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On the dimension of self-affine sets and measures with overlaps


Authors: Balázs Bárány, Michał Rams and Károly Simon
Journal: Proc. Amer. Math. Soc. 144 (2016), 4427-4440
MSC (2010): Primary 28A80; Secondary 28A78
DOI: https://doi.org/10.1090/proc/13121
Published electronically: June 10, 2016
MathSciNet review: 3531192
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Abstract: In this paper we consider diagonally affine, planar IFS $ \Phi =$
$ \{S_i(x,y)\!=\!(\alpha _ix+t_{i,1},\beta _iy+t_{i,2})\}_{i=1}^m$. Combining the techniques of Hochman and Feng and Hu, we compute the Hausdorff dimension of the self-affine attractor and measures and we give an upper bound for the dimension of the exceptional set of parameters.


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Additional Information

Balázs Bárány
Affiliation: MTA-BME Stochastics Research Group, Budapest University of Technology and Economics, P.O. Box 91, 1521 Budapest, Hungary — and — University of Warwick, Mathematics Institute, Coventry CV4 7AL, United Kingdom
Email: balubsheep@gmail.com

Michał Rams
Affiliation: Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-656 Warszawa, Poland
Email: rams@impan.pl

Károly Simon
Affiliation: Department of Stochastics, Institute of Mathematics, Budapest University of Technology and Economics, P.O. Box 91, 1521 Budapest, Hungary
Email: simonk@math.bme.hu

DOI: https://doi.org/10.1090/proc/13121
Keywords: Self-affine measures, self-affine sets, Hausdorff dimension.
Received by editor(s): April 27, 2015
Received by editor(s) in revised form: December 23, 2015
Published electronically: June 10, 2016
Additional Notes: The research of the first and third authors was partially supported by the grant OTKA K104745. The research of the first author was partially supported by the grant EP/J013560/1. The second author was supported by National Science Centre grant 2014/13/B/ST1/01033 (Poland). This work was partially supported by the grant 346300 for IMPAN from the Simons Foundation and the matching 2015-2019 Polish MNiSW fund.
Communicated by: Nimish A. Shah
Article copyright: © Copyright 2016 American Mathematical Society