Urysohn-type theorem under a dynamical constraint: Non-compact case

Abstract

We address the following question raised by M. Entov and L. Polterovich: Suppose $ V $ a vector field on the manifold $ M $ that generates a complete flow $ \varphi_t:M\to M $. If $ A,B\subset M $ are closed, and $ T>0 $, when is it possible to find a smooth function $ \theta:X\to \mathbb{R} $ such that$ V\theta\leq 1\text{, }\theta|A\leq 0\text{, and }\theta|B>T, $where $ V\theta $ is the derivative of $ \theta $ in the direction of $ V $. We solved this problem with $ A $ compact in [5]. In this work, we generalize to the case where $ A $ is not compact.