Metric spaces admitting only trivial weak contractions

• Autorzy: Richárd Balka
• Języki publikacji: EN

Abstrakty

EN

If (X,d) is a metric space then a 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 ≠ y. We determine the simplest non-closed sets X ⊆ ℝⁿ 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 $F_{σ}$ set F ⊆ ℝ such that every weak contraction f: F → F is constant. Similarly, there exists a non-closed $G_{δ}$ 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.

Kategorie tematyczne

• 54H05: Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
• 03E15: Descriptive set theory
• 28A78: Hausdorff and packing measures
• 47H10: Fixed-point theorems
• 54H25: Fixed-point and coincidence theorems

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2013
• Tom: 221
• Numer: 1
• Strony: 83-94

Daty

wydano

2013

Twórcy

autor

Richárd Balka

• Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, PO Box 127, 1364 Budapest, Hungary

Identyfikatory

DOI

10.4064/fm221-1-4