  The present paper explores the spectral cocycle, defined by A. Bufetov and B.
Solomyak, in the special case of bijective substitutions on two letters, the
most prominent example being the Thue-Morse substitution. We derive an explicit
subexponential behaviour of the deviations from the expected exponential
behavior. Moreover, these sharp bounds will be exploited to prove that the top
Lyapunov exponent is greater or equal to the top exponent of the subshift
topological factors after a renormalization. In order to obtain such results
for the substitutive subshift factors, we define a special kind of sum, which
is a multiple version of the twisted Birkhoff sum. For the particular case of
the Thue-Morse substitution, we derive that the exponent is zero, we give an
explicit subexponential bound for the twisted Birkhoff sums and we do the same
for subshift topological factors.
