Topological invariants and Lipschitz equivalence of fractal squares

Ruan, Huo-Jun; Wang, Yan

doi:10.1016/j.jmaa.2017.02.012

Cited by 33 publications

(25 citation statements)

References 24 publications

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“…4. By a result of Whyburn [23], they are homeomorphic, and [21] conjectures these two fractal squares are not Lipschitz equivalent (see also [17]). To show two sets are not Lipschitz equivalent, the main method is to construct a certain Lipschitz invariant to distinct them.…”

Section: E Fmentioning

confidence: 99%

“…A non-empty compact set satisfying the set equation F = ∪ d∈D F+d n is called a fractal square if n ≥ 2 and D ⊂ {0, 1, • • • , n − 1} 2 . Ruan and Wang [21] studied fractal squares of ratio 1/3 and with 7 or 8 branches, which are all connected fractals. Their method is to find various connectivity properties, which depend on very careful observations.…”

Section: Introductionmentioning

confidence: 99%

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Lipschitz Invariance of Critical Exponents on Besov Spaces

sci¹

2020

ATA

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In this paper we prove that the critical exponents of Besov spaces on a compact set possessing an Ahlfors regular measure is an invariant under Lipschitz transforms. Under mild conditions, the critical exponent of Besov spaces of certain selfsimilar sets coincides with the walk dimension, which plays an important role in the analysis on fractals. As an application, we show examples having different critical exponents are not Lipschitz equivalent.

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Section: E Fmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Lipschitz Invariance of Critical Exponents on Besov Spaces

sci¹

2020

ATA

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“…There are several works devoted to the Lipschitz classification of non-totally disconnected fractal squares with contraction ratio 1/3, that is, a kind of Sierpinski carpets ( [10,13,17,23]), but the problem is unsolved in case of the fractal squares with 5 branches. Using neighbor automaton, Rao-Zhu [16] proved that F 1 F 2 in Figure 1, but it is not known whether F j , j = 2, 3, 4, 5 are Lipschitz equivalent or homeomorphic.…”

Section: Introductionmentioning

confidence: 99%

Finite state automata and homeomorphism of self-similar sets

Huang¹,

Wen²,

Yang³

et al. 2021

Preprint

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The topological and metrical equivalence of fractals is an important topic in analysis. In this paper, we use a class of finite state automata, called Σ-automaton, to construct psuedo-metric spaces, and then apply them to the study of classification of self-similar sets. We first introduce a notion of topology automaton of a fractal gasket, which is a simplified version of neighbor automaton; we show that a fractal gasket is homeomorphic to the psuedo-metric space induced by the topology automaton. Then we construct a universal map to show that psuedo-metric spaces induced by different automata can be bi-Lipschitz equivalent. As an application, we obtain a rather general sufficient condition for two fractal gaskets to be homeomorphic or Lipschitz equivalent.

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“…Rao, Ruan and Xi [22] gave an affirmative answer to this question by introducing the graph‐directed technique to prove certain self‐similar sets being Lipschitz equivalence. Although considerable efforts [5, 10, 15–18, 20–25, 27–38] have been made in the study of this issue, it is still a long way to understand the Lipschitz equivalence of self‐similar sets.…”

Section: Introductionmentioning

confidence: 99%