Parameter spaces of cosine and sine families

Definitions and mathematical statement

For $\lambda\in\mathbb C\setminus\{0\}$, consider the entire maps \[ S_\lambda(z)=\lambda\sin z,\qquad C_\lambda(z)=\lambda\cos z. \] Their Fatou sets are the open sets of normality of their iterates, and their Julia sets are the complements in $\mathbb C$. Each map has precisely the two finite critical values $\lambda$ and $-\lambda$, and no finite asymptotic values. A parameter is hyperbolic when both critical values are attracted to attracting periodic cycles. Connected components of this parameter locus are the hyperbolic components.

The source asks for a description of the complex parameter spaces of these two families. This is a classification program: determine how regions with specified critical-orbit behavior are organized, how their boundaries and bifurcations fit together, and which parameters have Julia set equal to the whole plane. In particular, the geometry of curves of parameters with escaping critical orbits and their accumulation must be related to the stable components.

The two families are to be studied in their given one-parameter normalizations. A classification of only real $\lambda$, or only parameters with an attracting fixed point, is a restricted case. The singular values move with the parameter, so their iterates provide natural data for organizing the loci. The requested description should specify the dynamical meaning of its parameter regions and boundary identifications, rather than give only a numerical picture. Complete classification is the broad target; individual structural theorems are partial progress toward it.

Short English statement

Describe the complex parameter spaces of the sine and cosine families through their critical orbits and bifurcations.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 5300064 (AMR-052-0064), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Stony Brook - Open Problems in Dynamical Systems, 1992, Devaney P6. Original source indexed by AMR-052-0064. https://www.math.stonybrook.edu/open-problems-dynamical-systems.
• S3 Devaney, Open Questions in Non-Rational Complex Dynamics, trigonometric-family question, in Bielefeld–Lyubich (1992). https://www.math.stonybrook.edu/preprints/ims92-7.pdf.