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arXiv · 1504.00626

Uniformly Lipschitzian group actions on hyperconvex spaces

Andrzej Wiśnicki·Jacek Wośko
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Abstract

Suppose that $\{T_{a}:a\in G\}$ is a group of uniformly $L$-Lipschitzian mappings with bounded orbits $\left\{T_{a}x:a\in G\right\}$ acting on a hyperconvex metric space $M$. We show that if $L<\sqrt{2}$, then the set of common fixed points $Fix \, G$ is a nonempty H\"older continuous retract of $M$. As a consequence, it follows that all surjective isometries acting on a bounded hyperconvex space have a common fixed point. A fixed point theorem for $L$-Lipschitzian involutions and some generalizations to the case of $\lambda$-hyperconvex spaces are also given.

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Andrzej Wiśnicki, Jacek Wośko. 2015-04-02. Uniformly Lipschitzian group actions on hyperconvex spaces. https://doi.org/10.1090/proc/13016

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