31 research outputs found
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
FACTOR AND PALINDROMIC COMPLEXITY OF THUE-MORSE’S AVATARS
Two infinite words that are connected with some significant univoque numbers are studied. It is shown that their factor and palindromic complexities almost coincide with the factor and palindromic complexities of the famous Thue-Morse word
Complexity and fractal dimensions for infinite sequences with positive entropy
The complexity function of an infinite word on a finite alphabet is
the sequence counting, for each non-negative , the number of words of length
on the alphabet that are factors of the infinite word . The goal of
this work is to estimate the number of words of length on the alphabet
that are factors of an infinite word with a complexity function bounded by
a given function with exponential growth and to describe the combinatorial
structure of such sets of infinite words. We introduce a real parameter, the
{\it word entropy} associated to a given function and we determine
the fractal dimensions of sets of infinite sequences with complexity function
bounded by in terms of its word entropy. We present a combinatorial proof
of the fact that is equal to the topological entropy of the subshift
of infinite words whose complexity is bounded by and we give several
examples showing that even under strong conditions on , the word entropy
can be strictly smaller than the limiting lower exponential growth
rate of .Comment: 24 page
Bispecial factors in circular non-pushy D0L languages
We study bispecial factors in fixed points of morphisms. In particular, we
propose a simple method of how to find all bispecial words of non-pushy
circular D0L-systems. This method can be formulated as an algorithm. Moreover,
we prove that non-pushy circular D0L-systems are exactly those with finite
critical exponent.Comment: 18 pages, 5 figure
Towards a statement of the S-adic conjecture through examples
The -adic conjecture claims that there exists a condition such that a
sequence has a sub-linear complexity if and only if it is an -adic sequence
satisfying Condition for some finite set of morphisms. We present an
overview of the factor complexity of -adic sequences and we give some
examples that either illustrate some interesting properties or that are
counter-examples to what could be believed to be "a good Condition ".Comment: 2
Factor frequencies in languages invariant under more symmetries
The number of frequencies of factors of length in a recurrent aperiodic
infinite word does not exceed 3\Delta \C(n), where \Delta \C (n) is the
first difference of factor complexity, as shown by Boshernitzan. Pelantov\'a
together with the author derived a better upper bound for infinite words whose
language is closed under reversal. In this paper, we further diminish the upper
bound for uniformly recurrent infinite words whose language is invariant under
all elements of a finite group of symmetries and we prove the optimality of the
obtained upper bound.Comment: 13 page
Characterization of infinite LSP words and endomorphisms preserving the LSP property
Answering a question of G. Fici, we give an -adic characterization of
thefamily of infinite LSP words, that is, the family of infinite words having
all their left special factors as prefixes.More precisely we provide a finite
set of morphisms and an automaton such that an infinite word is
LSP if and only if it is -adic and one of its directive words is
recognizable by .Then we characterize the endomorphisms that preserve
the property of being LSP for infinite words.This allows us to prove that there
exists no set of endomorphisms for which the set of infinite LSP words
corresponds to the set of -adic words. This implies that an automaton is
required no matter which set of morphisms is used.Comment: arXiv admin note: text overlap with arXiv:1705.0578