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2014 | 12 | 4 | 593-602
Tytuł artykułu

Countable contraction mappings in metric spaces: invariant sets and measure

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Wydawca
Czasopismo
Rocznik
Tom
12
Numer
4
Strony
593-602
Opis fizyczny
Daty
wydano
2014-04-01
online
2014-01-17
Bibliografia
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  • [9] Mauldin R.D., Infinite iterated function systems: theory and applications, In: Fractal Geometry and Stochastics, Progr. Probab., 37, Birkhäuser, Basel, 1995, 91–110 http://dx.doi.org/10.1007/978-3-0348-7755-8_5
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Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-013-0371-0
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