Some characterizations of Sturmian words in terms of the lexicographic order

In this paper we present three new characterizations of Sturmian words based on the lexicographic ordering of their factors

Cyclic Complexity of Words

We introduce and study a complexity function on words $c_x(n),$ called \emph{cyclic complexity}, which counts the number of conjugacy classes of factors of length $n$ of an infinite word $x.$ We extend the well-known Morse-Hedlund theorem to the setting of cyclic complexity by showing that a word is ultimately periodic if and only if it has bounded cyclic complexity. Unlike most complexity functions, cyclic complexity distinguishes between Sturmian words of different slopes. We prove that if $x$ is a Sturmian word and $y$ is a word having the same cyclic complexity of $x,$ then up to renaming letters, $x$ and $y$ have the same set of factors. In particular, $y$ is also Sturmian of slope equal to that of $x.$ Since $c_x(n)=1$ for some $n\geq 1$ implies $x$ is periodic, it is natural to consider the quantity $\liminf_{n\rightarrow \infty} c_x(n).$ We show that if $x$ is a Sturmian word, then $\liminf_{n\rightarrow \infty} c_x(n)=2.$ We prove however that this is not a characterization of Sturmian words by exhibiting a restricted class of Toeplitz words, including the period-doubling word, which also verify this same condition on the limit infimum. In contrast we show that, for the Thue-Morse word $t$, $\liminf_{n\rightarrow \infty} c_t(n)=+\infty.$Comment: To appear in Journal of Combinatorial Theory, Series

A new factorization of mechanical words

In this paper, we introduce a morphism on Sturmian words which is tightly related to the coefficients of a particular continued fraction the ceiled continued fraction. This morphism will be applied to factorize periodic Sturmian words called Christoffel words, as well as, to characterize and construct discrete geometric objects called cellular line

Convex Dynamics and Applications

This paper proves a theorem about bounding orbits of a time dependent dynamical system. The maps that are involved are examples in convex dynamics, by which we mean the dynamics of piecewise isometries where the pieces are convex. The theorem came to the attention of the authors in connection with the problem of digital halftoning. \textit{Digital halftoning} is a family of printing technologies for getting full color images from only a few different colors deposited at dots all of the same size. The simplest version consist in obtaining grey scale images from only black and white dots. A corollary of the theorem is that for \textit{error diffusion}, one of the methods of digital halftoning, averages of colors of the printed dots converge to averages of the colors taken from the same dots of the actual images. Digital printing is a special case of a much wider class of scheduling problems to which the theorem applies. Convex dynamics has roots in classical areas of mathematics such as symbolic dynamics, Diophantine approximation, and the theory of uniform distributions.Comment: LaTex with 9 PostScript figure

Piecewise Linear Models for the Quasiperiodic Transition to Chaos

We formulate and study analytically and computationally two families of piecewise linear degree one circle maps. These families offer the rare advantage of being non-trivial but essentially solvable models for the phenomenon of mode-locking and the quasi-periodic transition to chaos. For instance, for these families, we obtain complete solutions to several questions still largely unanswered for families of smooth circle maps. Our main results describe (1) the sets of maps in these families having some prescribed rotation interval; (2) the boundaries between zero and positive topological entropy and between zero length and non-zero length rotation interval; and (3) the structure and bifurcations of the attractors in one of these families. We discuss the interpretation of these maps as low-order spline approximations to the classic ``sine-circle'' map and examine more generally the implications of our results for the case of smooth circle maps. We also mention a possible connection to recent experiments on models of a driven Josephson junction.Comment: 75 pages, plain TeX, 47 figures (available on request