On the classification of fractal square dendrites

Abstract

We consider the classification of fractal square dendrites $K$ based on the types of the self-similar boundary $\partial K$ and the main tree $\gamma$ of such dendrites. We show that the self-similar boundary of a fractal square dendrite $K$ may be of 5 possible types and may consist of 3,4 or 6 points. We prove that the main trees of fractal square dendrites belong to 7 possible classes. Bearing in mind the placement and orders of the points of $\partial K$ with respect to the main tree $\gamma$, this results in 16 possible types of main trees for non-degenerate fractal square dendrites.