In order to prove the existence of a fractal induced by a given mapping family, we have to show that the induced invariance operator has a fixed point in the fractal space. Hutchinson handled this problem by showing that the invariance operator becomes a hyper-contraction provided that the underlying family consists of contractions. Then the Blaschke Completeness Theorem and the Banach Contraction Principle complete the proof.
In this talk, we are going to present an alternative approach to fractals by completely avoiding the use of the fractal space. A main tool of our method is a shrinking technique, which allows us to replace bounded domains by totally bounded ones. As applications, we present fixed point and fractal results for various kinds of mapping families, and illustrate the method with such examples that are not covered by the Hutchinson Theorem. In the proofs, the Knaster--Tarski Fixed Point Theorem, the Kantorovich Iteration, and the Brouwer Fixed Point Theorem play the key role.