The important development in the study of discrete dynamical systems was Newhouse’s use of persistent homoclinic tangencies to show that a large set of \(C^2\)-diffeomorphisms of compact surfaces have infinitely many coexisting periodic attractors, or sinks [S. E. Newhouse, Prog. Math. 8, 1-114 (1980; Zbl 0444.58001)]. In the present article this result is obtained for various spaces of holomorphic maps of two variables.