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

Juan Marshall-Maldonado

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.

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

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.
Paper Structure (22 sections, 29 theorems, 91 equations)

This paper contains 22 sections, 29 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Substitutions
  4. Thue-Morse sequence and bijective subtitutions
  5. Conjugacy and factor list of a substitution subshift
  6. Solutions of $s_2(k+a)-s_2(k)=d$
  7. Mixing coefficients and bounded law of iterated logarithm
  8. Spectral cocycle and top Lyapunov exponents
  9. Results on binary constant length substitutions
  10. Twisted Birkhoff sums and twisted correlations
  11. Results
  12. Top Lyapunov exponent of a general substitution
  13. The Thue-Morse top Lyapunov exponent
  14. Top Lyapunov exponent for a binary constant length substitution
  15. Upper bounds for twisted correlations: the Thue-Morse case
  16. ...and 7 more sections

Key Result

Theorem 1.1

There exists a positive constant $B$ such that for almost all $\omega$, there is a positive integer $n_0(\omega)$ such that for all $n\geq n_0(\omega)$,

Theorems & Definitions (53)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Theorem 2.4: see lind_marcus_1995, Theorem 6.2.9
  • Theorem 2.5: durand2000linearly
  • Theorem 2.6: durand1999substitutional
  • Theorem 2.7: durand2022decidability
  • ...and 43 more