A new proof of no-wandering domain theorems without quasiconformal techniques

In one complex dimension, we use hyperbolic area to prove the absence of wandering Fatou components for rational maps of degree at least two and for transcendental entire functions whose singular sets are compact and have all their accumulation points in the Fatou set. As corollaries, we recover Sullivan’s theorem for rational maps and the theorem of Eremenko–Lyubich and Goldberg–Keen for entire functions with finitely many singular values. The proof uses neither quasiconformal deformation nor Teichmüller theory.