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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: [email protected]) Department of Analysis, Eötvös Loránd University, Pázmány Péter sétány 1/c, H-1117 Budapest, Hungary (email: [email protected], [email protected])
TAMÁS KELETI
Affiliation:
Department of Analysis, Eötvös Loránd University, Pázmány Péter sétány 1/c, H-1117 Budapest, Hungary (email: [email protected], [email protected])
ANDRÁS MÁTHÉ
Affiliation:
Department of Analysis, Eötvös Loránd University, Pázmány Péter sétány 1/c, H-1117 Budapest, Hungary (email: [email protected], [email protected])
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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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References

[1]Bandt, C. and Graf, S.. Self-similar sets VII. A characterization of self-similar fractals with positive Hausdorff measure. Proc. Amer. Math. Soc. 114(4) (1992), 995–1001.Google Scholar
[2]Bedford, T.. Crinkly curves, Markov partitions and box dimension in self-similar sets. PhD Thesis, University of Warwick, 1984.Google Scholar
[3]Davies, R. O.. Sets which are null or non-sigma-finite for every translation-invariant measure. Mathematika 18 (1971), 161–162.CrossRefGoogle Scholar
[4]Elekes, M. and Keleti, T.. Borel sets which are null or non-σ-finite for every translation invariant measure. Adv. Math. 201 (2006), 102–115.CrossRefGoogle Scholar
[5]Falconer, K. J.. Classes of sets with large intersection. Mathematika 32(2) (1985), 191–205.CrossRefGoogle Scholar
[6]Falconer, K. J.. The Geometry of Fractal Sets (Cambridge Tracts in Mathematics, 85). Cambridge University Press, Cambridge, 1985.CrossRefGoogle Scholar
[7]Falconer, K. J.. Techniques in Fractal Geometry. Wiley, New York, 1997.Google Scholar
[8]Feng, D. and Wang, Y.. On the structures of generating iterated function systems of Cantor sets. Preprint,  http://math.cuhk.edu.hk/∼djfeng/djfeng1.html.Google Scholar
[9]Furstenberg, H.. Intersections of Cantor sets and transversality of semigroups. Problems in Analysis (Symposia of Salomon Bochner, Princeton University, Princeton, NJ, 1969). Princeton University Press, Princeton, NJ, 1970, pp. 41–59.Google Scholar
[10]Gatzouras, D. and Lalley, S.. Hausdorff and box dimensions of certain self-affine fractals. Indiana Univ. Math. J. 41 (1992), 533–568.Google Scholar
[11]Hutchinson, J. E.. Fractals and self-similarity. Indiana Univ. Math. J. 30(5) (1981), 713–747.CrossRefGoogle Scholar
[12]Igudesman, K.. Lacunary self-similar fractal sets and its application to intersection of Cantor sets. Lobachevskii J. Math. 12 (2003), 41–50.Google Scholar
[13]Järvenpää, M.. Hausdorff and packing dimensions, intersection measures, and similarities. Ann. Acad. Sci. Fenn. Math. 24(1) (1999), 165–186.Google Scholar
[14]Keleti, T.. A 1-dimensional subset of the reals that intersects each of its translates in at most a single point. Real Anal. Exchange 24 (1998/1999), 843–844.CrossRefGoogle Scholar
[15]Kenyon, R. and Peres, Y.. Measures of full dimension on affine-invariant sets. Ergod. Th. & Dynam. Sys. 16 (1996), 307–323.CrossRefGoogle Scholar
[16]Lagarias, J. C. and Wang, Y.. Self-affine tiles in ℝn. Adv. Math. 121(1) (1996), 21–49.CrossRefGoogle Scholar
[17]Li., W. and Xiao, D.. Intersection of translations of Cantor triadic set. Acta Math. Sci. (English Ed.) 19(2) (1999), 214–219.CrossRefGoogle Scholar
[18]Máthé, A.. Önhasonló halmazok egybevágóság-invariáns mértékeiről (On isometry invariant measures of self-similar sets) (In Hungarian), Master Thesis, Eötvös Loránd University, 2005 (http://www.cs.elte.hu/math/diploma/math).Google Scholar
[19]Mattila, P.. On the structure of self-similar fractals. Ann. Acad. Sci. Fenn. Ser. A I Math. 7(2) (1982), 189–195.CrossRefGoogle Scholar
[20]Mattila, P.. Hausdorff dimensions and capacities of intersections of sets in n-space. Acta Math. 152(1–2) (1984), 77–105.CrossRefGoogle Scholar
[21]Mattila, P.. On the Hausdorff dimension and capacities of intersections. Mathematika 32(2) (1985), 213–217.CrossRefGoogle Scholar
[22]McMullen, C.. The Hausdorff dimension of general Sierpiński carpets. Nagoya Math. J. 96 (1984), 1–9.CrossRefGoogle Scholar
[23]de A. Moreira, C. G. T.. Stable intersections of Cantor sets and homoclinic bifurcations. Ann. Inst. H. Poincaré Anal. Non Linéaire 13(6) 741–781.CrossRefGoogle Scholar
[24]de A. Moreira, C. G. T. and Yoccoz, J.-C.. Stable intersections of regular Cantor sets with large Hausdorff dimensions. Ann. of Math. (2) 154(1) (2001), 45–96.CrossRefGoogle Scholar
[25]Nekka, F. and Li, J.. Intersection of triadic Cantor sets with their translates. I. Fundamental properties. Chaos Solitons Fractals 13(9) (2002), 1807–1817.CrossRefGoogle Scholar
[26]Peres, Y. and Solomyak, B.. Self-similar measures and intersections of Cantor sets. Trans. Amer. Math. Soc. 350(10) (1998), 4065–4087.CrossRefGoogle Scholar
[27]Peres, Y.. The packing measure of self-affine carpets. Math. Proc. Cambridge Philos. Soc. 115(3) (1994), 437–450.CrossRefGoogle Scholar
[28]Peres, Y.. The self-affine carpets of McMullen and Bedford have infinite Hausdorff measure. Math. Proc. Cambridge Philos. Soc. 116 (1994), 513–526.CrossRefGoogle Scholar
[29]Rudin, W.. Real and Complex Analysis, 3rd edn. McGraw-Hill, New York, 1987.Google Scholar
[30]Schief, A.. Spearation properties for self-similar sets. Proc. Amer. Math. Soc. 122(1) (1994), 111–115.CrossRefGoogle Scholar