Universality Under Composition of Holomorphic Self-Maps

Definitions and mathematical statement

Let $\Omega\subseteq{\mathbb{C}}^d$ be a domain, including the unit disk when $d=1$, and let $(\varphi_j)_{j\geq1}$, $(\vartheta_j)_{j\geq1}$ be holomorphic self-map sequences of $\Omega$. Use $d$ for ambient dimension to distinguish it from the sequence index. Put \[ H^\infty(\Omega)=\{g\in H(\Omega):\|g\|_\infty<\infty\},\qquad \psi_j=\varphi_j\circ\vartheta_j. \] The source assumes that each original sequence admits a universal function "relative to $H^\infty$" and asks for natural, interesting sufficient conditions or a composition principle ensuring the same for $(\psi_j)$.

The universality data must include an admissible generating class $\mathcal X$, a target class $\mathcal Y$, and a topology. For example, under compact-open approximation the assertion for $(\psi_j)$ would mean \[ \exists f\in\mathcal X\ \forall g\in\mathcal Y\ \forall K\subseteq\Omega \text{ compact}\ \forall\varepsilon>0\ \exists j\geq1:\quad \sup_{z\in K}|f(\psi_j(z))-g(z)|<\varepsilon. \] This display is a precise schema, not a choice of the missing source data. The two original sequences may have different universal generators; neither a common generator nor a relation between them is supplied. The composition is synchronized at the same index $j$, not the full family of independent pairs $\varphi_j\circ\vartheta_k$. For $C_\varphi f=f\circ\varphi$, the order is $C_{\varphi_j\circ\vartheta_j}=C_{\vartheta_j}C_{\varphi_j}$.

A material boundedness issue remains. If $f\in H^\infty(\Omega)$, then $\|f\circ\psi_j\|_\infty\leq\|f\|_\infty$ for every $j$. Its orbit therefore cannot be dense in all of $H^\infty(\Omega)$, even for compact-open approximation, since constants of larger modulus cannot be approximated at one point. The intended formulation might use a bounded target ball, a different generator space, or another convention. The record does not specify which. Uniform norm density is also distinct from compact-open density. These choices cannot be inferred merely from the phrase relative to $H^\infty$.

The question seeks conditions producing meaningful positive cases, not an explicit assertion that separate universality always passes to synchronized composition. The supplied assessment dated 2026-08-21 reports necessary-and-sufficient results in important eventually-injective planar cases but says they do not settle this broad formulation. No eventual injectivity or one-dimensional restriction has been imposed on the general question.

Short English statement

When do two universal holomorphic self-map sequences remain universal after same-index composition? The target space, generator class, and topology must first be fixed; a bounded generator cannot approximate all bounded holomorphic functions of arbitrary norm. The record asks for useful composition conditions rather than an unconditional closure claim.

Sources

• Local UnsolvedMath registry, permanent ID $30000923$, exact source code catalogue entry, proposed year 2008; no individual proposer is supplied.
• Mini-Workshop: Complex Approximation and Universality (2008), Oberwolfach Reports, DOI 10.4171/owr/2008/06, pp. 297--346. https://doi.org/10.4171/owr/2008/06.
• Supplied reference labelled "universal composition sequences": https://arxiv.org/abs/2111.05165.
• The supplied corrected_verified note says the full universality question was preserved. The open label and assessment checked 2026-08-21 are dated supplied triage requiring independent verification of theorem scope and current status.