Self-intersections in parametrized self-similar sets under translations and extensions of copies
K. G. Kamalutdinov Novosibirsk State University
Abstract:
We study the problem of pairwise intersections
$F_i(K_t)\cap F_j^t (K_t)$ of different copies of a self-similar set
$K_t$ generated by a system
$\mathcal F_t=\{F_1,\dots,F_m\}$ of contracting similarities in
$\mathbb R^n$, where one mapping
$F_j^t$ depends on a real or vector parameter
$t$. Two cases are considered: the parameter
$t\in \mathbb R^n$ specifies a translation of a mapping
$F_j^t(x) = G(x)+t$, and the parameter
$t\in (a,b)$ is the similarity coefficient of a mapping
$F_j^t(x)=tG(x)+h$, where
$0<a<b<1$ and
$G$ is an isometry of
$\mathbb R^n$. We impose some constraints on the similarity coefficients of mappings of the system
$\mathcal F_t$ and require that the similarity dimension of the system does not exceed some number
$s$. For such systems it is proved that the Hausdorff dimension of the set of parameters
$t$ for which the intersection
$F_i(K_t)\cap F_j^t(K_t)$ is nonempty does not exceed
$2s$. The obtained results are applied to the problem of checking the strong separation condition for a system
$\mathcal F_\tau=\{F_1^\tau,\dots, F_m^\tau\}$ of contraction similarities depending on a parameter vector
$\tau=(t_1,\dots,t_m)$. Two cases are considered:
$\tau$ is a vector of translations of mappings
$F_i^\tau(x)=G_i(x)+t_i$,
$t_i\in \mathbb R^n$, and
$\tau$ is a vector of similarity coefficients of mappings
$F_i^\tau(x)=t_i G_i(x)+h_i$,
$t_i\in(a,b)$, where
$0<a<b<1$ and all
$G_i$ are isometries in
$\mathbb R^n$. In both cases we find sufficient conditions for the system
$\mathcal F_\tau$ to satisfy the strong separation condition for almost all values of
$\tau$. We also consider the easier problem of the intersection
$A\cap f_t(B)$ for a pair of compact sets
$A$ and
$B$ in the space
$\mathbb R^n$. Two cases are considered:
$f_t(B)=B+t$ for
$t\in\mathbb R^n$, and
$f_t(B)=tB$ for
$t\in\mathbb R$, where the closure of
$B$ does not contain the origin. In both cases it is proved that the Hausdorff dimension of the set of parameters
$t$ for which the intersection
$A\cap f_t(B)$ is nonempty does not exceed
$\dim_H (A\times B)$. Consequently, when the dimension of the product
$A\times B$ is small enough, the empty intersection
$A\cap f_t(B)$ is guaranteed for almost all values of
$t$ in both cases.
Keywords:
self-similar fractal, general position, strong separation condition, Hausdorff dimension.
UDC:
517.518.114
MSC: 28A78,
28A80 Received: 22.03.2019
DOI:
10.21538/0134-4889-2019-25-2-116-124
Fulltext:
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