Self-Similar Sets 2. A Simple Approach to the Topological Structure of Fractals

✍ Scribed by Christoph Bandt; Karsten Keller

• Publisher: John Wiley and Sons
• Year: 1991
• Tongue: English
• Weight: 642 KB
• Volume: 154
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.19911540104

✦ Synopsis

Much has been written on fractals, but their topological structure has rarely been investigated, in spite of the fact that most classical fractals were introduced by topologists. We propose an easy description of the topology of certain strictly self-similar sets which turns out to include also many JULIA sets [4], and consider connectivity and ramification properties. We shall work out some tools for the definition of interior distances [5] and BRowNian motion [6, 7, 161 on fractals. Related ideas were used by GILBERT to describe radix representations of complex numbers [ll, 121. The basic idea of our approach is to project the obvious self-similarity properties of a CANTOR set to other spaces. The resulting concept of an invariant factor of a one-sided shift space is related to itineraries [17] and other notions of symbolic dynamics.