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

Результат исследований: Научные публикации в периодических изданияхстатья

1 Цитирования (Scopus)

### Аннотация

In the 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 can reach any point in the (possibly unbounded) ocean
provided that the fluid velocity field is incompressible, bounded and has
vanishing mean drift. This result extends some known global
controllability theorems
though being substantially nonconstructive. We give a fish a different
recipe of how to survive in a turbulent ocean, and show its relationship
to structural stability of dynamical systems by providing a constructive
way to change slightly the velocity field to produce conservative (in the
sense of not having wandering sets of positive measure) dynamics. In
particular, this leads to the extension of C.Pugh's closing lemma to
incompressible vector fields over unbounded domains. The results are
based on an extension of the Poincare recurrence theorem to some
$\sigma$-finite measures and on specially constructed Newtonian
potentials.
Язык оригинала английский 3442 3474 Journal of Differential Equations https://doi.org/10.1016/j.jde.2019.04.010 Опубликовано - 2019