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On the Lipschitz equivalence of Cantor sets

Part of: Classical measure theory

Published online by Cambridge University Press:  26 February 2010

K. J. Falconer  and
D. T. Marsh
K. J. Falconer
Affiliation:
School of Mathematics, University of Bristol, University Walk, Bristol. BS8 1TW
D. T. Marsh
Affiliation:
School of Mathematics, University of Bristol, University Walk, Bristol. BS8 1TW
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Abstract

We show that under certain circumstances quasi self-similar fractals of equal Hausdorff dimensions that are homeomorphic to Cantor sets are equivalent under Hölder bijections of exponents arbitrarily close to 1. By setting up algebraic invariants for strictly self-similar sets, we show that such sets are not, in general, equivalent under Lipschitz bijections.

MSC classification

Secondary: 28A99: None of the above, but in this section
Type
Research Article
Information
Mathematika , Volume 39 , Issue 2 , December 1992 , pp. 223 - 233
Copyright
Copyright © University College London 1992

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References

1.Bedford, T. J.. Application of dynamical systems theory to fractal sets: a study of cookie cutter sets. In Fractal Geometry and Analysis (Kluwer, Amsterdam, 1991).Google Scholar
2.Cooper, D. and Pignataro, T.. On the shape of Cantor sets. J. Differential Geom., 28 (1988), 203–221.CrossRefGoogle Scholar
3.Falconer, K. J.. Dimensions and measures of quasi self-similar sets. Proc. American Math. Soc., 108 (1989), 543–554.CrossRefGoogle Scholar
4.Falconer, K. J.. Fractal Geometry—Mathematical Foundations and Applications (Wiley, Chichester, 1990).CrossRefGoogle Scholar
5.Falconer, K. J. and Marsh, D. T.. Classification of quasi-circles by Hausdorff dimension. Nonlinearity, 2 (1989), 489–493.CrossRefGoogle Scholar
6.Hall, P. and Heyde, C. C.. Martingale Limit Theory and its Application (Academic Press, 1980).Google Scholar
7.Marsh, D. T.. PhD. Thesis (University of Bristol, 1989).Google Scholar
8.Ruelle, D.. Repellers for real analytic maps. J. Ergodic Theory and Dynamical Systems, 2 (1982), 99–108.CrossRefGoogle Scholar