Composite periodic entire functions

Definitions and mathematical statement

Let $f,g:\mathbb C\to\mathbb C$ be nonconstant entire functions. A nonzero number $T$ is a period of $f\circ g$ if $f(g(z+T))=f(g(z))$ for every $z\in\mathbb C$. Classify the pairs for which such a period exists.

The proposed classification says that every pair belongs, after affine changes of the intermediate variable, to at least one of the following types. The function $g$ is periodic; or there are $T,K\ne0$ with $g(z+T)=g(z)+K$ and $f(w+K)=f(w)$; or $g$ is a polynomial of degree two; or \[ g=P\circ h,\qquad \deg P=2,\qquad h(z+T)=h(z)+K,\qquad (f\circ P)(w+K)=(f\circ P)(w), \] where $h$ is entire and $T,K\ne0$. In considering a type, one may choose a suitable nonzero period, rather than fixing one particular period in advance.

An affine change of the intermediate variable means replacing $(f,g)$ by $(f\circ L^{-1},L\circ g)$ with $L(w)=aw+b$, $a\ne0$, which leaves the composition unchanged. The quadratic alternative is a necessary structural possibility for an already periodic composition, not the assertion that every outer function produces a periodic composition with every quadratic polynomial. Prove that these mechanisms exhaust all possibilities, or construct a pair outside them. Excluding constant functions removes compositions that are trivially periodic regardless of the inner function.

Short English statement

Must every periodic composition of nonconstant entire functions arise from the listed periodic, translation, or quadratic mechanisms?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3700014 (AMR-036-0014), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Eremenko - My favorite unsolved problems, Composite periodic entire functions conjecture. Original source indexed by AMR-036-0014. https://www.math.purdue.edu/~eremenko/uns1.html.
• S3 A. Eremenko, Composite periodic entire functions. Author-maintained primary problem note. https://www.math.purdue.edu/~eremenko/dvi/periodic.pdf.