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

Juan Marshall-Maldonado

Published 2022-10-21Version 1

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