An Urysohn-type theorem under a dynamical constraint
Résumé
We address the following question raised by M. Entov and L. Polterovich: given a continuous map f:X→X of a metric space X, closed subsets A,B⊂X, and an integer n≥1, when is it possible to find a continuous function θ:X→ℝ such that
θf−θ≤1,θ∣∣A≤0,andθ∣∣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.