{'fraction_non_alphanumeric': 0.0405004939084623, 'fraction_numerical': 0.016792887718142906, 'mean_word_length': 4.910505836575876, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
HABILITATION THESIS New results in the theory of Countable Iterated Function Systems

The goal of this thesis is to describe the contributions of the author in the theory of infinite iterated function systems which generalizes the classical Hutchinson-Barnsley theory of (finite) iterated function systems. These contributions are published in several hight impact international journals and represent new results achieved by the author after obtaining the Ph.D. title in 2001. Theory of fractals is now widely used in the modern approach in almost all areas of science. Deterministic and random fractals, within the framework of Iterated Function Systems (IFS), have been used to model and study a wide range of phenomena across many areas of science and technology: statistical physics, neuronal nets, image compression, natural sciences, learning automata and computer graphics. E.g., the use of fractals in image processing may give a considerable compression of relevant data (see for instance Senesi N., Wilkinson K.J. (2008) have been investigated how the fractal approach has been applied in order to understand the structure and reactivity of natural, environmental systems). The mathematics of iterated functions is particularly appealing and it is embedded in the general theory of dynamical systems. IFSs are among the basic methods for generating fractals. The term itself was introduced in 1985 by Michael Fielding Barnsley and Stephen Demko (Iterated function systems and the global construction of fractals, Proc. R. Soc. Lond. Ser. A 399 (1985), 243-275), but the essential concept is usually attributed to John E. Hutchinson (Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), 713-747). Edward R. Vrscay (1990) traces the idea further back to R.F. Williams (1971), who studied fixed points of finite compositions of contractions. A theory of (self-similar) fractal sets and measures as well as the concept of IFS were also developed by Masayoshi Hata (1985). Existence, convergence and uniqueness results were based on contraction properties of the fractal operator with respect to the Pompeiu-Hausdorff metric for sets and the Monge-Kantorovich metric for measures. In fact, in his famous paper, Hutchinson proved that, given an IFS, i.e. a set of self contractions (ωn) N n=1 on a complete metric space X, the set mapping SN : K(X) → K(X), SN(B) = N ∪ n=1 ωn(B) is a contraction on the hyperspace of all nonempty compact subsets of X endowed with Pompeiu-Hausdorff metric (this space being complete too). His unique fixed point A ∈ K(X) is called the attractor of the IFS. A is generally a fractal set. Last decades many authors had to deal with various extensions of Barnsley-Hutchinson’s classical framework for fractals to more general spaces, generalized contractions and infinite IFSs or, moreover, to multifunction systems. For infinite iterated function systems there are the remarkable works of Henning Fernau, Krzysztof Leśniak, Franklin Mendivil, Mariusz Urbanski, R. Daniel Mauldin and others.