SEPARATION PROPERTIES FOR BI-LIPSCHITZ ITERATED FUNCTION SYSTEMS

Zhiyong Zhu ()
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Zhiyong Zhu: School of science, Northwest A&F University, Yangling, Shannxi 712100, P. R. China

FRACTALS (fractals), 2022, vol. 30, issue 04, 1-15

Abstract: In this paper, we study the iterated function systems (IFSs) consisting of bi-Lipschitz mappings instead of conformal contractions, focusing on IFSs that do not satisfy the open set condition. We define a weak* separation condition (W*SC), which is strictly weaker than the weak separation condition of Lau and Ngai. By assuming the bounded distortion property, we show that the W*SC is equivalent to the identity limit criterion of Zerner for such IFSs. In particular, in the one-dimensional case, we show that the W*SC is also equivalent to Ahlfors–David regularity, the Assouad dimension is strictly less than 1 and positivity of the s-dimensional Hausdorff measure, where s is the zero of the topological pressure.

Keywords: Fractal; Bi-Lipschitz Infinite Iterated Function System; Weak* Separation Condition; Hausdorff Dimension; Assouad Dimension (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1142/S0218348X22500980

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