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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 ix + bi 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 = {ρk}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. © 2014 Versita Warsaw and Springer-Verlag Wien.

Registro:

Documento: Artículo
Título:Countable contraction mappings in metric spaces: Invariant sets and measure
Autor:Barrozo, M.F.; Molter, U.
Filiación:Departamento de Matemática, Universidad Nacional de San Luis, Ejército de Los Andes 950, 5700 San Luis, Argentina
IMASL-CONICET, Italia 1556, 5700 San Luis, Argentina
Departamento de Matemática, Universidad de Buenos Aires, Ciudad Universitaria, Pabellón I, 1428 Capital Federal, Argentina
IMAS-CONICET, Ciudad Universitaria, Pabellón 1, 1428 Buenos Aires, Argentina
Palabras clave:Contraction maps; Countable iterated function system; Invariant measure; Invariant set
Año:2014
Volumen:12
Número:4
Página de inicio:593
Página de fin:602
DOI: http://dx.doi.org/10.2478/s11533-013-0371-0
Título revista:Central European Journal of Mathematics
Título revista abreviado:Cent. Eur. J. Math.
ISSN:18951074
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_18951074_v12_n4_p593_Barrozo

Referencias:

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Citas:

---------- APA ----------
Barrozo, M.F. & Molter, U. (2014) . Countable contraction mappings in metric spaces: Invariant sets and measure. Central European Journal of Mathematics, 12(4), 593-602.
http://dx.doi.org/10.2478/s11533-013-0371-0
---------- CHICAGO ----------
Barrozo, M.F., Molter, U. "Countable contraction mappings in metric spaces: Invariant sets and measure" . Central European Journal of Mathematics 12, no. 4 (2014) : 593-602.
http://dx.doi.org/10.2478/s11533-013-0371-0
---------- MLA ----------
Barrozo, M.F., Molter, U. "Countable contraction mappings in metric spaces: Invariant sets and measure" . Central European Journal of Mathematics, vol. 12, no. 4, 2014, pp. 593-602.
http://dx.doi.org/10.2478/s11533-013-0371-0
---------- VANCOUVER ----------
Barrozo, M.F., Molter, U. Countable contraction mappings in metric spaces: Invariant sets and measure. Cent. Eur. J. Math. 2014;12(4):593-602.
http://dx.doi.org/10.2478/s11533-013-0371-0