In this paper we study bi-infinite words on two letters. We say that such a word has stiffness $k$ if the number of different subwords of length $n$ equals $n+k$ for all $n$ sufficiently large. The word is called $k$-balanced if the numbers of occurrences of the symbol a in any two subwords of the same length differ by at most $k$. In the present paper we give a complete description of the class of bi-infinite words of stiffness $k$ and show that the number of subwords of length $n$ from this class has growth order...

This paper consists of three parts. In the first part we prove a general theorem on the image of a language $K$ under a substitution, in the second we apply this to the special case when $K$ is the language of balanced words and in the third part we deal with recurrent -words of minimal block growth.

We study the extreme and exposed points of the convex set consisting of representing measures of the disk algebra, supported in the closed unit disk. A boundary point of this set is shown to be extreme (and even exposed) if its support inside the open unit disk consists of two points that do not lie on the same radius of the disk. If its support inside the unit disk consists of 3 or more points, it is very seldom an extreme point. We also give a necessary condition for extreme points to be exposed...

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