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 (X h , T h ) 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.
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Received June 1997, and in final form May 1998.
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Holton, C., Zamboni, L. Descendants of Primitive Substitutions . Theory Comput. Systems 32, 133–157 (1999). https://doi.org/10.1007/s002240000114
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DOI: https://doi.org/10.1007/s002240000114