Countable contraction mappings in metric spaces: invariant sets and measure
Open Mathematics (2014)
- Volume: 12, Issue: 4, page 593-602
- ISSN: 2391-5455
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topMaría Barrozo, and Ursula Molter. "Countable contraction mappings in metric spaces: invariant sets and measure." Open Mathematics 12.4 (2014): 593-602. <http://eudml.org/doc/269455>.
@article{MaríaBarrozo2014,
abstract = {We consider a complete metric space (X, d) and a countable number of contraction mappings on X, F = \{F i: i ∈ ℕ\}. We show the existence of a smallest invariant set (with respect to inclusion) for F. If the maps F i are of the form F i(x) = r i x + b i on X = ℝd, we prove a converse of the classic result on contraction mappings, more precisely, there exists a unique bounded invariant set if and only if r = supi r i is strictly smaller than 1. Further, if ρ = \{ρ k\}k∈ℕ is a probability sequence, we show that if there exists an invariant measure for the system (F, ρ), then its support must be precisely this smallest invariant set. If in addition there exists any bounded invariant set, this invariant measure is unique, even though there may be more than one invariant set.},
author = {María Barrozo, Ursula Molter},
journal = {Open Mathematics},
keywords = {Contraction maps; Countable iterated function system; Invariant set; Invariant measure; contraction maps; countable iterated function system; invariant set; invariant measure},
language = {eng},
number = {4},
pages = {593-602},
title = {Countable contraction mappings in metric spaces: invariant sets and measure},
url = {http://eudml.org/doc/269455},
volume = {12},
year = {2014},
}
TY - JOUR
AU - María Barrozo
AU - Ursula Molter
TI - Countable contraction mappings in metric spaces: invariant sets and measure
JO - Open Mathematics
PY - 2014
VL - 12
IS - 4
SP - 593
EP - 602
AB - We consider a complete metric space (X, d) and a countable number of contraction mappings on X, F = {F i: i ∈ ℕ}. We show the existence of a smallest invariant set (with respect to inclusion) for F. If the maps F i are of the form F i(x) = r i x + b i on X = ℝd, we prove a converse of the classic result on contraction mappings, more precisely, there exists a unique bounded invariant set if and only if r = supi r i is strictly smaller than 1. Further, if ρ = {ρ k}k∈ℕ is a probability sequence, we show that if there exists an invariant measure for the system (F, ρ), then its support must be precisely this smallest invariant set. If in addition there exists any bounded invariant set, this invariant measure is unique, even though there may be more than one invariant set.
LA - eng
KW - Contraction maps; Countable iterated function system; Invariant set; Invariant measure; contraction maps; countable iterated function system; invariant set; invariant measure
UR - http://eudml.org/doc/269455
ER -
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