On dendrites, generated by polyhedral systems and their ramification points

Abstract

The paper considers systems of contraction similarities in $\mathbb R^d$ sending a given polyhedron $P$ to polyhedra $P_i\subset P$, whose non-empty intersections are singletons and contain the common vertices of those polyhedra, while the intersection hypergraph of the system is acyclic. It is proved that the attractor $K$ of such system is a dendrite in $\mathbb R^d$. The ramification points of such dendrite fave finite order whose upper bound depends only on the polyhedron $P$, and the set of the cut points of the dendrite $K$ is equal to the dimension of the whole $K$ iff $K$ is a Jordan arc.