Fractals corresponding to subsets of sequence spaces
The best known method of construction of fractal sets is as a limit set of the convergent sequences generated by recursive compositions of contractions in a compact metric space. In the present paper the author considers a more general case. Namely, it is supposed that there is given a family \((F_{i})\), \(i\geq 1\), of finite sets of transformations of a metric space \((X,d)\). If one considers the sequences in \(X\) of the form \((f_{i}\circ\cdots\circ f_{j}(x))_{j=i}^\infty\), \(x\in X\), and \(f_{j}\in F_{j}\), \(j\geq i\), some of them may converge, while others diverge. The convergence and the limit is assumed to be independent of \(x\). If \(E_{i}\) is the family of limit sets associated to convergent sequences, then \(E_{i}=\cup_{f\in F_{i}}f(E_{i+1})\). The author discusses some conditions that ensure compactness of the sets \(E_{i}\), and gives a method of construction for an unbounded fractal in \(\mathbb R\). Finally, an application of contractive sequences in finance is presented.
scientific article; zbMATH DE number 1404457