Extremal words in morphic subshifts

Given an infinite word X over an alphabet A a letter b occurring in X, and a total order \sigma on A, we call the smallest word with respect to \sigma starting with b in the shift orbit closure of X an extremal word of X. In this paper we consider the extremal words of morphic words. If X = g(f^{\omega}(a)) for some morphisms f and g, we give two simple conditions on f and g that guarantees that all extremal words are morphic. This happens, in particular, when X is a primitive morphic or a binary pure morphic word. Our techniques provide characterizations of the extremal words of the Period-doubling word and the Chacon word and give a new proof of the form of the lexicographically least word in the shift orbit closure of the Rudin-Shapiro word.Comment: Replaces a previous version entitled "Extremal words in the shift orbit closure of a morphic sequence" with an added result on primitive morphic sequences. Submitte

Binary words containing infinitely many overlaps

We characterize the squares occurring in infinite overlap-free binary words and construct various alpha power-free binary words containing infinitely many overlaps.Comment: 9 page

Replica Symmetry Breaking in Short-Range Spin Glasses: Theoretical Foundations and Numerical Evidences

We discuss replica symmetry breaking (RSB) in spin glasses. We update work in this area, from both the analytical and numerical points of view. We give particular attention to the difficulties stressed by Newman and Stein concerning the problem of constructing pure states in spin glass systems. We mainly discuss what happens in finite-dimensional, realistic spin glasses. Together with a detailed review of some of the most important features, facts, data, and phenomena, we present some new theoretical ideas and numerical results. We discuss among others the basic idea of the RSB theory, correlation functions, interfaces, overlaps, pure states, random field, and the dynamical approach. We present new numerical results for the behaviors of coupled replicas and about the numerical verification of sum rules, and we review some of the available numerical results that we consider of larger importance (for example, the determination of the phase transition point, the correlation functions, the window overlaps, and the dynamical behavior of the system).Comment: 48 pages, 21 figures. v2: the published versio

Cost and dimension of words of zero topological entropy

Let $A^*$ denote the free monoid generated by a finite nonempty set $A.$ In this paper we introduce a new measure of complexity of languages $L\subseteq A^*$ defined in terms of the semigroup structure on $A^*.$ For each $L\subseteq A^*,$ we define its {\it cost} $c(L)$ as the infimum of all real numbers $\alpha$ for which there exist a language $S\subseteq A^*$ with $p_S(n)=O(n^\alpha)$ and a positive integer $k$ with $L\subseteq S^k.$ We also define the {\it cost dimension} $d_c(L)$ as the infimum of the set of all positive integers $k$ such that $L\subseteq S^k$ for some language $S$ with $p_S(n)=O(n^{c(L)}).$ We are primarily interested in languages $L$ given by the set of factors of an infinite word $x=x_0x_1x_2\cdots \in A^\omega$ of zero topological entropy, in which case $c(L)<+\infty.$ We establish the following characterisation of words of linear factor complexity: Let $x\in A^\omega$ and $L=$Fac$(x)$ be the set of factors of $x.$ Then $p_x(n)=\Theta(n)$ if and only $c(L)=0$ and $d_c(L)=2.$ In other words, $p_x(n)=O(n)$ if and only if Fac$(x)\subseteq S^2$ for some language $S\subseteq A^+$ of bounded complexity (meaning $\limsup p_S(n)<+\infty).$ In general the cost of a language $L$ reflects deeply the underlying combinatorial structure induced by the semigroup structure on $A^*.$ For example, in contrast to the above characterisation of languages generated by words of sub-linear complexity, there exist non factorial languages $L$ of complexity $p_L(n)=O(\log n)$ (and hence of cost equal to $0)$ and of cost dimension $+\infty.$ In this paper we investigate the cost and cost dimension of languages defined by infinite words of zero topological entropy

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-Hungerb\"uhler.Comment: 16 pages, submitte

Finite-Degree Predicates and Two-Variable First-Order Logic

We consider two-variable first-order logic on finite words with a fixed number of quantifier alternations. We show that all languages with a neutral letter definable using the order and finite-degree predicates are also definable with the order predicate only. From this result we derive the separation of the alternation hierarchy of two-variable logic on this signature