Metric spaces admitting only trivial weak contractions

Balka, Richárd (2013) Metric spaces admitting only trivial weak contractions. FUNDAMENTA MATHEMATICAE, 221 (1). pp. 83-94. ISSN 0016-2736

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Abstract

If (X, d) is a metric space then the map f : X → X is defined to be a weak contraction if d(f(x), f(y)) < d(x, y) for all x, y ∈ X, x 6= y. We determine the simplest non-closed sets X ⊆ R n in the sense of descriptive set theoretic complexity such that every weak contraction f : X → X is constant. In order to do so, we prove that there exists a non-closed Fσ set F ⊆ R such that every weak contraction f : F → F is constant. Similarly, there exists a non-closed Gδ set G ⊆ R such that every weak contraction f : G → G is constant. These answer questions of M. Elekes. We use measure theoretic methods, first of all the concept of generalized Hausdorff measure.

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