In this expository article we describe the two main methods of representing geodesics on surfaces of constant negative curvature by symbolic sequences and their development. A geometric method stems from a 1898 work of J. Hadamard and was developed by M. Morse in the 1920s. It consists of recording the successive sides of a given fundamental region cut by the geodesic and may be applied to all finitely generated Fuchsian groups. Another method, of arithmetic nature, uses continued fraction expansions of the end points of the geodesic at infinity and is even older—it comes from the Gauss reduction theory. Introduced to dynamics by E. Artin in a 1924 paper, this method was used to exhibit dense geodesics on the modular surface. For 80 years these classical works have provided inspiration for mathematicians and a testing ground for new methods in dynamics, geometry and combinatorial group theory. We present some of the ideas, results (old and recent), and interpretations that illustrate the multiple facets of the subject.

. We introduce a new complexity measure of a factorial formal language L: the growth rate of the set of minimal forbidden words. We prove some combinatorial properties of minimal forbidden words. As main result we prove that the growth rate of the set of minimal forbidden words for L is a topological invariant of the dynamical system defined by L. Classification: Automata and Formal Languages 1 Introduction Let L ae A be a factorial language, i.e. a language containing all factors of its words. A word w 2 A is a minimal forbidden word for L if w = 2 L and all proper factors of w belong to L. We denote by MF (L) the language of minimal forbidden words for L. It turns out (as also stressed by the results of this paper) that the combinatorial properties of MF (L) provide an usefull tool to investigate the structure of the language L or of the system that it describes. Consider, for instance, the case of locally testable factorial languages (cf [14]): they are characterized by th...

Abstract. Let q be an odd number and Sq,0(n) the difference between the number of k < n, k ≡ 0mod q, with an even binary digit sum and the corresponding number of k<n,k≡0mod q, with an odd binary digit sum. A remarkable theorem of Newman says that S3,0(n)> 0 for all n. Inthispaper it is proved that the same assertion holds if q is divisible by 3 or q =4 N +1. On the other hand, it is shown that the number of primes q ≤ x with this property is o(x / log x). Finally, analoga for “higher parities ” are provided. 1.

Abstract. Let β be a real number with 1 < β < 2. We prove that the language of the β-shift is ∆ 0 n iff β is a ∆n-real. The special case where n is 1 is the independently interesting result that the language of the β-shift is decidable iff β is a computable real. The “if ” part of the proof is non-constructive; we show that for Walters ’ version of the β-shift, no constructive proof exists. 1

Denote by (R; \Delta) the multiplicative semigroup of an associative algebra R over an infinite field, and let (R; ffi) represent R when viewed as a semigroup via the circle operation x ffi y = x + y + xy. In this paper we characterize the existence of an identity in these semigroups in terms of the Lie structure of R. Namely, we prove that the following conditions on R are equivalent: the semigroup (R; ffi) satisfies an identity; the semigroup (R; \Delta) satisfies a reduced identity; and, the associated Lie algebra of R satisfies the Engel condition. When R is finitely generated these conditions are each equivalent to R being upper Lie nilpotent.

. I survey some of the connections between formal languages and number theory. Topics discussed include applications of representation in base k, representation by sums of Fibonacci numbers, automatic sequences, transcendence in finite characteristic, automatic real numbers, fixed points of homomorphisms, automaticity, and k-regular sequences. Key words. finite automata, automatic sequences, transcendence, automaticity AMS(MOS) subject classifications. Primary 11B85, Secondary 11A63 11A55 11J81 1. Introduction. In this paper, I survey some interesting connections between number theory and the theory of formal languages. This is a very large and rapidly growing area, and I focus on a few areas that interest me, rather than attempting to be comprehensive. (An earlier survey of this area, written in French, is [1].) I also give a number of open questions. Number theory deals with the properties of integers, and formal language theory deals with the properties of strings. At the interse...

The border correlation function β: A ∗ → A ∗ , for A = {a, b}, specifies which conjugates (cyclic shifts) of a given word w of length n are bordered, in other words, β(w) = c0c1... cn−1, where ci = a or b according to whether the i-th cyclic shift σ i (w) of w is unbordered or bordered. Except for some special cases, no binary word w has two consecutive unbordered conjugates (σ i (w) and σ i+1 (w)). We show that this is optimal: in every cyclically overlap-free word every other conjugate is unbordered. We also study the relationship between unbordered conjugates and critical points, as well as, the dynamic system given by iterating the function β. We prove that, for each word w of length n, the sequence w, β(w), β 2 (w),... terminates either in b n or in the cycle of conjugates of the word ab k ab k+1 for n = 2k + 3.

Measured geodesic laminations is a remarkable abstraction (due to W. P. Thurston) of many otherwise unrelated phenomena occurring in differential geometry, complex analysis and geometric topology. In this article we focus on connections of geodesic laminations with the inductive limits of finite-dimensional semi-simple C ∗-algebras (AF C ∗-algebras). Our main result is a bijection between combinatorial presentation of such C ∗-algebras (so-called Bratteli diagrams) and measured geodesic laminations on compact surfaces. This link appears helpful indeed as it provides insights to the Teichmüller spaces, Thurston’s theory of surface homeomorphisms, Stallings ’ fibrations to the one side, and noncommutative (algebraic) geometry to the other. Key words and phrases: AF C ∗-algebras, complex curves, geometric topology

We start a general study of counting the number of occurrences of ordered patterns in words generated by morphisms. We consider certain patterns with gaps (classical patterns) and that with no gaps (consecutive patterns). Occurrences of the patterns are known, in the literature, as rises, descents, (non-)inversions, squares and p-repetitions. We give recurrence formulas in the general case, then deducing exact formulas for particular families of morphisms. Many (classical or new) examples are given illustrating the techniques and showing their interest.

this paper.