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Mandelbrot Cascades and Related Topics Read chapter Read first chapter

2014 | OriginalPaper | Chapter

Quasisymmetric Modification of Metrics on Self-Similar Sets

Author : Jun Kigami

Published in: Geometry and Analysis of Fractals

Publisher: Springer Berlin Heidelberg

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Abstract

Using the notions of scales and their gauge functions associated with self-similar sets, we give a necessary and sufficient condition for two metrics on a self-similar set being quasisymmetric to each other. As an application, we construct metrics on the Sierpinski carpet which is quasisymmetric with respect to the Euclidean metrics and obtain an upper estimate of the conformal dimension of the Sierpinski carpet.

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previous chapter Self Similar Sets, Entropy and Additive Combinatorics
next chapter Recent Progress on Dimensions of Projections
Footnotes
1
After completion of the preliminary version of this paper, B. Kleiner informed me that he had obtained better upper bound of \(\dim _{\mathcal {C}}(\mathrm{SC}, d_E)\) around 1999 by a different method in [Kle00]. His upper bound is about \(1.856685\ldots \). The author would like to express his gratitude to Professor Bruce Kleiner for his detailed comments.
 
Literature
[BM00]
Bonk, M., Merenkov, S.: Rigidity of square Sierpinski carpets, preprint.
[KL04]
[Kig00]
Kigami, J.: Resistance Forms, Quasisymmetric Maps and Heat Kernel Estimates. Memoirs of the American Mathematical Society. American Mathematical Society, Providence, RI (2012)
[Kle00]
[Kig01]
Kigami, J.: Analysis on Fractals, Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (2001)
[Kig09]
Kigami, J.: Volume doubling measures and heat kernel estimates on self-similar sets. Mem. Am. Math. Soc. 199(932), XXX (2009)
[MT10]
Mackay, J.M., Tyson, J.T.: Conformal Dimension, Theory and Application. University Lecture Series. American Mathematical Society, Providence, RI (2010)MATH
[Pan89]
Pansu, P.: Dimension conforme et sphère á l’infini des variétés à courbure nègative. Ann. Acad. Sci. Fenn. Ser. A I Math. 14, 177–212 (1989)MATHMathSciNet
[TV80]
[TW06]
Metadata
Title
Quasisymmetric Modification of Metrics on Self-Similar Sets
Author
Jun Kigami
Copyright Year
2014
Publisher
Springer Berlin Heidelberg
Book
Geometry and Analysis of Fractals

Print ISBN: 978-3-662-43919-7

Electronic ISBN: 978-3-662-43920-3

Copyright Year: 2014

DOI
https://doi.org/10.1007/978-3-662-43920-3_9

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