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

K. G. Kamalutdinov

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 25 (2019) no. 2, pp. 116-124

Résumé

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 $0$ 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 $0$ 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.

Détail

Keywords: self-similar fractal, general position, strong separation condition
Mots-clés : Hausdorff dimension.

Bibliographie

[1] Hutchinson J., “Fractals and self-similarity”, Indiana Univ. Math. J., 30:5 (1981), 713–747 | DOI | MR | Zbl

[2] Marstrand J. M., “Some fundamental geometrical properties of plane sets of fractional dimensions”, Proc. Lond. Math. Soc., s3-4:1 (1954), 257–302 | DOI | MR | Zbl

[3] Mattila P., “Hausdorff dimension and capacities of intersections of sets in $n$-space”, Acta Math., 152 (1984), 77–105 | DOI | MR | Zbl

[4] Falconer K. J., “Dimensions of intersections and distance sets for polyhedral norms”, Real Anal. Exchange, 30:2 (2004), 719–726 | DOI | MR

[5] Pollicott M., Simon K., “The Hausdorff dimension of $\lambda$-expansions with deleted digits”, Trans. Am. Math. Soc., 347:3 (1995), 967–983 | DOI | MR | Zbl

[6] Kamalutdinov K. G., Tetenov A. V., “Twofold Cantor sets in $\mathbb R$”, Sib. elektron. mat. izv., 15 (2018), 801–814 | DOI | MR | Zbl

[7] Falconer K. J., Fractal geometry: mathematical foundations and applications, 3rd ed., J. Wiley and Sons, N Y, 2014, 398 pp. | MR | Zbl

[8] Edgar G., Measure, Topology, and Fractal Geometry, 2nd ed., Springer-Verlag, New York, 2008, 272 pp. | DOI | MR | Zbl

[9] Kuratowski K., Topology, v. 2, Acad. Press, N Y; London, 1968, 608 pp.