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On the Ledrappier–Young formula for self-affine measures

Published online by Cambridge University Press:  03 August 2015

BALÁZS BÁRÁNY
BALÁZS BÁRÁNY*
Affiliation:
Budapest University of Technology and Economics, MTA-BME Stochastics Research Group, P.O.Box 91, 1521 Budapest, Hungary. Mathematics Institute, University of Warwick, Coventry CV4 7AL, U.K. e-mail: balubsheep@gmail.com
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Abstract

Ledrappier and Young introduced a relation between entropy, Lyapunov exponents and dimension for invariant measures of diffeomorphisms on compact manifolds. In this paper, we show that a self-affine measure on the plane satisfies the Ledrappier–Young formula if the corresponding iterated function system (IFS) satisfies the strong separation condition and the linear parts satisfy the dominated splitting condition. We give sufficient conditions, inspired by Ledrappier and by Falconer and Kempton, that the dimensions of such a self-affine measure is equal to the Lyapunov dimension. We show some applications, namely, we give another proof for Hueter–Lalley's theorem and we consider self-affine measures and sets generated by lower triangular matrices.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 159 , Issue 3 , November 2015 , pp. 405 - 432
Copyright
Copyright © Cambridge Philosophical Society 2015 

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