The saga of a fish: from a survival guide to closing lemmas

Journal of Differential Equations. 2019. Vol. 267. No. 6. P. 3442–3474.

Kryzhevich S., Eugene Stepanov

In a recent paper by D. Burago, S. Ivanov and A. Novikov, "A survival guide for feeble fish", it has been shown that a fish with limited velocity capabilities can reach any point in the (possibly unbounded) ocean provided that the fluid velocity field is incompressible, bounded and has vanishing mean drift. This brilliant result extends some known point-to-point global controllability theorems though being substantially non constructive. We will give a fish a different recipe of how to survive in a turbulent ocean, and show how this is related to structural stability of dynamical systems by providing a constructive way to change slightly a divergence free vector field with vanishing mean drift to produce a non dissipative (i.e.\ conservative in the sense of not having wandering sets of positive measure) dynamics. This immediately leads to closing lemmas for dynamical systems, in particular to C. Pugh's closing lemma, the extension of which to incompressible vector fields over a possibly unbounded domain we provide here. The results are based on an extension of the Poincar\'{e} recurrence theorem to some σσ-finite measures an on specially constructed Newtonian potentials.

Keywords: дифференциальные уравнения, динамические системы, controllability, управляемость, nonlinear dynamical systems, differentional equations, dynamical systems, structurally stable dynamical systems, структурно устойчивые динамические системы