$\Gamma$-Supercyclicity of Composition Operators and Weighted Shifts

Definitions and mathematical statement

For a linear operator $A:E\to E$ on a complex topological vector space and a set $\Gamma\subseteq{\mathbb{C}}$, define the $\Gamma$-orbit of $x\in E$ by \[ \operatorname{Orb}_{\Gamma}(A,x)=\{\gamma A^n x:\gamma\in\Gamma,\ n\in{\mathbb{Z}}_{\geq 0}\}. \] Using the usual convention, $A$ is $\Gamma$-supercyclic if there exists $x\in E\setminus\{0\}$ such that \[ \overline{\operatorname{Orb}_{\Gamma}(A,x)}^{\,E}=E. \] The closure uses the topology of the space on which $A$ acts. This convention makes explicit the orbit and density terminology; the supplied record does not specify these conventions or the underlying function and sequence spaces.

The intended setting is a dissipative composition dynamical system $(X,\mathcal B,\mu,f,T_f)$ of bounded distortion generated by a set $W$, together with its associated weighted shift $B_w$. Here the composition operator acts formally by $T_f\varphi=\varphi\circ f$. The supplied record does not give the precise definition of this class of systems, the function space on which $T_f$ acts, or the formula labelled $(\mathrm{WS})$ defining $B_w$. These are unresolved hypotheses, rather than definitions supplied by this draft.

Intended question. Characterize the subsets $\Gamma\subseteq{\mathbb{C}}$ for which the equivalence \[ T_f\text{ is }\Gamma\text{-supercyclic} \quad\Longleftrightarrow\quad B_w\text{ is }\Gamma\text{-supercyclic} \] holds for every system and associated shift in the intended class. In particular, is this equivalence valid for every $\Gamma\subseteq{\mathbb{C}}$ and every such system? No restrictions on $\Gamma$, such as nonemptiness, boundedness, or closure under multiplication, are supplied. The universal quantification over systems is the natural class-wide reading of the question; the record does not explicitly distinguish it from a characterization for a fixed pair $(T_f,B_w)$.

Source-reported special cases and qualification. The extracted passage attributes the equivalence for $\Gamma={\mathbb{R}}$ to reference [7, Theorem 4.5] and for $\Gamma={\mathbb{C}}$ to [7, Corollary 4.6], under the dissipative bounded-distortion hypotheses generated by $W$. It also attributes to [7, Theorem 4.8] a characterization of ${\mathbb{C}}$-supercyclicity, whose extracted condition reads \[ \text{for every }q\in{\mathbb{N}},\qquad \liminf_{n\to+\infty} \mu\left(f^{q-n}(W)\right)\, \mu\left(f^{q+n}(W)\right)=0. \] This formula records the apparent reading of the supplied extraction only. Its typography, hypotheses, the meaning of negative iterates, and the convention for ${\mathbb{N}}$ require checking against the cited report. It is not asserted here as a verified theorem or used to fill the missing definitions.

Short English statement

Determine which scalar sets $\Gamma\subseteq{\mathbb{C}}$ preserve $\Gamma$-supercyclicity between a dissipative composition operator of bounded distortion and its associated weighted shift. The supplied source reports the equivalence for ${\mathbb{R}}$ and ${\mathbb{C}}$ and asks about arbitrary $\Gamma$. The precise operator spaces, system hypotheses, and shift construction are missing, so this remains a qualified statement-only draft requiring mathematical review.

Sources

• Local UnsolvedMath registry: data/unsolvedmath/problems.json, permanent ID $30005899$, source problem code catalogue entry; supplied title: $\Gamma$-Supercyclicity of Composition Operators and Weighted Shifts. The record belongs to the Oberwolfach Reports Open Problems collection and gives proposed year 2024; no individual proposer is identified.
• Supplied report citation: Mini-Workshop: New Horizons in Linear Dynamics, Universality, and the Invariant Subspace Problem (2024), Oberwolfach Reports, DOI 10.4171/owr/2024/19, pp. 1069--1096. Supplied source URL and literature-source link, labelled "Oberwolfach Report 19/2024": https://doi.org/10.4171/owr/2024/19.
• The extracted report passage cites [7, Theorem 4.5], [7, Corollary 4.6], and [7, Theorem 4.8]. The bibliographic identity of reference [7] and the associated shift formula $(\mathrm{WS})$ are not included in the supplied record.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and states that the equivalence for arbitrary scalar multiplier sets remains open, with existing results covering several special multiplier sets. This is supplied dated curation derived from the Oberwolfach material, not an independently verified assessment of the literature or current status.
• The registry labels the statement corrected_verified and says it was reconstructed and checked against the official Oberwolfach report. That is a supplied verification claim; this draft does not independently verify it or claim to have consulted the report.