Descendants of Primitive Substitutions

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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $ {\cal D}(s) $ \end{document} , 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $ {\cal D}(s) $ \end{document} contains a recognizable substitution. Under certain conditions the set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $ \Omega (s) = \bigcap_{s^\prime \in {\cal D}(s)}{\cal D}(s^\prime ) $ \end{document} is nonempty.