Languages, D0L-systems, sets of curves, and surface automorphisms

Abstract

We introduce, extend and apply some relationships between formal language theory and surface theory. First we show how boundaries of languages topologized with the Cantor metric can be mapped to sets of curves on surfaces, namely laminations. Second we present how and when endomorphisms of free monoids, i.e. substitutions, can be mapped to automorphisms of surfaces, so that D0L-systems correspond to iterations of these automorphisms. Third, we apply these ideas to construct sets of non-periodic irreducible automorphisms of surfaces following [14] , [35] , [43] , so that the involved proofs do not use any differential or algebraic tools, but, accordingly, substitution and D0L-system theory of [11] , [12] , [31] , [32] , [37] —mainly the decidability for a D0L-language [12] to be strongly repetitive or not. This construction also yields effective symbolic descriptions of the stable sets of curves associated with these automorphisms.