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On contractive families and a fixed-point question of Stein

Tim D. Austin
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Abstract

In this paper we disprove the following conjectured generalization of the Contraction Mapping Theorem (due to J.D. Stein Jr.): Let X be a complete metric space and let F be a finite family of self-maps of X. Suppose there is a postive constant strictly less than 1 such that, for any two points x and y of X, some member of F contracts those points by a factor of at most that constant. Then some composition of members of F has a fixed point. We also show that the above does hold for a (continuous) commuting F containing only two maps. We conjecture that it holds for commuting F of any finite size.

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Tim D. Austin. 2006-08-21. On contractive families and a fixed-point question of Stein. https://arxiv.org/abs/math/0608523

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