  In this paper we study the dimension theory of planar self-affine sets
satisfying dominated splitting in the linear parts and strong separation
condition. The main results of this paper is the existence of dimension
maximizing Gibbs measures (K\"aenm\"aki 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.
