Eigenvalues of substitution dynamical systems

Abstract

This chapter is devoted to the description of eigenvalues of the dynamical system (X,μ,T) arising from a primitive and aperiodic substitution ζ. We distinguish the case of constant-length substitutions from the case of nonconstant-length ones. In the first case, the result is due to M. Dekking [68] and J.C. Martin [174] who made use of a different approach. The description of the eigenvalues in the general case has been essentially established by B. Host [119]. Another reformulation and a new proof, in somewhat more geometric terms, have been given ten years later [87] and we just outline the ideas. The third part raises the problem of pure point spectrum for substitution dynamical systems, with emphasize on the emblematic Pisot case. © 2010 Springer-Verlag Berlin Heidelberg.