54 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
Abelian bordered factors and periodicity
A finite word u is said to be bordered if u has a proper prefix which is also
a suffix of u, and unbordered otherwise. Ehrenfeucht and Silberger proved that
an infinite word is purely periodic if and only if it contains only finitely
many unbordered factors. We are interested in abelian and weak abelian
analogues of this result; namely, we investigate the following question(s): Let
w be an infinite word such that all sufficiently long factors are (weakly)
abelian bordered; is w (weakly) abelian periodic? In the process we answer a
question of Avgustinovich et al. concerning the abelian critical factorization
theorem.Comment: 14 page
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