Keplerian shear in ergodic theory

Abstract

Many integrable physical systems exhibit Keplerian shear. We look at this phenomenon from the point of view of ergodic theory, where it can be seen as mixing conditionally to an invariant σ-algebra. In this context, we give a sufficient criterion for Keplerian shear to appear in a system, investigate its genericity and, in a few cases, its speed. Some additional, non-Hamiltonian, examples are discussed. Résumé.-Le cisaillement keplérien est une propriété commune à de nombreux systèmes intégrables. Nous considérons ce phénomène du point de vue de la théorie ergodique, en tant que propriété de mélange conditionnellement à une tribu invariante. Dans ce contexte, nous donnons des conditions suffisantes assurant que le cisaillement keplérien apparaisse dans un système dynamique donné. De plus, nous discutons la généricité de ce phénomène et, dans certains cas, sa vitesse. Quelques exemples supplémentaires, qui ne sont pas de nature hamiltonienne, sont donnés. Proposition 1.2 (Corollary of Theorem 3.3 and Proposition 3.5).-Let M be a Riemannian manifold, d 1 and k ∈ [1, ∞]. Let v ∈ C k (M, R d), and put g t (x, y) := (x, y + tv(x)) for (x, y) ∈ M × T d. If: Vol M (d ξ, v = 0) = 0 ∀ ξ ∈ Z d \ {0}, then the invariant σ-algebra I is (up to completion) B(M) ⊗ {0, T d }, and the dynamical system (M × T d , Vol M ⊗ Leb T d , (g t)) exhibits Keplerian shear. Moreover, the above criterion is satisfied for a generic v ∈ C k (M, R d). We also study the rate of decay of conditional covariance for the geodesic flow on T 1 T d , and give non-trivial examples of non-Hamiltonian systems with Keplerian shear. Keplerian shear for the geodesic flow on the flat torus is related to two famous problems. The first is Landau's damping for plasma dynamics on a torus (see Landau's article [Lan46], and [MV11, Theorem 3.1] for a version which follows closely our formalism), where the effect is qualitatively similar, although the underlying mechanism is different. The second is Gauss's circle problem, which consists in TOME 3 (2020) where α U,V is C 1 and A U,V ∈ GL d (Z).