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September 2017 Scaling distances on finitely ramified fractals
Roberto Peirone
Kyoto J. Math. 57(3): 475-504 (September 2017). DOI: 10.1215/21562261-2017-0003

Abstract

In this article we study two problems about the existence of a distance d on a given fractal having certain properties. In the first problem, we require that the maps ψi defining the fractal be Lipschitz of prescribed constants less than 1 with respect to the distance d, and in the second one, we require that arbitrary compositions of the maps ψi be uniformly bi-Lipschitz of related constants. Both problems have been investigated previously by other authors. In this article, on a large class of finitely ramified fractals, we prove that these two problems are equivalent and give a necessary and sufficient condition for the existence of such a distance. Such a condition is expressed in terms of asymptotic behavior of the product of certain matrices associated to the fractal.

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Roberto Peirone. "Scaling distances on finitely ramified fractals." Kyoto J. Math. 57 (3) 475 - 504, September 2017. https://doi.org/10.1215/21562261-2017-0003

Information

Received: 19 November 2015; Accepted: 6 April 2016; Published: September 2017
First available in Project Euclid: 24 April 2017

zbMATH: 1375.28018
MathSciNet: MR3685052
Digital Object Identifier: 10.1215/21562261-2017-0003

Subjects:
Primary: 28A80
Secondary: 05C12 , 15B48

Keywords: distances on fractals , Fractals , joint spectral radius and subradius

Rights: Copyright © 2017 Kyoto University

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Vol.57 • No. 3 • September 2017
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