1,649 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
Languages invariant under more symmetries: overlapping factors versus palindromic richness
Factor complexity and palindromic complexity of
infinite words with language closed under reversal are known to be related by
the inequality for any \,. Word for which
the equality is attained for any is usually called rich in palindromes. In
this article we study words whose languages are invariant under a finite group
of symmetries. For such words we prove a stronger version of the above
inequality. We introduce notion of -palindromic richness and give several
examples of -rich words, including the Thue-Morse sequence as well.Comment: 22 pages, 1 figur
Canonical Representatives of Morphic Permutations
An infinite permutation can be defined as a linear ordering of the set of
natural numbers. In particular, an infinite permutation can be constructed with
an aperiodic infinite word over as the lexicographic order
of the shifts of the word. In this paper, we discuss the question if an
infinite permutation defined this way admits a canonical representative, that
is, can be defined by a sequence of numbers from [0, 1], such that the
frequency of its elements in any interval is equal to the length of that
interval. We show that a canonical representative exists if and only if the
word is uniquely ergodic, and that is why we use the term ergodic permutations.
We also discuss ways to construct the canonical representative of a permutation
defined by a morphic word and generalize the construction of Makarov, 2009, for
the Thue-Morse permutation to a wider class of infinite words.Comment: Springer. WORDS 2015, Sep 2015, Kiel, Germany. Combinatorics on
Words: 10th International Conference. arXiv admin note: text overlap with
arXiv:1503.0618
Generalized Thue-Morse words and palindromic richness
We prove that the generalized Thue-Morse word defined for
and as , where denotes the sum of digits in the base-
representation of the integer , has its language closed under all elements
of a group isomorphic to the dihedral group of order consisting of
morphisms and antimorphisms. Considering simultaneously antimorphisms , we show that is saturated by -palindromes
up to the highest possible level. Using the terminology generalizing the notion
of palindromic richness for more antimorphisms recently introduced by the
author and E. Pelantov\'a, we show that is -rich. We
also calculate the factor complexity of .Comment: 11 page
A Note on Symmetries in the Rauzy Graph and Factor Frequencies
We focus on infinite words with languages closed under reversal. If
frequencies of all factors are well defined, we show that the number of
different frequencies of factors of length n+1 does not exceed 2C(n+1)-2C(n)+1.Comment: 7 page
Enumeration and Decidable Properties of Automatic Sequences
We show that various aspects of k-automatic sequences -- such as having an
unbordered factor of length n -- are both decidable and effectively enumerable.
As a consequence it follows that many related sequences are either k-automatic
or k-regular. These include many sequences previously studied in the
literature, such as the recurrence function, the appearance function, and the
repetitivity index. We also give some new characterizations of the class of
k-regular sequences. Many results extend to other sequences defined in terms of
Pisot numeration systems
On the number of return words in infinite words with complexity 2n+1
In this article, we count the number of return words in some infinite words
with complexity 2n+1. We also consider some infinite words given by codings of
rotation and interval exchange transformations on k intervals. We prove that
the number of return words over a given word w for these infinite words is
exactly k.Comment: see also http://liafa.jussieu.fr/~vuillon/articles.htm
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