Descendants of Primitive Substitutions

Abstract

Let s = (A, τ) be a primitive substitution. To each decomposition of the form τ (h) = uhv we associate a primitive substitution D[(h,u)](s) defined on the set of return words to h . The substitution D[(h,u)](s) is called a descendant of s and its associated dynamical system is the induced system (Xh, Th) on the cylinder determined by h . We show that \( {\cal D}(s) \) , the set of all descendants of s , is finite for each primitive substitution s . We consider this to be a symbolic counterpart to a theorem of Boshernitzan and Carroll which states that an interval exchange transformation defined over a quadratic field has only finitely many descendants. If s fixes a nonperiodic sequence, then \( {\cal D}(s) \) contains a recognizable substitution. Under certain conditions the set \( \Omega (s) = \bigcap_{s^\prime \in {\cal D}(s)}{\cal D}(s^\prime ) \) is nonempty.