On the structure of self-similar sets

Abstract

We shall investigate topological properties of a uniquely determined compact set K such that K = ~a~Afx(K), where eachfx is a weak contraction of a complete metric space and A = { 1, 2, " ', m} or A = N. Such a set K is said to be self-similar. Many classical peculiar sets can be represented in this form. We shall also discuss the interesting problem presented by R . F. Williams.

Disk-like tiles and self-affine curves with noncollinear digits

“…We first give the following criterion of the connectedness. It was first proved in [7] and later rediscovered in [9] independently.…”

The OSC does not imply the SOSC for infinite iterated function systems

“…. , i p } (see [1]), there exists j 0 such that Fix(f j0 ) ∈ B(x 0 , ε). Without loss of generality we may assume that diam f j0 (U ) < c i κ 0 and…”

Computability of Self‐Similar Sets

“…Hata investigated self-similar curves [2]. We summarize here some of the results necessary to proceed out task.…”

“…Mandelbrot called a set constructed from some miniatures of the whole a self-similar set [5]. A more precise definition was discovered by Hutchinson [3] and generalized by Hata [2]. For any finitely many contractions T 0 , .…”