Bounded Fatou components of cosine functions

TL;DR

The paper develops a Yoccoz puzzle framework for the cosine family $f(z)=a e^z+b e^{-z}$ under a bounded post-critical set, proving that any bounded Fatou component not eventual to a Siegel disk is a Jordan domain and establishing the local connectivity of the Julia set. It introduces a detailed puzzle construction, introduces dynamics via dynamic rays with addresses, and shows how to modify unbounded puzzle pieces into bounded, thickened pieces to enable modulus arguments. A renormalization mechanism is obtained when a critical value escapes, yielding a quadratic-like map and allowing the Straightening Theorem to imply local connectivity of Fatou components. Combining these results with a dichotomy on critical orbits, the work concludes local connectivity of $J(f)$ in the renormalizable and non-renormalizable regimes, thereby advancing understanding of transcendental cosine dynamics and their Julia sets.

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 $J(f)$.

Theorems & Definitions (31)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.1
• Theorem 1.3
• Lemma 2.1
• proof
• Lemma 2.2: BW91
• Lemma 3.1: RS08, Theorem 4.1 & Proposition 4.3
• Remark 3.1
• Lemma 3.2: BR20, Theorem 1.4