Invariant sets for substitution

✍ Scribed by Taishin Nishida; Youichi Kobuchi

• Publisher: Elsevier Science
• Year: 1994
• Tongue: English
• Weight: 310 KB
• Volume: 49
• Category: Article
• ISSN: 0022-0000
• DOI: 10.1016/s0022-0000(05)80044-3

✦ Synopsis

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 Xo=X, Xt,..., (not necessarily distinct) in X such that xieF(xi+l ) (i>~O). Then any invariant set Z is characterized as Z=F+(K) for some set K of ascendable elements, where F + stands for Uk=t Fk. 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 eF+(u).