# American Institute of Mathematical Sciences

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

## An Urysohn-type theorem under a dynamical constraint

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

Received  January 2016 Revised  June 2016 Published  July 2016

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.
Citation: Albert Fathi. An Urysohn-type theorem under a dynamical constraint. Journal of Modern Dynamics, 2016, 10: 331-338. doi: 10.3934/jmd.2016.10.331
##### References:
 [1] L. Buhovsky, M. Entov and L. Polterovich, Poisson brackets and symplectic invariants, Selecta Math. (N.S.), 18 (2012), 89-157. doi: 10.1007/s00029-011-0068-9.  Google Scholar [2] M. Entov and L. Polterovich, Lagrangian tetragons and instabilities in Hamiltonian dynamics,, , ().   Google Scholar [3] A. Fathi and P. Pageault, Aubry-Mather theory for homeomorphisms, Ergodic Theory Dynam. Systems, 35 (2015), 1187-1207. doi: 10.1017/etds.2013.107.  Google Scholar [4] J. L. Kelley, General Topology, Graduate Texts in Mathematics, 27, Springer-Verlag, New York-Berlin, 1975.  Google Scholar

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##### References:
 [1] L. Buhovsky, M. Entov and L. Polterovich, Poisson brackets and symplectic invariants, Selecta Math. (N.S.), 18 (2012), 89-157. doi: 10.1007/s00029-011-0068-9.  Google Scholar [2] M. Entov and L. Polterovich, Lagrangian tetragons and instabilities in Hamiltonian dynamics,, , ().   Google Scholar [3] A. Fathi and P. Pageault, Aubry-Mather theory for homeomorphisms, Ergodic Theory Dynam. Systems, 35 (2015), 1187-1207. doi: 10.1017/etds.2013.107.  Google Scholar [4] J. L. Kelley, General Topology, Graduate Texts in Mathematics, 27, Springer-Verlag, New York-Berlin, 1975.  Google Scholar
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