quasi-periodic cocycles, or surface flows, as well as finite dimensional Hamiltonian systems. In an opposite direction, the study of the dynamical properties of some diagonal and unipotent actions on the space of lattices can be applied to arithmetics, namely to the theory of Diophantine approximations. We will mention in the last section some problems related to that topic. The field of quasi-periodic dynamics is very extensive and has a wide range of interactions with other mathematical domains. The list of questions we propose is naturally far from exhaustive and our choice was often motivated by our research involvements.