Lipschitz equivalence of generalized {1,3,5}-{1,4,5} self-similar sets

Xi, Lifeng; Ruan, Huo-Jun

doi:10.1007/s11425-007-0113-5

Cited by 33 publications

(31 citation statements)

References 13 publications

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“…David and Semmes [3] asked whether they are bilipschitz equivalence, and the question is called "{1, 3, 5}-{1, 4, 5} problem. " Rao, Ruan and Xi [15] gave an affirmative answer to the problem, then Xi et al [16,22,23] studied some generalizations, recently Xi and Xiong [25] dealt with the problem in high dimension.…”

Section: For Metric Spaces (Amentioning

confidence: 99%

Bilipschitz embedding of self-similar sets

Deng

Zhou

Xiong

et al. 2011

JAMA

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Section: For Metric Spaces (Amentioning

confidence: 99%

Bilipschitz embedding of self-similar sets

Deng

Zhou

Xiong

et al. 2011

JAMA

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“…We notice that for the self-similar sets considered in [20], [30], [33]- [36], the open set condition holds, which means two pieces of self-similar sets touch a little. In this paper, we will consider a quite different case according to complete overlaps.…”

Section: +mentioning

confidence: 98%

“…Rao, Ruan and Xi [20] proved that H 1 and H 2 are Lipschitz equivalent. Furthermore, Xi and Ruan [33] proved that for given r 1 , r 2 , r 3 ∈ (0, 1) with r 1 + r 2 + r 3 < 1, self-similar sets J 1 and J 2 are Lipschitz equivalent if and only if log r 1 / log r 2 ∈ Q, where…”

Section: +mentioning

confidence: 99%

Lipschitz equivalence of a class of self-similar sets with complete overlaps

Guo¹,

Li²,

Wang³

et al. 2012

Ann. Acad. Sci. Fenn. Math.

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“…1 some generalizations were made in [19,17]. For related works on Lipschitz equivalence of other fractals, see [10,12,14,16].…”

Section: Introductionmentioning

confidence: 99%

Lipschitz equivalence of Cantor sets and algebraic properties of contraction ratios

Rao

Ruan

Yang

2012

Trans. Amer. Math. Soc.

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Abstract. In this paper we investigate the Lipschitz equivalence of dust-like self-similar sets in R d . One of the fundamental results by Falconer and Marsh [On the Lipschitz equivalence of Cantor sets, Mathematika, 39 (1992), 223-233] establishes conditions for Lipschitz equivalence based on the algebraic properties of the contraction ratios of the self-similar sets. In this paper we extend the study by examining deeper such connections.A key ingredient of our study is the introduction of a new equivalent relation between two dust-like self-similar sets called a matchable condition. Thanks to a certain measure-preserving property of bi-Lipschitz maps between dust-like self-similar sets, we show that the matchable condition is a necessary condition for Lipschitz equivalence.Using the matchable condition we prove several conditions on the Lipschitz equivalence of dust-like self-similar sets based on the algebraic properties of the contraction ratios, which include a complete characterization of Lipschitz equivalence when the multiplication groups generated by the contraction ratios have full rank. We also completely characterize the Lipschitz equivalence of dust-like self-similar sets with two branches (i.e., they are generated by IFS with two contractive similarities). Some other results are also presented, including a complete characterization of Lipschitz equivalence when one of the self-similar sets has uniform contraction ratio.

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Lipschitz equivalence of generalized {1,3,5}-{1,4,5} self-similar sets

Abstract: This paper investigates the Lipschitz equivalence of generalized {1,3,5}-{1,4,5} self-similarand proves that D and E are Lipschitz equivalent if and only if there are positive integers m and n such that r m 1 = r n 3 .

Cited by 33 publications

References 13 publications

Bilipschitz embedding of self-similar sets

Bilipschitz embedding of self-similar sets

Lipschitz equivalence of a class of self-similar sets with complete overlaps

Lipschitz equivalence of Cantor sets and algebraic properties of contraction ratios

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