We consider a class of expanding maps of the 2-torus of the form $f(x,y)=(a(x,y), b(y))$ that are C$^2$-perturbations of linear ones. In that class we consider invariant sets $\Lambda$ possessing a simple Markov structure, and show there exist ergodic invariant measures supported on $\Lambda$ with Hausdorff dimension arbitrarily close to the Hausdorff dimension of $\Lambda$. For f and $\Lambda$ as above we also show that the Birkhoff exceptional set has full Hausdorff dimension.