Finite type in measure sense for self-similar sets with overlaps
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Mathematische Zeitschrift
Finite type in measure sense for self-similar sets with overlaps Juan Deng1 · Zhiying Wen2 · Lifeng Xi3,4 Received: 29 September 2019 / Accepted: 8 October 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract For self-similar sets with overlaps, we introduce a notion named the finite type in measure sense and reveal its intrinsic relationships with the weak separation condition and the generalized finite type. Keywords Self-similar set · Finite type in measure sense · Weak separation condition · Generalized finite type Mathematics Subject Classification 28A80
1 Introduction The separation condition for the iterated function system (IFS) plays an important role in the study of self-similar fractals with overlaps. We have separation conditions or structures such as the open set condition (OSC) by Hutchinson [8], the weak separation condition (WSC) by Lau and Ngai [12], the finite type (FT) by Ngai and Wang [21], and the generalized finite type (GFT) by Jin and Yau [10] and Lau and Ngai [13] independently. As shown in the
Supported by National Natural Science Foundation of China (Nos. 11831007, 11771226, 11871098, 11371329, 11301346) and K.C. Wong Magna Fund in Ningbo University.
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Lifeng Xi [email protected]; [email protected] Juan Deng [email protected] Zhiying Wen [email protected]
1
Department of Mathematics, ShenZhen University, ShenZhen 518000, People’s Republic of China
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Department of Mathematics, Tsinghua University, Beijing 100084, People’s Republic of China
3
Department of Mathematics, Ningbo University, Ningbo 315211, People’s Republic of China
4
Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education), School of Mathematics and Statistics, Hunan Normal University, Changsha 410081, Hunan, People’s Republic of China
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J. Deng et al.
survey paper [4], it is known that OSC ⇒ GFT and FT ⇒ GFT ⇒ WSC. We refer the reader to [3,5,6,14,15,18–20,22,24] for the study on separation conditions and [9,11,23] for self-similar sets with overlaps respectively. To characterize self-similar sets with overlaps, in this paper we introduce the separation condition named finite type in measure sense (Definition 1) in terms of Hausdorff measure rather than in topological and algebraic ways. We will study properties of this separation condition, and reveal its intrinsic relationships with WSC and GFT.
1.1 Main result m a family of contractive maps on Rn of the Let m be a positive integer and = {φi (x)}i=0 form
φi (x) = ρi Ri x + bi for i = 0, 1, . . . , m,
(1.1)
where ρi ∈ (0, 1), Ri is orthogonal and bi ∈ Rn for each i. Then is an iterated function system (IFS) on Rn and the attractor of is the unique compact set K ∈ R satisfying m φi (K ). (1.2) K = K = i=0
Without loss of generality, we always assume that K does not lie in any hyperplane. Denote ρ = min{ρi : i = 0, 1, . . . , m}. For any finite word I = i 1 i 2 · · · i t ∈ {0, 1, . . . , m}t , denote |I | the length of the word, and write φ I = φi1 ◦· · ·◦φ
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