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The combinatorial properties of the Fibonacci innnite word are of great interest in some aspects of mathematics and physics, such as number theory, fractal geometry, formal language, computational complexity , quasicrystals etc. In this note, we introduce the singular words of the Fibonacci innnite word and discuss their properties. We establish two… (More)
In this paper, we prove that a class of regular sequences can be viewed as projections of fixed points of uniform morphisms on a countable alphabet, and also can be generated by countable states automata. Moreover, we prove that the regularity of some regular sequences is invariant under some codings.
The infinite Fibonacci sequence F, which is an extension of the classic Fibonacci sequence to the infinite alphabet N, is the fixed point of the morphism φ: (2i) 7→ (2i)(2i+ 1) and (2i+ 1) 7→ (2i+ 2) for all i ∈ N. In this paper, we study the growth order and digit sum of F, and give several decompositions of F using singular words.
We study the structure of invertible substitutions on three-letter alphabet. We show that there exists a finite set S of invertible substitutions such that any invertible substitution can be written as I w • σ 1 • σ 2 • · · · • σ k , where I w is the inner automorphism associated with w, and σ j ∈ S for 1 ≤ j ≤ k. As a consequence, M is the matrix of an… (More)
We consider the complexities of substitutive sequences over a binary alphabet. By studying various types of special words, we show that, knowing some initial values, its complexity can be completely formulated via a recurrence formula determined by the characteristic polynomial.