![]() Durand, Contributions à l'étude des suites et systèmes dynamiques substitutifs, thèse de doctorat, Université d'Aix-Marseille II, 1996. Tomiyama, Bounded topological equivalence and C*-algebras, J. Handelmann, Orbit equivalence and ordered cohomology, Israel J. The combinatorial properties of the Fibonacci innite word are of great interest in some aspects of mathematics and physics, such as number the- ory, fractal geometry, formal language, computational complexity, quasicrys- tals etc. Moreover, we also give new proofs for the results on special words and the power of the factors. By using these results, we discuss the local isomorphism of the Fibonacci word and the overlap properties of the factors. We establish two decompositions of the Fibonacci word in singular words and their consequences. In this note, we introduce the singular words of the Fibonacci innite word and discuss their properties. The combinatorial properties of the Fibonacci innite word are of great interest in some aspects of mathematics and physics, such as num- ber theory, fractal geometry, formal language, computational complex- ity, quasicrystals etc. The second application concerns the topological perfectness of some families of infinite words. In the firts we give a characterization of ultimately periodic infinite words. We further show that the number ϑ2 is tight for this result. In this case a repetition which is a square does not suffices and the golden ratio ϑ (more precisely its square ϑ2 = 2.618 …) surprisingly appears as a threshold for establishing a connection between local and global periods of the word. We here take into account a different notion of local period by considering, for any position in the word, the shortest repetition “immediately to the left” from that position. The Critical Factorization theorem states, roughly speaking, a connection between local and global periods of a word the local period at any position in the word is there defined as the shortest repetition (a square) “centered” in that position. We prove a periodicity theorem on words that has strong analogies with the Critical Factorization theorem. The results show that while there is some similarity it is possible for the square root map to exhibit quite different behavior compared to the Sturmian case. The main results are characterizations of periodic points and the limit set. In this paper, we study these dynamical systems in more detail and compare their properties to the Sturmian case. In our earlier work, we introduced another type of subshift of optimal squareful words which together with the square root map form a dynamical system. The dynamics of the square root map on a Sturmian subshift are well understood. We proved that the square root map preserves the languages of Sturmian words (which are optimal squareful words). We apply our results to the symbolic square root map $\sqrt\) of s is the infinite word \(X_1 X_2 \cdots \) obtained by deleting half of each square. This result can be interpreted as a yet another characterization for standard Sturmian words. A particular and remarkable consequence is that a word $w$ is a standard word if and only if its reversal is a solution to the word equation and $\gcd(|w|, |w|_1) = 1$. We apply a method developed by the second author for studying word equations and prove that there are exactly two families of solutions: reversed standard words and words obtained from reversed standard words by a simple substitution scheme. We consider solutions of the word equation $X_1^2 \dotsm X_n^2 = (X_1 \dotsm X_n)^2$ such that the squares $X_i^2$ are minimal squares found in optimal squareful infinite words. We also give a characterization of 2-repetitive sequences and solve the values of M(α) for 1≤α≤15/7. In this paper, we study optimal 2-repetitive sequences and optimal 2 -repetitive sequences, and show that Sturmian words belong to both classes. We call the everywhere α-repetitive sequences witnessing this property optimal. In both cases, the number of distinct minimal α-repetitions (or α -repetitions) occurring in the sequence is finite.A natural question regarding global regularity is to determine the least number, denoted by M(α), of distinct minimalα-repetitions such that an α-repetitive sequence is not necessarily ultimately periodic. If each repetition is of order strictly larger than α, then the sequence is called everywhereα -repetitive. Such a sequence is defined by the property that there exists an integer N≥2 such that every length-N factor has a repetition of order α as a prefix. We consider this topic by studying everywhereα-repetitive sequences. Local constraints on an infinite sequence that imply global regularity are of general interest in combinatorics on words. ![]()
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