A.V. Tetenov, M. Samuel, D.A. Vaulin. On dendrites generated by polyhedral systems and their ramification points ... P. 281-291

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.

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

The paper was received by the Editorial Office on June 27, 2017

Andrei Viktorovich Tetenov, Dr. Phys.-Math. Sci., Prof., Gorno-Altaisk State University, Gorno-
Altaisk, 649000 Russia, e-mail: atet@mail.ru .

Mary Samuel, Department of Mathematics, Bharata Mata College, Kochi, India,
e-mail: marysamuel2000@gmail.com.

Dmitrii Alekseevich Vaulin, Gorno-Altaisk State University, Gorno-Altaisk, 649000 Russia,
e-mail: d_warrant@mail.ru .


1.   Aseev V.V., Tetenov A.V., Kravchenko A.S. On self-similar Jordan curves on the plane. Sib. Math. J., 2003, vol. 44, no. 3, pp. 379-386. doi: 10.1023/A:1023848327898.

2.   Bandt C., Stahnke J. Self-similar sets 6. Interior distance on deterministic fractals. Preprint, Greifswald, 1990.

3.   Bandt C.,  Keller K. Self-similar sets 2. A simple approach to the topological structure of fractals. Math. Nachrichten, 1991, vol. 154, pp. 27-39. doi: 10.1002/mana.19911540104.

4.   Barnsley M.F. Fractals Everywhere Academic Press, 1988, 396 p. ISBN: 0-12-079062-9.

5.   Charatonik J., Charatonik W. Dendrites. Aportaciones Mat. Comun., 1998, vol. 22, pp. 227-253.

6.   Croydon D. Random fractal dendrites. Ph.D. Thesis, St. Cross College, University of Oxford, Trinity, 2006. 161 p.

7.   Hata M. On the structure of self-similar sets. Japan. J. Appl. Math, 1985, vol. 3, pp. 381-414. doi: 10.1007/BF03167083.

8.   Kigami J. Harmonic calculus on limits of networks and its application to dendrites. J. Funct. Anal., 1995, vol. 128, no. 1, pp. 48-86. doi: 10.1006/jfan.1995.1023.

9.   Kigami J. Analysis on fractals. Cambridge: Cambridge Univer. Press, 2001, Ser. Cambridge Tracts in Math., vol. 143, 226 p.  ISBN: 0-521-79321-1.

10.   Strichartz R.S. Isoperimetric estimates on Sierpinski gasket type fractals. Trans. Amer. Math. Soc., 1999, vol. 351, no. 5, pp. 1705-1752. doi: 10.1090/S0002-9947-99-01999-6.

11.   Tetenov A. V. Self-similar Jordan arcs and the graph directed systems of similarities. Siberian Math. J., 2006, vol. 47, no. 5, pp. 940-949. doi: 10.1007/s11202-006-0105-7.

12.   Tetenov A.V. Structural theorems in the theory of self-similar fractals: Habilitation Thesis. Gorno-Altaisk state university, Gorno-Altaisk, 2011.  216 p.

13.   Zeller. R. Branching dynamical systems and slices through fractals, Ph.D. Thesis,  University of Greifswald, 2015.