Closing and connecting lemmas for conservative flows in Euclidean spaces

Sergey Kryzhevicz

Saint-Petersburg State University, Russia e Università di Nova Gorica, Slovenia

Let V be a bounded Lipschitz continuous vector field in Rd. Suppose that V is divergence – free and satisfies the so-called small mean drift condition V(x+y)dy =0. (1) L→∞x∈Rd L [0,L]d authors demonstrate that the system x ̇ = V (x) (2) is controllable. Namely, for any σ > 0 there exist positive values C1 and C2 such that for any two points x0,y0 ∈ Rd there is a value τ(x0,y0) ∈ (0,C1|x − y| + C2) and a continuous function α : [0,τ(x0,y0)] → Rd such that φα(τ(x0,y0),0,x0) = y0. Here φα(τ(t2,t1,ξ) is the solution of the system x ̇ = V(x)+α(t). In other words, system (2) is chain transitive. However, points of the Euclidean space may be wandering. In our talk we discuss possible generalisations of this result on controlability (we are mostly interested in possible analogs of Pugh’s closing lemma [3]). First of all, as it follows from results of [1], if system (2), additionally to what supposed above, does not have non-hyperbolic periodic solutions then for any ε > 0 and any points x0,y0 ∈ Rd there exists an ε - small (in C1-norm) vector field W(x) such that the point y0 belongs to the positive semi-trajectory of the point x0 with respect to the perturbed system x ̇ =V(x)+W(x). (3) We demonstrate that assumption on hyperbolicity of periodic orbits can be avoided at least for an analog of Closing Lemma. Theorem 1. Let the vector field V be bounded, uniformly Lipschitz continuous, divergence-free and satisfy condition (1). Then for any ε > 0 and any x0 ∈ Rd there exists a perturbation W (x) such that ∥W ∥C 1 < ε and the point x0 is periodic with respect to system (3). In the proof, we use an original techniques of construct for system (3) probabil- ity invariant measures, distributed everywhere, and applying Poincar ́e’s Recurrence Theorem and Pugh’s lemma. This work was supported by the Russian Foundation for Basic Researches (grant 15-01-03797). References [1] Bonatti Ch., Crovisier S. R ́ecurrence et g ́en ́ericit ́e, Invent. Math. 158, 33–104 (2004). [2] Burago D., Ivanov S., Novikov A., A survival guide for feeble fish, Algebra and Analysis, 29, 49–59 (2017). [3] Pugh C., The closing lemma. Amer. J. Math. 89, 956–1009, (1967).