  Deviation of ergodic sums is studied for substitution dynamical systems with
a matrix that admits eigenvalues of modulus 1. We consider the corresponding
eigenfunctions, and in Theorem 1.1 we prove that the limit inferior of the
ergodic sums is bounded for every point in the phase space. In Theorem 1.2, we
prove existence of limit distributions along certain exponential subsequences
of times for substitutions of constant length. Under additional assumptions, we
prove that ergodic integrals satisfy the Central Limit Theorem (Theorem 1.3,
Theorem 1.9).
