Leaves of laminations can be symbolically represented by deforming them into paths of labeled embedded carrier graphs, including train tracks. Here, we describe and characterize the languages of two-way infinite words coming from this kind of coding, called lamination languages, first, by using carrier graph sequences, and second, by using word combinatorics. These characterizations generalize those existing for interval exchange transformations. We also show that lamination languages have ultimately affine factor complexity, and we present effective techniques to build these languages.

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This paper introduces a generalized way of defining languages of infinite words by topological means. It focuses on languages generated by iterating morphisms on free semi-groups (also called substitutions or D0L-systems). The main result is an effective construction of the boundary of such languages which leads to a bijective mapping of the boundary onto a regular language. Among the obtained properties are the uncountability of the boundary and the strict quasiperiodicity of its words. We also investigate the decidability of the boundary equality problem and the dynamical system induced by the inversed morphism.

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