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

DOI:
https://doi.org/10.1090/tran/7103

Published electronically:
July 7, 2017

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Abstract | References | Similar Articles | Additional Information

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

Email:
balubsheep@gmail.com

**Michał Rams**

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

Email:
rams@impan.pl

DOI:
https://doi.org/10.1090/tran/7103

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