54 research outputs found
On winning shifts of marked uniform substitutions
The second author introduced with I. T\"orm\"a a two-player word-building
game [Playing with Subshifts, Fund. Inform. 132 (2014), 131--152]. The game has
a predetermined (possibly finite) choice sequence , ,
of integers such that on round the player chooses a subset
of size of some fixed finite alphabet and the player picks
a letter from the set . The outcome is determined by whether the word
obtained by concatenating the letters picked lies in a prescribed target
set (a win for player ) or not (a win for player ). Typically, we
consider to be a subshift. The winning shift of a subshift is
defined as the set of choice sequences for which has a winning strategy
when the target set is the language of . The winning shift mirrors
some properties of . For instance, and have the same entropy.
Virtually nothing is known about the structure of the winning shifts of
subshifts common in combinatorics on words. In this paper, we study the winning
shifts of subshifts generated by marked uniform substitutions, and show that
these winning shifts, viewed as subshifts, also have a substitutive structure.
Particularly, we give an explicit description of the winning shift for the
generalized Thue-Morse substitutions. It is known that and have the
same factor complexity. As an example application, we exploit this connection
to give a simple derivation of the first difference and factor complexity
functions of subshifts generated by marked substitutions. We describe these
functions in particular detail for the generalized Thue-Morse substitutions.Comment: Extended version of a paper presented at RuFiDiM I
More on the dynamics of the symbolic square root map
In our earlier paper [A square root map on Sturmian words, Electron. J.
Combin. 24.1 (2017)], we introduced a symbolic square root map. Every optimal
squareful infinite word contains exactly six minimal squares and can be
written as a product of these squares: . The square
root of is the infinite word obtained by
deleting half of each square. We proved that the square root map preserves the
languages of Sturmian words (which are optimal squareful words). The dynamics
of the square root map on a Sturmian subshift are well understood. 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. In this paper,
we study these dynamical systems in more detail and compare their properties to
the Sturmian case. The main results are characterizations of periodic points
and the limit set. 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.Comment: 22 pages, Extended version of a paper presented at WORDS 201
Characterization of repetitions in Sturmian words: A new proof
We present a new, dynamical way to study powers (that is, repetitions) in
Sturmian words based on results from Diophantine approximation theory. As a
result, we provide an alternative and shorter proof of a result by Damanik and
Lenz characterizing powers in Sturmian words [Powers in Sturmian sequences,
Eur. J. Combin. 24 (2003), 377--390]. Further, as a consequence, we obtain a
previously known formula for the fractional index of a Sturmian word based on
the continued fraction expansion of its slope.Comment: 9 pages, 1 figur
Privileged Words and Sturmian Words
This dissertation has two almost unrelated themes: privileged words and Sturmian words. Privileged words are a new class of words introduced recently. A word is privileged if it is a complete first return to a shorter privileged word, the shortest privileged words being letters and the empty word. Here we give and prove almost all results on privileged words known to date. On the other hand, the study of Sturmian words is a well-established topic in combinatorics on words. In this dissertation, we focus on questions concerning repetitions in Sturmian words, reproving old results and giving new ones, and on establishing completely new research directions.
The study of privileged words presented in this dissertation aims to derive their basic properties and to answer basic questions regarding them. We explore a connection between privileged words and palindromes and seek out answers to questions on context-freeness, computability, and enumeration. It turns out that the language of privileged words is not context-free, but privileged words are recognizable by a linear-time algorithm. A lower bound on the number of binary privileged words of given length is proven. The main interest, however, lies in the privileged complexity functions of the Thue-Morse word and Sturmian words. We derive recurrences for computing the privileged complexity function of the Thue-Morse word, and we prove that Sturmian words are characterized by their privileged complexity function. As a slightly separate topic, we give an overview of a certain method of automated theorem-proving and show how it can be applied to study privileged factors of automatic words.
The second part of this dissertation is devoted to Sturmian words. We extensively exploit the interpretation of Sturmian words as irrational rotation words. The essential tools are continued fractions and elementary, but powerful, results of Diophantine approximation theory. With these tools at our disposal, we reprove old results on powers occurring in Sturmian words with emphasis on the fractional index of a Sturmian word. Further, we consider abelian powers and abelian repetitions and characterize the maximum exponents of abelian powers with given period occurring in a Sturmian word in terms of the continued fraction expansion of its slope. We define the notion of abelian critical exponent for Sturmian words and explore its connection to the Lagrange spectrum of irrational numbers. The results obtained are often specialized for the Fibonacci word; for instance, we show that the minimum abelian period of a factor of the Fibonacci word is a Fibonacci number. In addition, we propose a completely new research topic: the square root map. We prove that the square root map preserves the language of any Sturmian word. Moreover, we construct a family of non-Sturmian optimal squareful words whose language the square root map also preserves.This construction yields examples of aperiodic infinite words whose square roots are periodic.Siirretty Doriast
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