Asian Journal of Mathematics

Volume 22 (2018)

Number 6

Rigidity theorem of graph-directed fractals

Pages: 1127 – 1156



Li-Feng Xi (Department of Mathematics, Ningbo University, Ningbo, Zhejiang, China)

Ying Xiong (Department of Mathematics, South China University of Technology, Guangzhou, China)


In this paper, we identify two fractals if and only if they are bilipschitz equivalent. Fix a ratio $r$, for dust-like graph-directed sets with ratio $r$ and integer characteristic, we obtain a rigid theorem that these graph-directed sets are uniquely determined by their Hausdorff dimension (or integer characteristic) in the sense of bilipschitz equivalence. Using this rigidity theorem, we show that in a suitable class of self-similar sets, two totally disconnected self-similar sets without complete overlaps are bilipschitz equivalent. We also provide an algorithm to test complete overlaps in polynomial time.


fractal, bilipschitz equivalence, graph-directed sets, self-similar set

2010 Mathematics Subject Classification


This work is supported by NSF of China (Nos. 11871227, 11831007, 11771226, 11471124, 11371329, 11071224, 11101159), K.C.Wong Magna Fund in Ningbo University, NCET, The Chinese University of Hong Kong and Morningside Center of Mathematics.

Received 15 March 2016

Accepted 5 April 2017

Published 6 February 2019