An Urysohn-type theorem under a dynamical constraint

Albert  Fathi

doi:10.3934/jmd.2016.10.331

Article Contents

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

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An Urysohn-type theorem under a dynamical constraint

Albert Fathi^1,

1.
UMPA, ENS-Lyon, 46 allée d’Italie, 69364 Lyon Cedex 7

Received: December 31, 2015

Revised: May 31, 2016

Published: June 2016

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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.

Citation:

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References

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[2]	M. Entov and L. Polterovich, Lagrangian tetragons and instabilities in Hamiltonian dynamics, arXiv:1510.01748.
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[4]	J. L. Kelley, General Topology, Graduate Texts in Mathematics, 27, Springer-Verlag, New York-Berlin, 1975.