On dendrites generated by polyhedral systems and their ramification points

A. V. Tetenov ; M. Samuel ; D. A. Vaulin

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 23 (2017) no. 4, pp. 281-291

Résumé

The methods of construction of self-similar dendrites in $\mathbb R^d$ and their geometric properties are considered. These issues have not yet been studied in the theory of self-similar fractals. We construct and analyze a class of $P$-polyhedral dendrites $K$ in $\mathbb R^d$, which are defined as attractors of systems $S=\{S_1, \ldots, S_m\}$ of contracting similarities in $\mathbb R^d$ sending a given polyhedron $P$ to polyhedra $P_i\subset P$ whose pairwise intersections either are empty or are singletons containing common vertices of the polyhedra, while the hypergraph of pairwise intersections of the polyhedra $P_i$ is acyclic. We prove that there is a countable dense subset $G_S(V_P)\subset K$ such that for any of its points $x$ the local structure of a neighbourhood of $x$ in $K$ is defined by some disjoint family of solid angles with vertex $x$ congruent to the angles at the vertices of $P$. Therefore, the ramification points of a $P$-polyhedral dendrite $K$ have finite order whose upper bound depends only on the polyhedron $P$. We prove that the geometry and dimension of the set $CP(K)$ of the cutting points of $K$ are defined by its main tree, which is a minimal continuum in $K$ containing all vertices of $P$. That is why the dimension $\dim_HCP(K)$ of the set $CP(K)$ is less than the dimension $\dim_H(K)$ of $K$ and $\dim_HCP(K)=\dim_H(K)$ if and only if $K$ is a Jordan arc.

Détail

Keywords: self-similar set, polyhedral system, main tree
Mots-clés : dendrite, ramification point, Hausdorff dimension.