Iterated function systems generated by strict contractions and place-dependent probabilities

• Autorzy: Zaharopol, R.
• Języki publikacji: EN

Abstrakty

EN

Let (w[sub l], w[sub 2],...,w[sub k];p[sub 1],p[sub 2],...p[sub k]) be an iterated function system (IFS for short) with continuous place-dependent probabilities, defined on a metric space (X, d). Assume that every closed ball in X is compact. Our main result is that the IFS has an attractive probability measure whenever the following three conditions are satisfied: (1) w[sub i] : X --> X is a strict contraction for every i = 1,...,k. (2) sum[...]p[sub i](x)p[sub i](y) > 0 for every x, y [belongs to] X. (3) There exists p > 0 in R such that [...] for every x, y [belongs to] X and j = l, 2,...,k. Note that we do not require the p[sub i]'s to be even uniformly continuous. This research was motivated by a question of Barnsley, Demko, Elton, and Geronimo, [1, p. 373], concerning IFS which satisfy only condition (1). We construct a family C of IFS which we use to answer the question. Our main result allows us to distinguish IFS in C which possess attractive probabilities.

Słowa kluczowe

PL

funkcja Lapunowa; operator Markowa; prawdopodobieństwo atraktywne; układ funkcji iterowany; warunek Diniego

EN

attractive probabilities; Dini condition; iterated function systems; Lyapunov functions; Markov operators

• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2000
• Tom: Vol. 48, no 4
• Strony: 429--438
• Opis fizyczny: Bibliogr. 4 poz.

Twórcy

autor

Zaharopol, R.

• Department of Mathematical Sciences, Binghamton University (S.U.N.Y. At Bing- Hamton), Binghamton, Ny, 13902-6000, U.S.A.,

Bibliografia

• [1] M. F. Barnsley, S. G. Demko, J. H. Elton, J. S. Geronimo, Invariant measures for Markov processes arising from iterated function systems with place dependent probabilities, Ann. Inst. H. Poincaré, Probab. Statist., 24 (1988) 367-394 and 25 (1989) 589-590.
• [2] D. L. Cohn, Measure theory, Birkhäuser, Boston 1980.
• [3] A. Lasota, J. A. Yorke, Lower bound technique for Markov operators and iterated function systems, Random Comput. Dynamics, 2 (1994) 41-77.
• [4] R. Zaharopol, On the existence of attractive probability measures for certain iterated function systems, to be published.