Metric spaces admitting only trivial weak contractions

Richárd Balka. "Metric spaces admitting only trivial weak contractions." Fundamenta Mathematicae 221.1 (2013): 83-94. <http://eudml.org/doc/282965>.

@article{RichárdBalka2013,
abstract = {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.},
author = {Richárd Balka},
journal = {Fundamenta Mathematicae},
keywords = {contraction; weak contraction; fixed point; Borel class; Hausdorff measure; gauge; balanced set},
language = {eng},
number = {1},
pages = {83-94},
title = {Metric spaces admitting only trivial weak contractions},
url = {http://eudml.org/doc/282965},
volume = {221},
year = {2013},
}

TY - JOUR
AU - Richárd Balka
TI - Metric spaces admitting only trivial weak contractions
JO - Fundamenta Mathematicae
PY - 2013
VL - 221
IS - 1
SP - 83
EP - 94
AB - 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.
LA - eng
KW - contraction; weak contraction; fixed point; Borel class; Hausdorff measure; gauge; balanced set
UR - http://eudml.org/doc/282965
ER -