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Dmitry Drozdov

Publications and source records attributed to Dmitry Drozdov.

4 recordsLinked to original sources
arXiv2024

On the intersection of fractal cubes

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.

math.MG↗
arXiv2023

On the classification of fractal square dendrites

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.

math.MG↗
arXiv2020

On the connected components of fractal cubes

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.

math.MG↗