Bounded Fatou components of cosine functions

TL;DR

This paper studies the dynamics of cosine functions, constructing Yoccoz puzzles, characterizing Fatou components, and proving local connectivity of the Julia set, advancing understanding of complex dynamical systems with bounded post-critical sets.

Contribution

It introduces a Yoccoz puzzle construction for cosine functions with bounded post-critical sets and characterizes Fatou components and renormalizability, providing new insights into their complex dynamics.

Findings

Fatou components are Jordan domains if bounded and not eventually Siegel disks

Cosine functions are renormalizable when critical values escape to infinity

Julia sets are locally connected under certain conditions

Abstract

We constructed Yoccoz puzzle for cosine functions $f(z)=ae^z+be^{-z}$ 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 $J(f)$.