Attractors Of Generalized Iterated Function Systems

Iterated Function Systems (IFS) originates from dynamical system. Dynamical theory deals with iteration of one map but IFS theory deals with iteration of many maps. IFS theory’s root was very early but the beginning of active development was J.Hutchinson’s paper([21]). He studied self-similarity of fractal sets using system of finite number of similar contraction maps of R". M.Barnsley called a finite set of contraction mapps as an iterated function systems and systemized IFS theory like today’s, applied to the research of fractals theory and made wide applications including fractal image compressions, fractal interpolation and so on. Now IFS is not only one of importance tools of studies and applications of fractal theory but also ideas of IFS that give a great influence to various fields including the theory of complex dynamical systems. IFS theory is also applied to autoregressive time series, image processing theory and stochastic dynamical system. The research and extensions of IFS theory have very important theoretical values.The theory of finite IFS consisting of contraction mappings has been studied very thoroughly. Some extensions of the spaces and the contractions concern many researchers. The extensions mainly include the following respects:finite IFS to infinite IFS; compact metric space to incompact metric space (Polish space); IFS to Generalized IFS; Lipschitz contraction to some contractive-type conditions (φ-contraction. Meir-Keeler type mapping and contractive).There are total three chapters in this paper.The first chapter makes researches on the existence of an invariant measure and asymptotically stable for Markov operator defined by infinite IFS on Polish spaces. The existence of invariant probabilities, ergodicity and asymptotic stationary properties are very important subjects of Markov operator theory. IFS is a special kind of Markov chain, and the Markov operator defined by IFS is simple. So we can use Markov operator theory to investigate IFS. T.Szarek [13] provides a new sufficient conditions for the existence of an invariant measure for Markov operators defined on Polish spaces, and applies the criterion to finite IFS. He provides the sufficient conditions of existence of an invariant measure and asymptotically stable for Markov operator defined by finite IFS. The first chapter makes researches on infinite IFS. We use T.Szarek’s investigation methods and use the properties of globally concentrating, semi-concentrating and tight to obtain the sufficient conditions of existence of an invariant measure and asymptotically stable for Markov operator defined by the infinite IFS.The second chapter offers the existence of the attractor of generalized countable iteration function systems (GCIFS) consisting of general contractive-type mappings. GCIFS are a generalization of iterated function systems by considering contractions from X×X into X instead of contractions from the metric space X to itself. A.Mihail, R.Miculescu [27] and N.A.Secelean [29] provide some properties of GIFS consisting of Lipschitz contractions. N.A.Secelean [30] also recalls some contraction-type conditions (φ-contraction, Meir-Keeler type mapping and contractive) and proves the existence of the attractor of CIFS consisting of general contractions. Obviously, the Lipschitz contraction is the special case of the above mentioned contraction-type conditions. The second chapter extends N.A.Secelean [30]’s conclusions about CIFS to the case of generalized CIFS and provides the sufficient conditions of the existence of the attractor of GCIFS consisting of general contractions.The last chapter makes researches on the continuous dependence of attractors of (generalized) CIFS consisting of general contractions. The second chapter offers the existence of the attractor of GCIFS consisting of general contractions. In this chapter we discuss the continuous dependence of attractors. There is a family of CIFS μλ={(wn)λ|n≥1}, let Aλ is the attractor of CFS. then the continuous dependence of Aλ is a very interesting subject. L.C.Kong [40] makes researches on the existence and continuous dependence of attractors of hyperbolic iterated function systems (HIFS) and non-uniformly hyperbolic iterated function systems. N.A.Secelean [29] provides continuous dependence of attractors of GCIFS consisting of Lipschitz contractions. In this chapter, we extend the above conclusions, and provides continuous dependence of attractors for (generalized) CIFS consisting of general contractions.