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Hausdorff measures, Hölder continuous maps and self-similar fractals

Published online by Cambridge University Press:  24 October 2008

Miguel-Angel Martin  and
Pertti Mattila
Miguel-Angel Martin
Affiliation:
Departamento de Matemática Aplicada, E.T.S.I. Agronomos, E-28040 Madrid, Spain
Pertti Mattila
Affiliation:
Department of Mathematics, University of Jyväskylä, SF-40100 Jyväskylä, Finland
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Extract

Let f: A → ℝn be Hölder continuous with exponent α, 0 < α ≼ 1, where A ⊂ ℝm has finite m-dimensional Lebesgue measure. Then, as is easy to see and well-known, the s-dimensional Hausdorif measure HS(fA) is finite for s = m/α. Many fractal-type sets fA also have positive Hs measure. This is so for example if m = 1 and f is a natural parametrization of the Koch snow flake curve in ℝ2. Then s = log 4/log 3 and α = log 3/log 4. In this paper we study the question of what s-dimensional sets in can intersect some image fA in a set of positive Hs measure where A ⊂ ℝm and f: A → ℝn is (m/s)-Hölder continuous. In Theorem 3·3 we give a general density result for such Holder surfacesfA which implies for example that Hs(E fA) = 0 for any totally disconnected self-similar set. E in this situation. In Theorem 32 we shall first show that such fA has positive s-dimensional lower density H8 almost everywhere.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 114 , Issue 1 , July 1993 , pp. 37 - 42
Copyright
Copyright © Cambridge Philosophical Society 1993

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References

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