On the group of isometries of planar IFS fractals*

Abstract

Abstract For any iterated function system (IFS) on R 2 , let K be the attractor. Consider the group of all isometries on K. If K is a self-similar or self-affine set, it is proven that the group must be finite. If K is a bi-Lipschitz IFS fractal, the necessary and sufficient conditions for the infiniteness (or finiteness) of the group are given. For the finite case, the computation of the size of the group is also discussed.