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An Urysohn-type theorem under a dynamical constraint

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  • 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.
    Mathematics Subject Classification: Primary: 37B99; Secondary: 37C10.


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    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.


    M. Entov and L. Polterovich, Lagrangian tetragons and instabilities in Hamiltonian dynamics, arXiv:1510.01748.


    A. Fathi and P. Pageault, Aubry-Mather theory for homeomorphisms, Ergodic Theory Dynam. Systems, 35 (2015), 1187-1207.doi: 10.1017/etds.2013.107.


    J. L. Kelley, General Topology, Graduate Texts in Mathematics, 27, Springer-Verlag, New York-Berlin, 1975.

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