Attractors of iterated function systems and Markov operators

Abstract

This paper contains a review of results concerning “generalized” attractors for a large class of iterated function systems $\left\{w_i:i\in I\right\}$ acting on a complete separable metric space. This generalization, which originates in the Banach contraction principle, allows us to consider a new class of sets, which we call semi-attractors (or semifractals). These sets have many interesting properties. Moreover, we give some fixed-point results for Markov operators acting on the space of Borel measures, and we show some relations between semi-attractors and supports of invariant measures for such Markov operators. Finally, we also show some relations between multifunctions and transition functions appearing in the theory of Markov operators.