On Homeomorphism Group of Self-Similar Sets-Finite Group Case
by
Katsuhiko Yoshida
Graduate Course of Applied Mathematics and Informatics, Graduate School of Science and Technology, Ryukoku University, Seta, Otsu 520-2194, Japan

We show a sufficient condition in order that the homeomorphism group of a self-similar set is finite. We also give many examples of such self-similar sets which are 1-dimensional locally connected continua without cut points. For any dihedral group and any cyclic group, we construct self-similar sets whose homeomorphism group is the given group. In particular, we can construct self-similar sets such that the homeomorphism onto itself is only the identity.

The definition of self-similar sets was introduced by Hutchinson in 1981. But, before that, the existence of such sets had already been known. The Sierpi\'nski Carpet, which is also called the universal plane curve, is famous as one of them. There exist many results about self-similar sets. By Krasinkiewicz's result on its homogeneity, we get immediately that the group of all homeomorphisms from universal plane curve onto itself is infinite. Naturally, the group of all homeomorphisms of Koch curve is also infinite, since it is homeomorphic to a simple arc. Furthermore, there also exist some examples of self-similar sets with such property. But as for Sierpi\'nski Gasket, we have the result that the order of its homeomorphism group is only six. We are interested in the property determining its homeomorphism group, about self-similar sets.

Let X be a complete metric space. For self-similar sets, we introduce notions cyclic, single-critical and nested. Further, we introduce the index of a one-step miniature Ind_K(i), for each of contractions f_i of X. Then our main result is the following theorem.

Theorem  Let { f_i }_{1 <= i <= N} be a finite set of one-to-one contractions of X. Suppose that the self-similar set K = K(f₁, f₂, ... , f_N) is nested, single-critical and cyclic. If K satisfies Ind_K(i) =/= N/2 for any symbol i in { 1, 2, ... , N }, then the group of all homeomorphisms of K is finite.

Date received: June 30, 1999