Periodic and fixed points of multivalued maps on Euclidean spaces

Details Periodic and fixed points of multivalued maps on Euclidean spaces Periodic and fixed points of multivalued maps on Euclidean spaces

2012-02-07

arxiv.org/abs/1202.1544v1

Mathematics

General Topology

13 pages

Scientific paper

We show, in particular, that a multivalued map $f$ from a closed subspace $X$ of $\mathbb R^n$ to ${\rm exp}_k(\mathbb R^n)$ has a point of period exactly $M$ if and only if its continuous extension $\tilde f: \beta X\to {\rm exp}_k(\beta \mathbb R^n)$ has such a point. The result also holds if one repace $\mathbb R^n$ by a locally compact Lindel\"of space of finite dimension. We also show that if $f$ is a colorable map froma normal space $X$ to the space ${\mathcal K}(X)$ of all compact subsets of $X$ then its extension $\tilde f:\beta X\to {\mathcal K}(\beta X)$ is fixed-point free.

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