Let \( f:\mathbb {C}\rightarrow \widehat{\mathbb {C}} \) be a transcendental map, and let U be an attracting or parabolic basin, or a doubly parabolic Baker domain. Assume U is simply connected. Then, we prove that periodic points are dense in \( \partial U \) , under certain hypothesis on the postsingular set. This generalizes a result by Przytycki and Zdunik for rational maps (Fund Math 145(1):65–77, 1994). Our proof uses techniques from measure theory, ergodic theory, conformal analysis, and inner functions. In particular, a result on the distortion of inner functions near the unit circle is provided, which is of independent interest.
