Codimension formulae for the intersection of fractal subsets of Cantor spaces
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- by Casey Donoven and Kenneth Falconer PDF
- Proc. Amer. Math. Soc. 144 (2016), 651-663 Request permission
Abstract:
We examine the dimensions of the intersection of a subset $E$ of an $m$-ary Cantor space $\mathcal {C}^m$ with the image of a subset $F$ under a random isometry with respect to a natural metric. We obtain almost sure upper bounds for the Hausdorff and upper box-counting dimensions of the intersection, and a lower bound for the essential supremum of the Hausdorff dimension. The dimensions of the intersections are typically $\max \{\dim E +\dim F -\dim \mathcal {C}^m, 0\}$, akin to other codimension theorems. The upper estimates come from the expected sizes of coverings, whilst the lower estimate is more intricate, using martingales to define a random measure on the intersection to facilitate a potential theoretic argument.References
- Laurent Bartholdi, Rostislav Grigorchuk, and Volodymyr Nekrashevych, From fractal groups to fractal sets, Fractals in Graz 2001, Trends Math., Birkhäuser, Basel, 2003, pp. 25–118. MR 2091700
- Kenneth Falconer, Fractal geometry, 3rd ed., John Wiley & Sons, Ltd., Chichester, 2014. Mathematical foundations and applications. MR 3236784
- Jean-Pierre Kahane, Sur la dimension des intersections, Aspects of mathematics and its applications, North-Holland Math. Library, vol. 34, North-Holland, Amsterdam, 1986, pp. 419–430 (French). MR 849569, DOI 10.1016/S0924-6509(09)70272-7
- Pertti Mattila, Hausdorff dimension and capacities of intersections of sets in $n$-space, Acta Math. 152 (1984), no. 1-2, 77–105. MR 736213, DOI 10.1007/BF02392192
- Pertti Mattila, On the Hausdorff dimension and capacities of intersections, Mathematika 32 (1985), no. 2, 213–217 (1986). MR 834491, DOI 10.1112/S0025579300011001
- Pertti Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge, 1995. Fractals and rectifiability. MR 1333890, DOI 10.1017/CBO9780511623813
- C. A. Rogers, Hausdorff measures, Cambridge University Press, London-New York, 1970. MR 0281862
- S. J. Taylor, Introduction to measure and integration, Cambridge University Press, Cambridge, 1998.
Additional Information
- Casey Donoven
- Affiliation: Mathematical Institute, University of St Andrews, North Haugh, St Andrews, Fife KY16 9SS, United Kingdom
- Email: cd65@st-andrews.ac.uk
- Kenneth Falconer
- Affiliation: Mathematical Institute, University of St Andrews, North Haugh, St Andrews, Fife KY16 9SS, United Kingdom
- MR Author ID: 65025
- Email: kjf@st-andrews.ac.uk
- Received by editor(s): September 30, 2014
- Received by editor(s) in revised form: January 15, 2015, and January 16, 2015
- Published electronically: June 26, 2015
- Communicated by: Jeremy Tyson
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 651-663
- MSC (2010): Primary 28A80; Secondary 20E08, 60G57
- DOI: https://doi.org/10.1090/proc12730
- MathSciNet review: 3430842