Abstract
The aim of this note is to present an elementary way to fractals which completely avoids advanced analysis and uses purely naive set theory. The approach relies on fixed point methods, where the role of the Banach contraction principle is replaced by a slightly improved version of the Knaster–Tarski fixed point theorem.
Highlights
Fractals definitely belong to the most popular mathematical inventions, they have no unified definition in the technical literature
The family T is frequently called iterated function system (IFS), while fractals are termed the attractors of the underlying IFS
Hutchinson’s fundamental result [4] guarantees the unique resolvability of (1). His brilliant approach interprets the invariance equation as a fixed point problem, which has exactly one solution in the complete metric space of fractals according to the Blaschke Theorem and the Banach Contraction Principle
Summary
Fractals definitely belong to the most popular mathematical inventions, they have no unified definition in the technical literature. Hutchinson’s fundamental result [4] guarantees the unique resolvability of (1) His brilliant approach interprets the invariance equation as a fixed point problem, which has exactly one solution in the complete metric space of fractals according to the Blaschke Theorem and the Banach Contraction Principle. Our approach provides a minimalists’ fractal theory in a double sense: a minimal theoretical setup leads to the minimum solution of the invariance equation. This elementary way completely avoids the advanced tools of analysis, and simultaneously reflects the method and the beauty of the fixed point approach
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