ITERATED FUNCTION SYSTEMS. A CRITICAL SURVEY

In the last 30 years or so, the phrase ‘iterated function system’ has become more and more frequent in mathematical papers and in very many publications of applied people. As in many other instances, the notion of an iterated function system (IFS) is not a new one. Actually, we are faced with the renaming of an old concept, as shown in the first section of the present paper. However, it should be accepted that the study of this notion has been very much deepened under its new clothes. This survey is thus intended to present the state of the art of the IFS notion, its connections with other concepts, as well as to point out to some open problems. The paper is divided into six sections and two appendices. The first section sketches a historical perspective starting from the simplest case of a finite number of self-mappings. Section 2 introduces the general case of an arbitrary family of self-mappings obeying an i.i.d. mechanism. In Section 3 the existence and uniqueness of a stationary distribution are studied while in Section 4 almost sure convergence properties of the backward process are proved. Section 5 is devoted to a study of the support of the stationary distribution. Section 6 takes up the more general case of an arbitrary family of self-mappings obeying a strictly stationary mechanism instead of an i.i.d. one as in the five previous sections. The appendices collect some classical concepts and results on metrics and distances in metric spaces that we are using in the paper.

[1]  An Iterated Random Function with Lipschitz Number One , 2003 .

[2]  Masayoshi Hata,et al.  On the structure of self-similar sets , 1985 .

[3]  R. Tweedie,et al.  Locally contracting iterated functions and stability of Markov chains , 2001, Journal of Applied Probability.

[4]  J. Elton.  A multiplicative ergodic theorem for lipschitz maps , 1990 .

[5]  J. Doob.  Stochastic processes , 1953 .

[6]  N. Sidorov.  Combinatorics of linear iterated function systems with overlaps , 2007, math/0703607.

[7]  G. O'Brien.  The Comparison Method for Stochastic Processes , 1975 .

[8]  P. Erdös.  On a Family of Symmetric Bernoulli Convolutions , 1939 .

[9]  Mariusz Urbański,et al.  Dimensions and Measures in Infinite Iterated Function Systems , 1996 .

[10]  J. Hoffman-Jorgensen.  Probability with a View Towards Statistics, Volume II , 1994 .

[11]  D. Stoyan,et al.  Monotonicity Properties of Second Order Characteristics of Stochastically Monotone MARKOV Chains , 1978 .

[12]  G. Letac.  A contraction principle for certain Markov chains and its applications , 1986 .

[13]  R. Zaharopol.  Invariant Probabilities of Markov-Feller Operators and Their Supports , 2005 .

[14]  Persi Diaconis,et al.  Iterated Random Functions , 1999, SIAM Rev..

[15]  Vilmos Komornik,et al.  Unique Developments in Non-Integer Bases , 1998 .

[16]  P. Erdös.  On the Smoothness Properties of a Family of Bernoulli Convolutions , 1940 .

[17]  Mariusz Urbański,et al.  Graph Directed Markov Systems: Geometry and Dynamics of Limit Sets , 2003 .

[18]  G. Winkler,et al.  RANDOM DYNAMICAL SYSTEMS ON ORDERED TOPOLOGICAL SPACES , 2006 .

[19]  Krishna B. Athreya,et al.  Perfect sampling for Doeblin chains , 2000 .

[20]  K. Lau,et al.  A Generalized Finite Type Condition for Iterated Function Systems , 2007 .

[22]  Nikita Sidorov,et al.  On the Fine Structure of Stationary Measures in Systems Which Contract-on-Average , 2000 .

[24]  Wei Biao Wu,et al.  Limit theorems for iterated random functions , 2004, Journal of Applied Probability.

[25]  O. Hernández-Lerma,et al.  Markov chains and invariant probabilities , 2003 .

[26]  D. Freedman,et al.  Invariant Probabilities for Certain Markov Processes , 1966 .

[27]  Michael F. Barnsley,et al.  Fractals everywhere , 1988 .

[28]  M. Barnsley,et al.  Recurrent iterated function systems , 1989 .

[29]  M. Barnsley,et al.  A new class of markov processes for image encoding , 1988, Advances in Applied Probability.

[30]  Jim Freeman.  Probability Metrics and the Stability of Stochastic Models , 1991 .

[31]  Y. Kifer.  Ergodic theory of random transformations , 1986 .

[32]  B. Solomyak.  On the random series $\sum \pm \lambda^n$ (an Erdös problem) , 1995 .

[33]  Y. Peres,et al.  Sixty Years of Bernoulli Convolutions , 2000 .

[34]  Radu Theodorescu,et al.  Random systems with complete connections , 1969 .

[35]  K. Parthasarathy,et al.  Probability measures on metric spaces , 1967 .

[36]  G. Letac,et al.  Explicit stationary distributions for compositions of random functions and products of random matrices , 1991 .

[37]  Edwin Hewitt,et al.  Real And Abstract Analysis , 1967 .