Contribution to the study of substitutive dynamical systems
Juan Marshall
I2M, Aix-Marseille Université
Date(s) : 02/03/2022 iCal
10h30 - 12h00
This thesis studies the spectrum of systems associated to substitutions, in particular the continuous spectrum. We have based the analysis on the study of the spectral cocycle and twisted Birkhoff sums (and integrals). These tools have been widely used in many recent works to ensure quantitative rates of weak mixing and spectrum singularity in settings such as substitution subshifts, S-adic systems, translations surfaces, deterministic and random substitutive tilings and interval exchange transformations.
The first results are obtained in the case of suspension flows over Salem type substitutions. We prove Hölder decays for correlation measures in the spectral parameters belonging to the algebraic field arising from the Salem number. The proof is based in a fine analysis of the distribution modulo 1 of the sequence (ηα n ), n≥0 , where η ∈ Q(α) and α is the corresponding Salem number.
The second set of results are related to the Thue-Morse substitution. We study the behavior of the top Lyapunov exponents of the spectral cocycle associated to the Thue-Morse substitution and its topological factors. We prove that for all topological factors the top Lyapunov exponent is zero, and we also give the sub-exponential behavior of the twisted Birkhoff sums.
Emplacement
Saint-Charles - FRUMAM (2ème étage)
Catégories
