How to construct asymptotically stable iterated function systems
The paper is devoted to the following problem: given a countable or finite family \(\{S_i, \, i\in I\}\) of continuous transformations \(S_i:X\to X\), is there exists a family \(\{p_i, \, i\in I\}\), such that \(\sum_{i\in I} p_i=1\), and the iterated function system \(\{(S_i, p_i), \, i\in I\}\) is asymptotically stable. The answer to this problem was given in two following cases. 1. If for every \(i\in I\) a transformation \(S_i\) is Lipschitzian on every bounded subset of \(X\), and for at least one element of \(I\) (say, \(j\in I\)), \(S_j\) is a generalized contraction with a fixed point \(c\in X\). Then there exists a family \(\{p_i, \, i\in I\}\) of continuous functions such that \(p_i\in (0,1)\) for all \(x\neq c\), and the family \(\{(S_i, p_i), \, i\in I\}\) is asymptotically stable iterated function system. 2. Assume that for every \(i\in I\) a transformation \(S_i\) is Lipschitzian with the constant 1, and for at least \(j\in I\) \(S_j\) is a generalized contraction. Then if \(\{p_i, \, i\in I\}\) is an arbitrary sequence of numbers with \(p_0>0\), \(\sum_{i\in I}p_i=1\), then the iterated function system \(\{(S_i, p_i), \, i\in I\}\) is asymptotically stable. See also \textit{A. Lasota} and \textit{J. Myjak} [Bull. Pol. Acad. Sci., Math. 46, No. 2, 179--196 (1998; Zbl 0921.28007)], \textit{A. Lasota} and \textit{M. Mackey} [Chaos, fractals, and noise: Stochastic aspects of dynamics. Applied Mathematical Sciences. 97. New York, NY: Springer-Verlag (1994; Zbl 0784.58005)], \textit{T. Szarek} [Ann. Pol. Math. 67, No. 3, 247--257 (1997; Zbl 0903.60052)] for related results.
scientific article; zbMATH DE number 5344885