The saga of a fish: from a survival guide to closing lemmas (Q2420520)

A vector field \(V\) in \(\mathbb{R}^d\) is said to have vanishing mean drift if \[ \lim_{l\to\infty}\sup_{x\in\mathbb{R}^d}\left|l^{-d}\int_{[0,l]^d}V(x+y)\;dy\right|=0. \] The first main result of the authors is as follows. Let \(V\) be a bounded incompressible vector field \(V\) in \(\mathbb{R}^d\) with vanishing mean drift. Then for any \(\epsilon>0\) there exists a vector field \(W_{\epsilon}\) with \(\|W_{\epsilon}\|_{\infty}<\epsilon\) such that any point \(x_0\in\mathbb{R}^d\) is nonwandering for the system \[ \dot{x}=V(x)+W_{\epsilon}(x). \] They also prove an analog of the Pugh closing lemma: For any \(x_0\in\mathbb{R}^d\) and any \(\epsilon>0\) there exists a vector field \(W_{\epsilon}\) with \(\|W_{\epsilon}\|_{\mathrm{Lip}}<\epsilon\) such that the trajectory of \(x_0\) in the above system is periodic.