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In this paper, we set up a "dictionary" between discrete Schr\"odingeroperators and the holomorphic dynamics on certain affine cubic surfaces,building on previous work by Cantat, Damanik and Gorodetski.  Paragraphs 3.2.2 and 3.2.3 were written by Serge Cantat as an appendix.

Dynamical Green Functions and Discrete Schr\"odinger Operators with Potentials Generated by Primitive Invertible Substitution