Periodic boundary points for simply connected Fatou components of transcendental maps

Abstract Let $$ f:\mathbb {C}\rightarrow \widehat{\mathbb {C}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>:</mml:mo> <mml:mi>C</mml:mi> <mml:mo>→</mml:mo> <mml:mover> <mml:mi>C</mml:mi> <mml:mo>^</mml:mo> </mml:mover> </mml:mrow> </mml:math> 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 $$ \partial U $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>∂</mml:mi> <mml:mi>U</mml:mi> </mml:mrow> </mml:math> , 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.