239 research outputs found
Extremal words in morphic subshifts
Given an infinite word X over an alphabet A a letter b occurring in X, and a
total order \sigma on A, we call the smallest word with respect to \sigma
starting with b in the shift orbit closure of X an extremal word of X. In this
paper we consider the extremal words of morphic words. If X = g(f^{\omega}(a))
for some morphisms f and g, we give two simple conditions on f and g that
guarantees that all extremal words are morphic. This happens, in particular,
when X is a primitive morphic or a binary pure morphic word. Our techniques
provide characterizations of the extremal words of the Period-doubling word and
the Chacon word and give a new proof of the form of the lexicographically least
word in the shift orbit closure of the Rudin-Shapiro word.Comment: Replaces a previous version entitled "Extremal words in the shift
orbit closure of a morphic sequence" with an added result on primitive
morphic sequences. Submitte
Extremal properties of (epi)Sturmian sequences and distribution modulo 1
Starting from a study of Y. Bugeaud and A. Dubickas (2005) on a question in
distribution of real numbers modulo 1 via combinatorics on words, we survey
some combinatorial properties of (epi)Sturmian sequences and distribution
modulo 1 in connection to their work. In particular we focus on extremal
properties of (epi)Sturmian sequences, some of which have been rediscovered
several times
Suffix conjugates for a class of morphic subshifts
Let A be a finite alphabet and f: A^* --> A^* be a morphism with an iterative
fixed point f^\omega(\alpha), where \alpha{} is in A. Consider the subshift (X,
T), where X is the shift orbit closure of f^\omega(\alpha) and T: X --> X is
the shift map. Let S be a finite alphabet that is in bijective correspondence
via a mapping c with the set of nonempty suffixes of the images f(a) for a in
A. Let calS be a subset S^N be the set of infinite words s = (s_n)_{n\geq 0}
such that \pi(s):= c(s_0)f(c(s_1)) f^2(c(s_2))... is in X. We show that if f is
primitive and f(A) is a suffix code, then there exists a mapping H: calS -->
calS such that (calS, H) is a topological dynamical system and \pi: (calS, H)
--> (X, T) is a conjugacy; we call (calS, H) the suffix conjugate of (X, T). In
the special case when f is the Fibonacci or the Thue-Morse morphism, we show
that the subshift (calS, T) is sofic, that is, the language of calS is regular
Suffix conjugates for a class of morphic subshifts
Let A be a finite alphabet and f: A^* --> A^* be a morphism with an iterative fixed point f^\omega(\alpha), where \alpha{} is in A. Consider the subshift (X, T), where X is the shift orbit closure of f^\omega(\alpha) and T: X --> X is the shift map. Let S be a finite alphabet that is in bijective correspondence via a mapping c with the set of nonempty suffixes of the images f(a) for a in A. Let calS be a subset S^N be the set of infinite words s = (s_n)_{n\geq 0} such that \pi(s):= c(s_0)f(c(s_1)) f^2(c(s_2))... is in X. We show that if f is primitive and f(A) is a suffix code, then there exists a mapping H: calS --> calS such that (calS, H) is a topological dynamical system and \pi: (calS, H) --> (X, T) is a conjugacy; we call (calS, H) the suffix conjugate of (X, T). In the special case when f is the Fibonacci or the Thue-Morse morphism, we show that the subshift (calS, T) is sofic, that is, the language of calS is regular.https://www.cambridge.org/core/journals/ergodic-theory-and-dynamical-systems/article/suffix-conjugates-for-a-class-of-morphic-subshifts/A531E7B26F382EDAF8455382C9C1DC9
Substitutive systems and a finitary version of Cobham's theorem
We study substitutive systems generated by nonprimitive substitutions and
show that transitive subsystems of substitutive systems are substitutive. As an
application we obtain a complete characterisation of the sets of words that can
appear as common factors of two automatic sequences defined over
multiplicatively independent bases. This generalises the famous theorem of
Cobham.Comment: 23 pages. v2: incorporates referee's comments, updated references, to
appear in Combinatoric
Episturmian words: a survey
In this paper, we survey the rich theory of infinite episturmian words which
generalize to any finite alphabet, in a rather resembling way, the well-known
family of Sturmian words on two letters. After recalling definitions and basic
properties, we consider episturmian morphisms that allow for a deeper study of
these words. Some properties of factors are described, including factor
complexity, palindromes, fractional powers, frequencies, and return words. We
also consider lexicographical properties of episturmian words, as well as their
connection to the balance property, and related notions such as finite
episturmian words, Arnoux-Rauzy sequences, and "episkew words" that generalize
the skew words of Morse and Hedlund.Comment: 36 pages; major revision: improvements + new material + more
reference
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