Construction of dynamical systems of iterated function systems and some generalizations

Abstract

Fractals are objects with mathematical patterns, consisting mostly of irregular shapes, newlinethat repeat and does not terminate on finite scale magnification. Most of the fractals newlineidentified are found to follow a relatively simple recursive mathematical formula in newlinecomparison to the complexity of its structure. Fractal structures are often considered newlineas the manifestation of chaos arising in dynamical systems. The similarity in pat- newlineterns followed by fractals to the minute scale is termed in mathematical literature newlineas self-similarity. There are several methods available to generate these objects newlinemathematically, of which a widely used method is the method of iterated function newlinesystems(IFS). A quantum of theory has been developed in mathematical analysis con- newlinecerning the complex geometrical patterns of fractals. It is recognized as an emerging newlinebranch of mathematical sciences and is called the fractal theory. newlineA dynamical system consists of a state-space with different states, a time-space newlinethrough which the states can change following a rule defined by a function called newlineevolution operator. An iterated function system can be identified with a dynamical newlinesystem with the underlying state space as a complete metric space, and the evolution newlinefunctions as contractions. There have been many generalizations to the idea of iterated newlinefunction system. This work develops theory towards constructing a dynamical system newlineof iterated function systems. It also explores more on the recent generalizations to newlinethe theory of iterated function systems. newlineThis work introduces the space of iterated function systems consisting of a fixed newlinenumber of contractions. It also gives an ordering of an IFS with respect to another newlineIFS in the space. Further, it defines several types of sequences of IFSs, such as newlinedecreasing, eventually decreasing, Cauchy, convergent, etc. Later, this basic structure newlineis used to develop certain results connecting the attractors of IFSs in a sequence of IFSs. newline