New Aspects of Fractal on Higher Dimensions

Abstract

Fractals are geometric forms characterized by irregular or non-smooth features, where each constituent element possesses statistical properties identical to the overall structure. Barnsley [13], employed Hutchinson's operator to construct a Fractal Interpolation Function (FIF) through a specific Iterated Function System (IFS). The FIF can be characterized as a continuous function that interpolates particular data points and possesses a graph that serves as the attractor, forming a fractal set via an IFS. By appropriately defining IFS, it is possible to create diverse forms of fractal functions, encompassing both non-self-affine and partially self-affine (as well as partially non-self-affine) FIFs. Navascue̒s [54], introduced an extensive approach to defining non-affine fractal variations of classical trigonometric approximations. The author has created a non-self-affine fractal function fα that approximates and interpolates a given continuous function f [53, 54]. The function fα is commonly referred to as an α-fractal function or non-affine fractal function that produces a set of continuous functions corresponding to the original function f at various scale vectors α=(α1,α2,…,αN)ϵ (-1, 1)N. Chand et al. [28, 78], developed a constructive approach to constrained interpolation problems from a fractal perspective. The essence of fractal geometry lies at the core of dimension theory. Kolmogorov [48], introduced the concept of fractal dimension, which was later validated by Mandelbrot [50], as a way to quantify irregular patterns or fractal sets. The present work is mainly devoted to the study of the IFS on the sphere (Euclidean space) and the IFS on the real projective space (non-Euclidean space) and the quantization of Cantor-like set on the real projective plane. It is organized into six chapters. Chapter [1], provides an overview of the basics definition, as well as some useful results that are relevant to our work. In Chapter [2], an IFS on the sphere is defined, and a metric (known as a Stereographic metric) on it is established to demonstrate the contractive property of this IFS. The topological and geometric properties of the attractor of such an IFS are described. Relations between various dimensions of a non-empty compact subset of R as well as a non-empty compact subset of the sphere with respect to the various metrics are investigated. The dimensions of a non-empty compact subset on the sphere and its projection on the plane with respect to different metrics are compared. Finally, we construct a fractal path whose image is the attractor of an IFS on the sphere. In Chapter [3], an IFS on the real projective plane is constructed in such a way that its attractor is a graph of a self-referential function. To achieve this, a decomposition of the real projective plane which avoids a hyperplane is provided. In this regard, a new metric is defined in this space to demonstrate its completeness. In Chapter [4], we considered an IFS that operates on the non-Euclidean real projective plane with a linear structure. The fractal dimension of the associated curve is then investigated, both as a subset of the real projective plane and as a subset of the Euclidean space. Finally, this chapter proposes a dual-real projective iterated function system on the dual of the real projective plane. In Chapter [5], a non-affine real projective fractal function (RPFF) within the framework of the real projective interval is constructed. The approximation results were obtained using the constructed non-affine RPFF on the projective interval. In this context, initially, the classical approximation result on the projective interval is established. Ultimately, an Iterated Function System (IFS) on the dual of the real projective plane is formulated, and designed to yield an attractor in the form of a graph of a non-affine fractal function. In Chapter [6], an IFS is designed on the real projective line such that its attractor is a Cantor-like set on the real projective line. The existence of a probability measure associated with this IFS has been demonstrated, and the upper bound of the quantization error for this probability measure has been estimated.