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Resonance between Cantor sets

Published online by Cambridge University Press:  01 February 2009

YUVAL PERES  and
PABLO SHMERKIN
YUVAL PERES
Affiliation:
Microsoft Research, Redmond and Departments of Statistics and Mathematics, University of California, Berkeley, USA (email: peres@stat.berkeley.edu)
PABLO SHMERKIN
Affiliation:
Departments of Mathematics and Statistics, University of Jyväskylä, Finland (email: shmerkin@maths.jyu.fi)
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Abstract

Let Ca be the central Cantor set obtained by removing a central interval of length 1−2a from the unit interval, and then continuing this process inductively on each of the remaining two intervals. We prove that if log b/log a is irrational, then where dim is Hausdorff dimension. More generally, given two self-similar sets K,K′ in ℝ and a scaling parameter s>0, if the dimension of the arithmetic sum K+sK′ is strictly smaller than dim (K)+dim (K′)≤1 (‘geometric resonance’), then there exists r<1 such that all contraction ratios of the similitudes defining K and K′ are powers of r (‘algebraic resonance’). Our method also yields a new result on the projections of planar self-similar sets generated by an iterated function system that includes a scaled irrational rotation.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 1 , February 2009 , pp. 201 - 221
Copyright
Copyright © 2008 Cambridge University Press

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References

[1]Cabrelli, C. A., Hare, K. E. and Molter, U. M.. Sums of Cantor sets yielding an interval. J. Aust. Math. Soc. 73(3) (2002), 405–418.CrossRefGoogle Scholar
[2]Eroǧlu, K. I.. On planar self-similar sets with a dense set of rotations. Ann. Acad. Sci. Fenn. Math. 32(2) (2007), 409–424.Google Scholar
[3]Eroǧlu, K. I.. On the arithmetic sums of Cantor sets. Nonlinearity 20 (2007), 1145–1161.CrossRefGoogle Scholar
[4]Falconer, K.. Fractal Geometry (Mathematical Foundations and Applications). John Wiley & Sons, Chichester, 1990.Google Scholar
[5]Falconer, K.. Techniques in Fractal Geometry. John Wiley & Sons, Chichester, 1997.Google Scholar
[6]Furstenberg, H.. Intersections of Cantor sets and transversality of semigroups. Problems in Analysis (Symp. Salomon Bochner, Princeton University, Princeton, NJ, 1969). Princeton University Press, Princeton, NJ, 1970, pp. 41–59.Google Scholar
[7]Keane, M., Simon, K. and Solomyak, B.. The dimension of graph directed attractors with overlaps on the line, with an application to a problem in fractal image recognition. Fund. Math. 180(3) (2003), 279–292.CrossRefGoogle Scholar
[8]Keane, M., Smorodinsky, M. and Solomyak, B.. On the morphology of γ-expansions with deleted digits. Trans. Amer. Math. Soc. 347(3) (1995), 955–966.Google Scholar
[9]Kemeny, J. G. and Snell, J. L.. Finite Markov Chains (The University Series in Undergraduate Mathematics). D. Van Nostrand, Princeton, NJ, 1960.Google Scholar
[10]Kenyon, R.. Projecting the one-dimensional Sierpiński gasket. Israel J. Math. 97 (1997), 221–238.CrossRefGoogle Scholar
[11]Lagarias, J. C. and Wang, Y.. Tiling the line with translates of one tile. Invent. Math. 124(1–3) (1996), 341–365.CrossRefGoogle Scholar
[12]Mattila, P.. Fractals and rectifiability. Geometry of Sets and Measures in Euclidean Spaces (Cambridge Studies in Advanced Mathematics, 44). Cambridge University Press, Cambridge, 1995.CrossRefGoogle Scholar
[13]Mendes, P. and Oliveira, F.. On the topological structure of the arithmetic sum of two Cantor sets. Nonlinearity 7(2) (1994), 329–343.CrossRefGoogle Scholar
[14]Moreira, C. G. T. d. A.. Sums of regular Cantor sets, dynamics and applications to number theory. Period. Math. Hungar. 37(1–3) (1998), 55–63. (International Conference on Dimension and Dynamics, Miskolc, 1998.)CrossRefGoogle Scholar
[15]Moreira, C. G. T. d. A. 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
[16]Palis, J.. Homoclinic orbits, hyperbolic dynamics and dimension of Cantor sets. The Lefschetz Centennial Conference, Part III (Mexico City, 1984) (Contemporary Mathematics, 58). American Mathematical Society, Providence, RI, 1987, pp. 203–216.CrossRefGoogle Scholar
[17]Peres, Y. and Schlag, W.. Smoothness of projections, Bernoulli convolutions, and the dimension of exceptions. Duke Math. J. 102(2) (2000), 193–251.CrossRefGoogle Scholar
[18]Peres, Y., Schlag, W. and Solomyak, B.. Sixty years of Bernoulli convolutions. Fractal Geometry and Stochastics, II (Greifswald/Koserow, 1998) (Progress in Probability, 46). Birkhäuser, Basel, 2000,pp. 39–65.CrossRefGoogle Scholar
[19]Peres, Y. and Solomyak, B.. Self-similar measures and intersections of Cantor sets. Trans. Amer. Math. Soc. 350(10) (1998), 4065–4087.CrossRefGoogle Scholar
[20]Pollicott, M. and Simon, K.. The Hausdorff dimension of λ-expansions with deleted digits. Trans. Amer. Math. Soc. 347(3) (1995), 967–983.Google Scholar
[21]Rams, M.. Generic behavior of iterated function systems with overlaps. Pacific J. Math. 218(1) (2005), 173–186.CrossRefGoogle Scholar
[22]Shmerkin, P.. Moreira’s theorem on the arithmetic sum of dynamically defined Cantor sets. Unpulished. Available at http://arxiv.org/abs/0807.3709, 2008.Google Scholar
[23]Solomyak, B.. On the measure of arithmetic sums of Cantor sets. Indag. Math. (N.S.) 8(1) (1997), 133–141.CrossRefGoogle Scholar
[24]Spitzer, F.. Principles of Random Walk, 2nd edn(Graduate Texts in Mathematics, 34). Springer, New York, 1976.CrossRefGoogle Scholar