7 research outputs found
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
Infinite permutations vs. infinite words
I am going to compare well-known properties of infinite words with those of
infinite permutations, a new object studied since middle 2000s. Basically, it
was Sergey Avgustinovich who invented this notion, although in an early study
by Davis et al. permutations appear in a very similar framework as early as in
1977. I am going to tell about periodicity of permutations, their complexity
according to several definitions and their automatic properties, that is, about
usual parameters of words, now extended to permutations and behaving sometimes
similarly to those for words, sometimes not. Another series of results concerns
permutations generated by infinite words and their properties. Although this
direction of research is young, many people, including two other speakers of
this meeting, have participated in it, and I believe that several more topics
for further study are really promising.Comment: In Proceedings WORDS 2011, arXiv:1108.341
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
Ergodic Infinite Permutations of Minimal Complexity
International audienceAn infinite permutation can be defined as a linear ordering of the set of natural numbers. Similarly to infinite words, a complexity p(n) of an infinite permutation is defined as a function counting the number of its factors of length n. For infinite words, a classical result of Morse and Hedlind, 1940, states that if the complexity of an infinite word satisfies p(n) ≤ n for some n, then the word is ultimately periodic. Hence minimal complexity of aperiodic words is equal to n + 1, and words with such complexity are called Sturmian. For infinite permutations this does not hold: There exist aperiodic permutations with complexity functions of arbitrarily slow growth, and hence there are no permutations of minimal complexity. In the paper we introduce a new notion of ergodic permutation, i.e., a permutation which 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 the minimal complexity of an ergodic permutation is p(n) = n, and that the class of ergodic permutations of minimal complexity coincides with the class of so-called Sturmian permutations, directly related to Sturmian words
On automatic infinite permutations
An infinite permutation α is a linear ordering of N. We study properties
of infinite permutations analogous to those of infinite words, and show some resemblances
and some differences between permutations and words. In this paper, we try to extend to
permutations the notion of automaticity. As we shall show, the standard definitions which
are equivalent in the case of words are not equivalent in the context of permutations. We
investigate the relationships between these definitions and prove that they constitute a
chain of inclusions. We also construct and study an automaton generating the Thue-Morse
permutation