The aim of this note is to study a pair of length functions on a free group, associated to a point in the unprojectivized Outer space. We discuss various characteristics which compare them and describe how these may be unified by a “Manhattan curve” analogous to those used to study pairs of hyperbolic surfaces. Our approach involves the thermodynamic formalism used in the ergodic theory of hype...

We study length-k-overlap-free binary infinite words, i.e., binary infinite words which can contain only overlaps xyxyx with |x| ≤ k− 1. We prove that no such word can be generated by a morphism, except if k = 1. On the other hand, for every k ≥ 2, there exist length-k-overlap-free binary infinite words which are not length-(k−1)-overlap-free. As an application, we prove that, for every non-neg...

A set of codewords is x free if it is both pre x free and su x free no codeword in the set is a pre x or a su x of any other A set of codewords fx x xng over a t letter alphabet is said to be complete if it satis es the Kraft inequality with equality so that X

In this paper we investigate languages containing at most a bounded number of words of each length. We rst show that the context-free languages for which the number of words of every length is bounded by a xed polynomial are exactly the bounded context-free languages in the sense of Ginsburg G]. Thus, we present a length characterization for bounded context-free languages. We then study slender...