  We consider the deviation of Birkhoff sums along fixed orbits of substitution
dynamical systems. We show distributional convergence for the Birkhoff sums of
eigenfunctions of the substitution matrix. For noncoboundary eigenfunctions
with eigenvalue of modulus 1, we obtain a central limit theorem. For other
eigenfunctions, we show convergence to distributions supported on Cantor sets.
We also give a new criterion for such an eigenfunction to be a coboundary, as
well as a new characterization of substitution dynamical systems with bounded
discrepancy
