Valeurs propres des systèmes dynamiques définis par des substitutions de longuer variable. (Eigenvalues of dynamical systems defined by substitutions of non-constant length). (French) Zbl 0625.28011

Host, B.

Ergodic Theory Dyn. Syst. 6, 529-540 (1986).

Dynamical systems defined by substitutions are strictly ergodic systems obtained as the closed orbit (under the shift) of a single sequence u in a shift space over a finite set A. The sequence u is generated by iterating a substitution, i.e., a map \(\zeta\) from A to the set of all non-empty words over A. Such systems can be considered as one of the simplest generalizations of periodic systems. However, their structure is still not completely understood in the case where the words \(\zeta\) (a), \(a\in A\), do not all have the same length. The author makes an important contribution to the solution of this problem by giving a way (at least in principle) to determine the eigenvalues of the \(L^ 2\) operator associated with the dynamical system, generalizing and simplifying previous work in this field.

Reviewer: F.M.Dekking

MSC:

28D10	One-parameter continuous families of measure-preserving transformations
54H20	Topological dynamics (MSC2010)
28D05	Measure-preserving transformations

Keywords:

substitutions of non-constant length; eigenvalues of dynamical systems; Dynamical systems defined by substitutions; strictly ergodic systems; shift space; periodic systems

References:

[1]	DOI: 10.1007/BF01824809 · Zbl 0256.54026
[2]	DOI: 10.1007/BF00534241 · Zbl 0348.54034
[3]	DOI: 10.2307/1993856 · Zbl 0121.18002