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The present paper explores the spectral cocycle defined in [BS20b] in thespecial case bijective substitutions on two letters, the most prominent examblebeing the Thue-Morse substitution. We derive an explicit sub-exponentialbehaviour of the deviations from the expected exponential behavior. Moreover,this sharp bounds will be exploited to prove the top Lyapunov exponent isgreater or equal to the top exponent for the subshifts topological factors. Inorder to obtain such results for the substitutive subshifts factors, we definea special kind of sum, which is multiple version of the twisted Birkhoff sum.For the particular case the Thue-Morse substitution, we derive that theexponent is zero, we give an explicit sub-exponential bound for the tiwstedBirkhoff sums and the same for its subshifts topological factors.

Lyapunov exponents of the spectral cocycle for topological factors of bijective substitutions on two letters