We consider domains D in the plane such that TD = D for some T > 1: D is T-automorphic. If D is any unbounded domain with Dirichlet boundary, there is always a positive harmonic function which is zero at every finite boundary point (Martin function). For automorphic domains, we show that D always has a Martin function of finite order, and if ĥD is Lipschitz, this generates the cone of Martin functions. However, there exist T-automorphic domains with infinitely many non-proportional Martin functions; still the space of functions of finite order remains one-dimensional.

This work (joint with V. Azarin and P. Poggi-Corradini) has applications to non-self-adjoint operators on the torus, and to entire functions of classes more general than functions of completely regular growth.