C. Holton, L. Q. Zamboni, Descendants of primitive substitutions, Theory Comput. Systems 32 (1999) 133--157.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....TILINGS 3 W . Originally, primitive substitutive sequences were characterized by Durand [6] as those which have a nite number of recodings when one considers the recodings for every initial word in the sequence. Later the condition was expanded to include recodings for arbitrary words [9]. In higher dimensions, the return words are more dicult to mimic, as the notion of the next copy of a set of tiles in T is unclear. Vorono tiles give a convenient geometric solution: two occurrences are next to one another if their Vorono cells are adjacent. Pseudo self similar tilings ....

C. Holton and L. Zamboni, Descendants of primitive substitutions, Theory Comput. Systems 32 (1999), 133-157.

....factors of length n. Other equivalent characterizations of Sturmian words are mainly about representations of straight lines and rotations over the unit circle. 3. 2 Return words and derived words The notions of return words and derived words were introduced by Durand [7] and Holton and Zamboni [11]. Let x be an infinite word and u be a recurrent factor of x of length . The factor v is a return word of u if there exist i; j 2 N; i j, such that v = x[i; j) x[i; i ) x[j; j ) u and jx[i; j )j u = 2. In other words, we define the set of return words of u to be the set of all ....

C. Holton, L. Q. Zamboni, Descendants of primitive substitutions, Theory Comput. Systems 32 (1999) 133--157.

....that in a minimal symbolic dynamical system (X ; T ) any point in X in the orbit of a primitive substitutive sequence is primitive substitutive. The second statement of Lemma 2 uses a characterization of primitive substitutive sequences discovered independently by F. Durand[5] and by the authors[6]. We end the paper with a few examples. 2. Proof of Theorem 1 We begin by establishing some notation which will be used in the proof of Theorem 1. For w 2 A n and 1 i n put jwj = n (the length of w) w i = the ith letter of w (so that w = w 1 w 2 : wn ) and w(i) w 1 w 2 : w i ....

C. Holton, L.Q. Zamboni, Descendants of primitive substitutions, Theory Comput. Systems 32 (1999), 133-157.

....system on the infinite path space of the graph is just the continuous extension of the Vershik map. Other constructions of this kind exist, for example, 15, 18, 19] A primitive substitutive sequence is, by definition, an image under a morphism of a fixed point of a primitive substitution (see [13, 21]) Directed graphs give a new way to characterize primitive substitutive sequences. As part of the characterization, we show that a point in the minimal shift arising from a primitive substitution is primitive substitutive if and only if it is represented by an eventually periodic path in the ....

....of a primitive substitution under a letter to letter morphism. Durand [13] showed that primitive substitutive sequences are precisely those minimal sequences which admit only finitely many derived sequences on prefixes (see Theorem 6. 5 below) This description was generalized by Holton Zamboni in [21] to obtain a characterization of primitive substitutive subshifts. In this section we extend the notion of primitive substitutive to bi infinite sequences in the natural way, and derive an alternative characterization in terms of the associated directed graph. A two sided infinite sequence is ....

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C. Holton, L.Q. Zamboni, Descendants of primitive substitutions, Theory Comput. Systems, 32 (1999), p. 133--157.

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C. Holton and L.Q. Zamboni, Descendants of primitive substitutions, Theory Comput. Systems, 32 (1999), 133-157.

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C. Holton, L. Q. Zamboni, Descendants of primitive substitutions, Theory Comput. Systems 32 (1999) 133--157.

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