arXiv subjects
Publications and source records attributed to Kirill Kamalutdinov.
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.
This was a long-standing question since 90-ies whether one-point intersection property for a self-similar set implies open set condition. We answer this question negatively. We give an example of a totally disconnected self-similar set $K\subset \mathbb R$ which does not have open set condition and has minimal overlap of its pieces, that is, all intersections of its pieces $K_i\cap K_j,i\neq j$ are empty except only one, which is a single point.
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.
It was proved in 2007 by C.Bandt and H.Rao that if a system $S = \{S_1 , ..., S_m \}$ of contraction similarities in $R^2$ with a connected attractor $K$ has the finite intersection property, then it satisfies OSC. We construct a self-simiilar Jordan arc in $R^3$, defined by a system $S$ , which does not satisfy OSC and at the same time has one-point intersection property.
We propose a construction which transforms a self-similar zipper in $\mathbb R^n$ to a self-affine zipper $\mathbb R^{n+1}$ whose attractor is a smooth curve.
We consider dense 2-generator multiplicative subgroups in $\mathbb C$ and show that for each point $z\in \mathbb C$ the set of limit values for the arguments of the powers of each generator at the point $z$ is either finite or is $[-π,π]$