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Dimension maximizing measures for self-affine systems

Authors: Balázs Bárány and Michał Rams
Journal: Trans. Amer. Math. Soc. 370 (2018), 553-576
MSC (2010): Primary 28A80; Secondary 37C45
Published electronically: July 7, 2017
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Abstract: In this paper we study the dimension theory of planar self-affine sets satisfying dominated splitting in the linear parts and the strong separation condition. The main result of this paper is the existence of dimension maximizing Gibbs measures (Käenmäki measures). To prove this phenomena, we show that the Ledrappier-Young formula holds for Gibbs measures and we introduce a transversality type condition for the strong-stable directions on the projective space.

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Additional Information

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

Michał Rams
Affiliation: Institute of Mathematics, Polish Academy of Sciences, ul. Sńiadeckich 8, 00-656 Warszawa, Poland

Keywords: Self-affine measures, self-affine sets, Hausdorff dimension.
Received by editor(s): August 24, 2015
Received by editor(s) in revised form: April 12, 2016
Published electronically: July 7, 2017
Additional Notes: The research of the first author was supported by the grants EP/J013560/1 and OTKA K104745. The second author was supported by National Science Centre grant 2014/13/B/ST1/01033 (Poland).
Article copyright: © Copyright 2017 American Mathematical Society

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