Dynamics of fractals in euclidean and measure spaces

Abstract

In this thesis, we discuss the constructions of the generalized Cantor sets which are the prototypical fractals and also discuss the Markov operators defined on separable complete metric space. We show that these special types of sets are Borel set as well as Borel measurable whose Lebesgue measures are zero. We formulate Iterated Function System of the Generalized Cantor Sets (IFSGCS) using affine transformation and fixed points method. We discuss the Hausdorff dimension of the invariant set for iterated function system of generalized Cantor sets. We also formulate Iterated Function System with probabilities of the Generalized Cantor Sets (IFSPGCS). We show their invariant measures using Markov operators and Barnsley-Hutchinson multifunction. We observe that these functions satisfy the sweeping properties of Markov operator. In addition, we show that these iterated function systems with probabilities are non-expansive and asymptotically stable if the Markov operator has the corresponding property. Further we study two dimensional fractals such as the Koch snowflake, the Koch curve, the Sierpiński triangles, the Sierpiński carpet, the box fractal and also three dimensional fractals such as the Menger sponge and the Sierpinski tetrahedron. We show fractal and topological dimensions and Lebesgue measures of those fractals. We formulate iterated function system of higher dimensional fractals such as the square fractals, the Menger sponge, the Sierpinski tetrahedron and the octahedron fractal. We also discuss the Hausdorff dimension of the invariant set for iterated function system of those fractals.