Bounded Fatou components of cosine functions

Abstract

We constructed Yoccoz puzzle for cosine functions $f(z)=a{e}^z+b{e}^{-z}$ with bounded post-critical set, and proved that a Fatou component is a Jordan domains if it is bounded and is not eventually a Siegal disk. We proved that $f$ is renormalizable if a critical value escapes to $\infty$. Finally, we obtained the local connectivity of $J(f)$.