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On the Lipschitz equivalence of self‐affine sets
Author(s) -
Luo Jun Jason
Publication year - 2019
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201800041
Subject(s) - mathematics , lipschitz continuity , affine transformation , iterated function system , iterated function , equivalence (formal languages) , euclidean geometry , norm (philosophy) , pure mathematics , affine arithmetic , discrete mathematics , mathematical analysis , fractal , geometry , political science , law
Recently Lipschitz equivalence of self‐similar sets onR d has been studied extensively in the literature. However for self‐affine sets the problem is more awkward and there are very few results. In this paper, we introduce a w ‐Lipschitz equivalence by repacing the Euclidean norm with a pseudo‐norm w . Under the open set condition, we prove that any two totally disconnected integral self‐affine sets with a common matrix are w ‐Lipschitz equivalent if and only if their digit sets have equal cardinality. The main methods used are the technique of pseudo‐norm and Gromov hyperbolic graph theory on iterated function systems.