Dynamics of Iterated Functions Systems: Hausdorff Dimension and Related Topics
Dai, Mingde
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https://hdl.handle.net/2142/86945
Description
Title
Dynamics of Iterated Functions Systems: Hausdorff Dimension and Related Topics
Author(s)
Dai, Mingde
Issue Date
1997
Doctoral Committee Chair(s)
Palmore, Julian
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Language
eng
Abstract
This thesis explores the Hausdorff dimension of fractal sets generated by iterated function systems (IFS). Hutchinson studied IFS and obtained a useful formula to compute the Hausdorff dimension of fractals generated by a finite family of disjoint similar IFS, where by disjoint we mean the IFS satisfies the open set condition. In the first part of this thesis, we extend Hutchinson's formula by removing the open set condition. We introduce a systematic approach which is always feasible to obtain a nontrivial positive lower bound and sometimes are able to derive the precise value of the Hausdorff dimension of fractals generated by a finite family of overlapping similar IFS. By overlapping we mean the IFS does not satisfy the open set condition. The second part of the thesis utilizes the affine IFS to construct, based on piecewise linear, continuous functions, a hierarchy of continuous, nowhere differentiable functions whose graphs are fractals. The Hausdorff dimension of the graphs of this hierarchy of continuous, nowhere differentiable functions forms the open interval between 1 and 2. Unlike the previous result, our construction procedure can be based on any continuous function rather than a piecewise linear, continuous function. The last part of this thesis deals with fat fractal sets whose Hausdorff measures are positive values. We generalize the one-dimensional fat Cantor set to R$\sp2$ and R$\sp{n}$ and compute the fat fractal exponents. Such a typical class of fat fractal sets provides a concrete example of how the fat fractal exponent describes and quantifies fat fractal sets.
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