The author continues previous work of S. Cantat [Duke Math. J. 149, No. 3, 411–460 (2009 Zbl 1181.37068)] and D. Damanik and A. Gorodetski [Nonlinearity 22, No. 1, 123–143 (2009 Zbl 1154.82312), Geom. Funct. Anal. 22, No. 4, 976–989 (2012 Zbl 1256.81035)]. He considers discrete Schrödinger operators on \(\ell^2(\mathbb{Z})\) and holomorphic dynamics of several complex variables on certain affine cubic surfaces and he gives a dictionary between these two objects. For example, the almost-sure spectrum \(\Sigma_\kappa\) of the Schrödinger operator corresponds to the Julia set of an automorphism \(f\) on \(\mathcal{S}_{4+\kappa^2}\) and the Lyapunov exponent \(\gamma_k\) of the Schrödinger operator corresponds to the dynamical Green function \(G_f^+\). In order to achieve this goal, the author makes use of potential theory. In particular, a detailed description of the dynamical Green function is obtained and also basic results concerning the equilibrium measure and the Green function of compact subsets of \(\mathbb{C}\) are used to transfer statements from the dynamical context to the Schrödinger operator. This dictionary gives new insight in several recent theorems on this subject.