Some Questions Around Quasi-Periodic Dynamics

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.

Global rigidity for ultra-differentiable quasiperiodic cocycles and its spectral applications

“…We point out that, Theorem 1.2 is a semi-local result in the terminology of [FK18], i.e., the smallness of the perturbation ε * does not depend on the frequency α. One should not expect that ε * is independent of ρ f (in terms of γ, τ ) as this is not true in the C ∞ topology (or even Gevrey class) [AK16].…”

“…One should not expect that ε * is independent of ρ f (in terms of γ, τ ) as this is not true in the C ∞ topology (or even Gevrey class) [AK16]. To this end, we mention another open problem of Fayad-Krikorian [FK18]: Is the semi-local version of the almost reducibility conjecture true for cocycles in quasi-analytic classes? In the analytic topology, it has been established in [HY12,YZ13].…”

Approximate Identities and Lagrangian Poincaré Recurrence

“…The definition extends to flows in an obvious way. Approximate identities have been extensively studied in dynamics, although usually from a perspective different than ours; see, e.g., [3,5,9,10,26] and references therein. In this section we focus on C 0 -and C 1 -a.i.…”

“…cannot be mixing or topologically mixing, and a.i. 's are often studied in connection with mixing properties, [3,10]. For instance, the horocycle flow is mixing and hence not a C 0 -a.i.…”

“…The notion of an approximate identity and several variants of the definition are spelled out in Section 2. Approximate identities can have interesting and non-trivial dynamics, and the main examples of such maps are the so-called pseudo-rotations; see [3,5,10,16,17]. Yet, in some way, these maps resemble actions of compact abelian groups on manifolds.…”