Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 42, 2017, 473-486

THE DIMENSION OF PROJECTIONS OF SELF-AFFINE SETS AND MEASURES

Kenneth Falconer and Tom Kempton

University of St Andrews, Mathematical Institute
North Haugh, St Andrews, Fife, KY16 9SS, Scotland; kjf 'at' st-andrews.ac.uk

University of St Andrews, Mathematical Institute
North Haugh, St Andrews, Fife, KY16 9SS, Scotland; tmwk 'at' st-andrews.ac.uk

Abstract. Let E be a plane self-affine set defined by affine transformations with linear parts given by matrices with positive entries. We show that if μ is a Bernoulli measure on E with dim_Hμ = dim_Lμ, where dim_H and dim_L denote Hausdorff and Lyapunov dimensions, then the projection of μ in all but at most one direction has Hausdorff dimension min{dim_Hμ,1}. We transfer this result to sets and show that many self-affine sets have projections of dimension min{dim_HE,1} in all but at most one direction.

2010 Mathematics Subject Classification: Primary 28A30.

Key words: Dimension, projection, self-affine set, Furstenberg measure, r-scale entropy.

Reference to this article: K. Falconer and T. Kempton: The dimension of projections of self-affine sets and measures. Ann. Acad. Sci. Fenn. Math. 42 (2017), 473-486.

Full document as PDF file

https://doi.org/10.5186/aasfm.2017.4232