Journal of Modern Dynamics (JMD)

An Urysohn-type theorem under a dynamical constraint

Pages: 331 - 338, Volume 10, 2016      doi:10.3934/jmd.2016.10.331

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Albert Fathi - UMPA, ENS-Lyon, 46 allée d’Italie, 69364 Lyon Cedex 7, France (email)

Abstract: We address the following question raised by M. Entov and L. Polterovich: given a continuous map $f:X\to X$ of a metric space $X$, closed subsets $A,B\subset X$, and an integer $n\geq 1$, when is it possible to find a continuous function $\theta:X\to\mathbb{R}$ such that \[ \theta f-\theta\leq 1, \quad \theta|A\leq 0, \quad\text{and}\quad \theta|B> n\,? \] To keep things as simple as possible, we solve the problem when $A$ is compact. The non-compact case will be treated in a later work.

Keywords:  Urysohn, constraint.
Mathematics Subject Classification:  Primary: 37B99; Secondary: 37C10.

Received: January 2016;      Revised: June 2016;      Available Online: July 2016.