Self-similar sets. 2. A simple approach to the topological structure of fractals.

*(English)*Zbl 0824.28007Much 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, and consider connectivity and ramification properties. We work out some tools for the definition of interior distances and Brownian motion on fractals. Related ideas were used by W. J. Gilbert [Can. J. Math. 34, 1335-1348 (1982; Zbl 0494.10005); Ann. Sci. Math. Que. 11, 65-77 (1987; Zbl 0633.10008)] to describe radix representations of complex numbers. 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 and other notions of symbolic dynamics.

[See also Part 1 in Math. Nachr. 142, 107-123 (1989; Zbl 0707.28004) and Part 7 in Proc. Am. Math. Soc. 114, No.4, 995-1001 (1992; Zbl 0823.28003)].

[See also Part 1 in Math. Nachr. 142, 107-123 (1989; Zbl 0707.28004) and Part 7 in Proc. Am. Math. Soc. 114, No.4, 995-1001 (1992; Zbl 0823.28003)].

##### MSC:

28A80 | Fractals |

54H20 | Topological dynamics (MSC2010) |

##### Keywords:

topology; self-similar sets; connectivity; ramification; interior distances; Brownian motion; fractals; Cantor set; symbolic dynamics
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\textit{C. Bandt} and \textit{K. Keller}, Math. Nachr. 154, 27--39 (1991; Zbl 0824.28007)

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##### References:

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