Ergodic Theory and Dynamical Systems

Research Article

Overlapping self-affine sets of Kakeya type

ANTTI KÄENMÄKIa1 and PABLO SHMERKINa1

a1 Department of Mathematics and Statistics, University of Jyväskylä, PO Box 35 (MaD), FI-40014, Finland (email: antakae@maths.jyu.fi, shmerkin@maths.jyu.fi)

Abstract

We compute the Minkowski dimension for a family of self-affine sets on ℝ2. Our result holds for every (rather than generic) set in the class. Moreover, we exhibit explicit open subsets of this class where we allow overlapping, and do not impose any conditions on the norms of the linear maps. The family under consideration was inspired by the theory of Kakeya sets.

(Received October 02 2007)

(Revised March 25 2008)