arXiv subjects
Publications and source records attributed to Andrei Tetenov.
Each self-similar dendrite K with a finite self-similar boundary defines a finite acyclic edge-labeled bipartite graph G, called the sprout of K. The paper shows that the sprout G determines the combinatorial properties of the dendrite K and its topological structure.
We prove that for any self-affine dendrite K generated by a polygonal system, there are constants C>0 and $λ\in(0, 1)$ such that for any x, y in K, the Jordan arc $γ$ in K with endpoints x, y satisfies the inequality $diam(γ)\le C |x-y|^λ$.
We consider the intersections of fractal k-cubes of order n and intersections of their respective opposite l-faces. The main result of the paper is the theorem on representation of such intersection as the attractor of a graph-directed system of similarities in terms of the sets of units corresponding to these cubes and intersections of pairs of l-faces. As a corollary, we prove dimension formula for the intersection and the condition of finiteness of its measure. Another corollary gives the conditions under which the intersections have the given cardinality. Applying these techniques, we obtain the conditions under which a fractal k-cube has the finite intersection property and the conditions under which the fractal cube is a dendrite.
We consider the classification of fractal square dendrites $K$ based on the types of the self-similar boundary $\partial K$ and the main tree $γ$ 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 $γ$, this results in 16 possible types of main trees for non-degenerate fractal square dendrites.
We find the conditions for bi-Lipschitz equivalence of self-similar Jordan arcs which are the attractors of self-similar zippers.
We consider self-similar continua possessing finite intersection property and prove intersection graph criterion for self-similar dendrites, finite order Theorem for such continua satisfying open set condition in $\mathbb{R}^n$ and parameter matching Theorem which states that all Jordan arcs starting from a intersection point in such continuum on a plane should have the same slope parameter at that point.
We consider self-affine arcs in $\mathbb R^2$ and prove that violation of "inner" weak separation property for such arcs implies that the arc is a parabolic segment. Therefore, if a self-affine Jordan arc is not a parabolic segment, then it is the attractor of some multizipper.
We show that a fractal cube $F$ in $\mathbb R^3$ may have an uncountable set $Q$ of connected components $K_α$ neither of which is contained in any plane, whereas the set $Q$ is a totally disconnected self-similar subset of the hyperspace $C(\mathbb R^3)$, isomorphic to a Cantor set.
We introduce some general and special formulations of general position theorem for parametrized families of fractals and explain the techniques of its application to prove the existence of self-similar sets with prescribed special properties.
We introduce a class of self-similar sets which we call {\em twofold Cantor sets} $K_{pq}$ in $\mathbb R$ which are totally disconnected, do not have weak separation property and at the same time have isomorphic self-similar structures.
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.
The pinwheel triangle of Conway and Radin is a standard example for tilings with self-similarity and statistical circular symmetry. Many modifications were constructed, all based on partitions of triangles or rectangles. The fractal example of Frank and Whittaker requires 13 different types of tiles. We present an example of a single tile with fractal boundary and very simple geometric structure which has the same symmetry and spectral properties as the pinwheel triangle.