On polynomial growth functions of D0L-systems
The aim of this paper is to prove that every polynomial function that maps the natural integers to the positive integers is the growth function of some D0L-system.
cs.DM↗
arXiv subjects
Julien Cassaigne
Publications and source records attributed to Julien Cassaigne.
Avoiding Three Consecutive Blocks of the Same Size and Same Sum
We show that there exists an infinite word over the alphabet {0, 1, 3, 4} containing no three consecutive blocks of the same size and the same sum. This answers an open problem of Pirillo and Varricchio from 1994.
cs.DM↗
Outer billiard outside regular polygons
We consider outer billiard outside regular convex polygons. We deal with the case of regular polygons with $\{3,4,5,6,10\}$ sides, and we describe the symbolic dynamics of the map and compute the complexity of the language.
math.DS↗
On possible growth of Toeplitz languages
We consider a new family of factorial languages whose subword complexity grows as $Θ(n^α)$, where $α$ is the root of some transcendent equation. Analytical methods and in particular, a corollary of the Wiener-Pitt theorem, are used to find the asymptotic growth of the complexity. Factorial languages considered are languages of arithmetical factors of some Toeplitz words. So, we describe a new family of words with an unusual growth of arithmetical complexity.
math.CO↗
Avoiding Abelian powers in binary words with bounded Abelian complexity
The notion of Abelian complexity of infinite words was recently used by the three last authors to investigate various Abelian properties of words. In particular, using van der Waerden's theorem, they proved that if a word avoids Abelian $k$-powers for some integer $k$, then its Abelian complexity is unbounded. This suggests the following question: How frequently do Abelian $k$-powers occur in a word having bounded Abelian complexity? In particular, does every uniformly recurrent word having bounded Abelian complexity begin in an Abelian $k$-power? While this is true for various classes of uniformly recurrent words, including for example the class of all Sturmian words, in this paper we show the existence of uniformly recurrent binary words, having bounded Abelian complexity, which admit an infinite number of suffixes which do not begin in an Abelian square. We also show that the shift orbit closure of any infinite binary overlap-free word contains a word which avoids Abelian cubes in the beginning. We also consider the effect of morphisms on Abelian complexity and show that the morphic image of a word having bounded Abelian complexity has bounded Abelian complexity. Finally, we give an open problem on avoidability of Abelian squares in infinite binary words and show that it is equivalent to a well-known open problem of Pirillo-Varricchio and Halbeisen-Hungerbühler.
math.CO↗
Asymptotic behavior of growth functions of D0L-systems
A D0L-system is a triple (A, f, w) where A is a finite alphabet, f is an endomorphism of the free monoid over A, and w is a word over A. The D0L-sequence generated by (A, f, w) is the sequence of words (w, f(w), f(f(w)), f(f(f(w))), ...). The corresponding sequence of lengths, that is the function mapping each non-negative integer n to |f^n(w)|, is called the growth function of (A, f, w). In 1978, Salomaa and Soittola deduced the following result from their thorough study of the theory of rational power series: if the D0L-sequence generated by (A, f, w) is not eventually the empty word then there exist a non-negative integer d and a real number b greater than or equal to one such that |f^n(w)| behaves like n^d b^n as n tends to infinity. The aim of the present paper is to present a short, direct, elementary proof of this theorem.
cs.DM↗
Diophantine properties of real numbers generated by finite automata
We study some diophantine properties of automatic real numbers and we present a method to derive irrationality measures for such numbers. As a consequence, we prove that the $b$-adic expansion of a Liouville number cannot be generated by a finite automaton, a conjecture due to Shallit.
math.NT↗
Palindrome complexity
We study the palindrome complexity of infinite sequences on finite alphabets, i.e., the number of palindromic factors (blocks) of given length occurring in a given sequence. We survey the known results and obtain new results for some sequences, in particular for Rote sequences and for fixed points of primitive morphisms of constant length belonging to the class P substitutions of Hof-Knill-Simon. We also give an upper bound for the palindrome complexity of a sequence in terms of its (block-)complexity.
math.CO↗