Invariant sets for substitution

摘要

A relation F on a set X defines a function on the power set of X. A subset of X is said to be invariant for F if it is a fixed point for F. An element x of X is called ascendable in X if there exists an infinite sequence x0=x, x1, …,. (not necessarily distinct) in X such that xi∈F(xi+1) (i⩾0). Then any invariant set Z is characterized as Z = F+(K) for some set K of ascendable elements, where F+ stands for ∪k=1Kk. In this note we prove that if F is a substitution over a finite alphabet, then for any invariant set Z there exists a set S of repeatable words such that Z=F+(S), in which a repeatable word u satisfies u∈F+(u).