Hostname: page-component-77f85d65b8-g4pgd Total loading time: 0 Render date: 2026-04-18T06:13:58.440Z Has data issue: false hasContentIssue false
  • English
  • Français

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)
Get access Rights & Permissions [Opens in a new window]

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.

Information

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

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