Abstract and Applied Analysis
Volume 2003 (2003), Issue 8, Pages 479-502
Attractors of iterated function systems and Markov operators
1Dipartimento di Matematica Pura ed Applicata, Università di L'Aquila, Via Vetoio, L'Aquila 67100, Italy
2AGH University of Science and Technology, Mickiewicza Avenue 30, Kraków 30-059, Poland
3Institut of Mathematics, Silesian University, Bankowa 14, 40-007 Katowice
and Department of Mathematics, Technical University of Rzeszów,W. Pola 6, Rzeszów 35-959, Poland
Received 22 December 2001
Copyright © 2003 Józef Myjak and Tomasz Szarek. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper contains a review of results concerning generalized
attractors for a large class of iterated function systems 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.