Does there exist a transcendental entire function f:C→C with a wandering Fatou component U such that the forward orbit of U is bounded, i.e. f^m(U)≠ f^n(U) for all distinct m,n≥ 0 and bigcup_n≥ 0f^n(U) is a bounded subset of C?

Absence of bounded-orbit wandering domains

The problem asks whether transcendental entire functions can have wandering domains whose entire forward orbit remains bounded. A September 2026 preprint by Nikolai Prochorov, Lasse Rempe, and James Waterman claims a negative answer and a broader exclusion theorem.

Known results

On July 31, 2023, Leticia Pardo-Simon and David J. Sixsmith constructed wandering domains spending asymptotically any prescribed proportion of time in a bounded domain, including proportions arbitrarily close to $1$ .
Their paper explicitly left entirely bounded forward orbits as a major open question.

September 2026 claimed resolution

The newly posted paper claims to answer the long-standing question negatively and to prove a broader exclusion result for certain locally defined analytic functions. The claim is newly posted and unrefereed, with no independent verification or reported objection in the retrieved sources.

Current status (as of September 2026): A preprint claims that bounded-orbit wandering domains do not exist, but the result remains unverified.

Absence of bounded-orbit wandering domains

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