# 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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