Self-intersections in parametrized self-similar sets under translations and extensions of copies

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Abstract

We study the problem of pairwise intersections F-i(K-t) boolean AND F-j(t) (K-t) of different copies of a self-similar set K-t generated by a system F-t = {F-1,..., F-m} of contracting similarities in R-n, where one mapping F-j(t) depends on a real or vector parameter t. Two cases are considered: the parameter t is an element of R-n specifies a translation of a mapping F-j(t)(x) = G(x) + t, and the parameter t is an element of (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 R-n. We impose some constraints on the similarity coefficients of mappings of the system 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) boolean AND 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 F-iota(iota) = {F-1(iota), ... , F-m(iota)} of contraction similarities depending on a parameter vector iota = (t1, ..., tm). Two cases are considered: iota is a vector of translations of mappings F-i(iota)(x) = G(i)(x) + t(i), t(i) is an element of R-n, and iota is a vector of similarity coefficients of mappings F-i(iota) (x) = t(i)G(i)(x) + h(i), t(i) is an element of (a, b), where 0 <a <b <1 and all Gi are isometries in Rn. In both cases we find sufficient conditions for the system F-iota to to satisfy the strong separation condition for almost all values of iota. We also consider the easier problem of the intersection A boolean AND f(t) (B) for a pair of compact sets A and B in the space R(n)are considered: f(t) (B) = B + t for t is an element of R-n, and f(t) (B) = tB for t is an element of 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 boolean AND f(t) (B) is nonempty does not exceed dim(H) ( A x B). Consequently, when the dimension of the product A x B is small enough, the empty intersection A boolean AND f(t) (B) is guaranteed for almost all values of t in both cases.. Two cases
Translated title of the contributionСамопересечения в параметризованных самоподобных множествах при сдвигах и растяжениях копий
Original languageEnglish
Pages (from-to)116-124
Number of pages9
JournalTrudy Instituta Matematiki i Mekhaniki UrO RAN
Volume25
Issue number2
DOIs
Publication statusPublished - 1 Jan 2019

Keywords

  • General position
  • Hausdorff dimension
  • Self-similar fractal
  • Strong separation condition

State classification of scientific and technological information

  • 27 MATHEMATICS

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