71 research outputs found
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
Cyclic Complexity of Words
We introduce and study a complexity function on words called
\emph{cyclic complexity}, which counts the number of conjugacy classes of
factors of length of an infinite word We extend the well-known
Morse-Hedlund theorem to the setting of cyclic complexity by showing that a
word is ultimately periodic if and only if it has bounded cyclic complexity.
Unlike most complexity functions, cyclic complexity distinguishes between
Sturmian words of different slopes. We prove that if is a Sturmian word and
is a word having the same cyclic complexity of then up to renaming
letters, and have the same set of factors. In particular, is also
Sturmian of slope equal to that of Since for some
implies is periodic, it is natural to consider the quantity
We show that if is a Sturmian word,
then We prove however that this is
not a characterization of Sturmian words by exhibiting a restricted class of
Toeplitz words, including the period-doubling word, which also verify this same
condition on the limit infimum. In contrast we show that, for the Thue-Morse
word , Comment: To appear in Journal of Combinatorial Theory, Series
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
Minimal complexity of equidistributed infinite permutations
An infinite permutation is a linear ordering of the set of natural numbers.
An infinite permutation can be defined by a sequence of real numbers where only
the order of elements is taken into account. In the paper we investigate a new
class of {\it equidistributed} infinite permutations, that is, infinite
permutations which can be defined by equidistributed sequences. Similarly to
infinite words, a complexity of an infinite permutation is defined as a
function counting the number of its subpermutations of length . For infinite
words, a classical result of Morse and Hedlund, 1938, states that if the
complexity of an infinite word satisfies for some , then the
word is ultimately periodic. Hence minimal complexity of aperiodic words is
equal to , and words with such complexity are called Sturmian. For
infinite permutations this does not hold: There exist aperiodic permutations
with complexity functions growing arbitrarily slowly, and hence there are no
permutations of minimal complexity. We show that, unlike for permutations in
general, the minimal complexity of an equidistributed permutation is
. The class of equidistributed permutations of minimal
complexity coincides with the class of so-called Sturmian permutations,
directly related to Sturmian words.Comment: An old (weaker) version of the paper was presented at DLT 2015. The
current version is submitted to a journa
Palindromic richness for languages invariant under more symmetries
For a given finite group consisting of morphisms and antimorphisms of a
free monoid , we study infinite words with language closed under
the group . We focus on the notion of -richness which describes words
rich in generalized palindromic factors, i.e., in factors satisfying
for some antimorphism . We give several
equivalent descriptions which are generalizations of know characterizations of
rich words (in the terms of classical palindromes) and show two examples of
-rich words
Non-Injectivity of Infinite Interval Exchange Transformations and Generalized Thue-Morse Sequences
In this paper we study the non-injectivity arising in infinite interval
exchange transformations. In particular, we build and analyze an infinite
family of infinite interval exchanges semi-conjugated to generalized Thue-Morse
subshifts, whose non-injectivity occurs at a characterizable finite set of
points.Comment: 23 pages, 8 figure
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