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Self-similar and self-affine sets: measure of the intersection of two copies

Published online by Cambridge University Press:  29 June 2009

MÁRTON ELEKES ,
TAMÁS KELETI  and
ANDRÁS MÁTHÉ
MÁRTON ELEKES
Affiliation:
Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, PO Box 127, H-1364 Budapest, Hungary (email: emarci@renyi.hu) Department of Analysis, Eötvös Loránd University, Pázmány Péter sétány 1/c, H-1117 Budapest, Hungary (email: elek@cs.elte.hu, amathe@cs.elte.hu)
TAMÁS KELETI
Affiliation:
Department of Analysis, Eötvös Loránd University, Pázmány Péter sétány 1/c, H-1117 Budapest, Hungary (email: elek@cs.elte.hu, amathe@cs.elte.hu)
ANDRÁS MÁTHÉ
Affiliation:
Department of Analysis, Eötvös Loránd University, Pázmány Péter sétány 1/c, H-1117 Budapest, Hungary (email: elek@cs.elte.hu, amathe@cs.elte.hu)
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Abstract

Let K⊂ℝd be a self-similar or self-affine set and let μ be a self-similar or self-affine measure on it. Let 𝒢 be the group of affine maps, similitudes, isometries or translations of ℝd. Under various assumptions (such as separation conditions, or the assumption that the transformations are small perturbations, or that K is a so-called Sierpiński sponge) we prove theorems of the following types, which are closely related to each other.

  • • (Non-stability) There exists a constant c<1 such that for every g∈𝒢 we have either μ(K∩g(K))<c⋅μ(K) or K⊂g(K).

  • • (Measure and topology) For every g∈𝒢 we have μ(K∩g(K))>0⟺∫ K(K∩g(K))≠0̸ (where ∫ K is interior relative to K).

  • • (Extension) The measure μ has a 𝒢-invariant extension to ℝd.

Moreover, in many situations we characterize those g for which μ(K∩g(K))>0. We also obtain results about those g for which g(K)⊂K or g(K)⊃K.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 2 , April 2010 , pp. 399 - 440
Copyright
Copyright © Cambridge University Press 2009

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