On hyperbolic graphs induced by iterated function systems
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AbstractFor any contractive iterated function system (IFS, including the Moran systems), we show that there is a natural hyperbolic graph on the symbolic space, which yields the Hölder equivalence of the hyperbolic boundary and the invariant set of the IFS. This completes the previous studies ([16]; [20] ; [30]) by eliminating superfluous conditions, and admits more classes of sets (e.g., the Moran sets). We also show that the bounded degree property of the graph can be used to characterize certain separation properties of the IFS (open set condition, weak separation condition); the bounded degree property is particularly important when we consider random walks on such graphs. This application and the other application to Lipschitz equivalence of self-similar sets will be discussed.
All Author(s) ListKa-Sing Lau, Xiang-Yang Wang
Journal nameAdvances in Mathematics
Volume Number313
Pages357 - 378
LanguagesEnglish-United States
KeywordsHyperbolic graphs, Hyperbolic boundaries, Iterated function systems, Self-similar sets, Open set condition, Weak separation condition

Last updated on 2021-18-01 at 00:57