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Paradoxical decompositions using Lipschitz functions

Part of: Classical measure theory

Published online by Cambridge University Press:  26 February 2010

M. Laczkovich
M. Laczkovich
Affiliation:
Department of Analysis, Eötvös Loránd University, Múzeum krt. 6–8, Budapest, Hungary.
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Extract

§1. Introduction and main results. A map f: A → R (A ⊂ R) is called piecewise contractive if there is a finite partition A = A1∪ … ∪ An such that the restriction f| Ai is a contraction for every i = 1, …, n. According to a theorem proved by von Neumann in [3], every interval can be mapped, using a piecewise contractive map, onto a longer interval. This easily implies that whenever A, B are bounded subsets of R with nonempty interior, then A can be mapped, using a piecewise contractive map, onto B (see [6], Theorem 7.12, p. 105). Our aim is to determine the range of the Lebesgue measure of B, supposing that the number of pieces in the partition of A is given. The Lebesgue outer measure will be denoted by λ. If I is an interval then we write |I| = λ(I).

MSC classification

Secondary: 28A99: None of the above, but in this section
Type
Research Article
Information
Mathematika , Volume 39 , Issue 2 , December 1992 , pp. 216 - 222
Copyright
Copyright © University College London 1992

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References

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