Narrow Results

We explore the relationship between the limit spaces of contracting self-similar groups and self-similar structures. We give the condition on a contracting group such that its limit space admits a self-similar structure, and also the condition such that this self-similar structure is post-critically finite (p.c.f.). We then give necessary and sufficient conditions for a p.c.f. self-similar structure to be the limit space of a contracting self-similar group. When these conditions hold we give a construction of the contracting group. Finally, we illustrate our results with several examples.

In this paper, we introduce touching networks including skeleton networks of Sierpinski gasket, Sierpinski tetrahedron, Sierpinski hexagon and Lindstr öm Snowflake. For touching networks, we establish the self-similar theory of limit metric space.