A Study On Boundaries Of Attractors In Iterated Function Systems

Abstract

Fractal geometry is one of the novel and emerging branches of mathematics. Self newlinesimilarity is an important property, exhibited by most of the fractals. Several forms newlineof self similarity have been discussed in the literature. Iterated Function System (IFS) newlineis a mathematical scheme to generate fractals. There are several variants of IFSs such newlineas condensation IFS, countable IFS, etc. newlineCertain properties of attractors in dynamical systems and their self similarity, newlineusing the concept of boundary are focussed in this work. Different types boundaries newlinesuch as similarity boundary and dynamical boundary are studied in detail and established newlinetheir relationship. Also these notions are extended to condensation IFSs. newlineMeasure theoretic and topological properties of these boundaries in dynamical systems newlineare analyzed. Self similar sets are characterized using the Hausdorff measure of newlinetheir boundaries. newlineFurther, the concept of dynamical boundary is generalized for attractors of newlineproduct dynamical systems. The relationships between fractals in product spaces and newlinetheir projections in Euclidian subspaces are studied by means of open set conditions newline(OSC). A characterization for the dynamical boundary of the attractor of a product newlineIFS to be finitely generated is provided. Estimations to dimensions of dynamical newlineboundary and overlapping set of an attractor in a product space are also obtained. newlineThis work also focus on the study of self similarity in product dynamical systems, newlineusing similarity boundary. The relationship between similarity boundary of an newlineattractor in a product space to one of its projection spaces is studied. The impact of newlineinverse invariance of similarity boundary on its coordinate iterated function system newlineis discussed. A characterization of fractals satisfying the strong open set condition, newlinerestricted to attractors in product spaces, is provided