01 / The result
The same behavior for every scalar set
A dissipative composition operator with bounded distortion is Γ-supercyclic exactly when its associated bilateral weighted shift is Γ-supercyclic, for every subset Γ of the complex numbers.
In the original separable setting
Tf is Γ-supercyclic⇔Bw is Γ-supercyclic
For every Γ ⊆ ℂ, with no regularity or algebraic assumption on Γ.
Here Γ-supercyclicity means that a vector has a dense orbit after each iterate may be multiplied by a scalar from Γ. The note addresses the question in Emma D’Aniello’s contribution, joint work with Martina Maiuriello, on page 1083 of Oberwolfach Report 19/2024.
The hypotheses are those of the report: a separable complex Lp space, 1 ≤ p < ∞; a bijective bimeasurable transformation inducing a bounded invertible composition operator; and a wandering set W of finite positive measure whose iterates partition the space, with bounded distortion. Separability is essential.
02 / The argument
Reduce to one weighted translation
- Use the wandering-set coordinates. Bounded distortion identifies the composition operator with bilateral translation on a weighted sequence space with coefficient space Lp(W).
- Recognize the scalar model. The associated weighted shift is conjugate to the scalar version of that translation.
- Apply the earlier criteria. Γ-supercyclicity passes between the scalar model and its version with any nonzero separable Banach coefficient space.
The paper provides the mathematical argument. Supporting diagnostics check finite identities and representative boundary cases; they are not a formal machine proof of the infinite-dimensional theorem.
03 / Attribution
The earlier work behind the answer
The scalar criterion was established by Arafat Abbar in 2019. The passage to Banach-valued sequences also follows from earlier criteria, including Abbar and Kuznetsova’s 2020 preprint, published in 2021. The note makes their consequence for the Oberwolfach question explicit; it claims no new scalar criterion or amplification mechanism.
The literature audit did not locate an earlier publication explicitly spelling out this particular application to the 2024 question. That does not establish priority for the application.