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

TL;DR

This paper investigates the spectral cocycle of bijective two-letter substitutions, especially Thue-Morse, establishing bounds on deviations, Lyapunov exponents, and their relation to topological factors with explicit subexponential estimates.

Contribution

It provides explicit subexponential bounds on deviations and Lyapunov exponents for bijective substitutions, notably for Thue-Morse, linking spectral properties to topological factors.

Findings

Top Lyapunov exponent is at least as large as that of topological factors.

Explicit subexponential bounds for twisted Birkhoff sums in Thue-Morse.

The Lyapunov exponent for Thue-Morse is zero.

Abstract

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.