Baranov–Lishanskii–Papathanasiou infinite-lattice hypercyclicity problem

The problem asks for a necessary-and-sufficient hypercyclicity criterion for backward shifts on the infinite lattice, especially $\mathbb{N}\times\mathbb{N}$. The exact criterion is explicitly described as open in the retrieved literature.

Known results

• If the bounded weight satisfies $\limsup_{i+j\to\infty}|\mu_{v_{i,j}}|^{1/(i+j)}<2$, the shift is mixing and therefore hypercyclic.
• If $|\mu_{v_{i,j}}|\ge c\,2^{i+j}$ for some $c>0$, the shift is not hypercyclic.
• Weights depending on only one coordinate yield mixing under boundedness.
• The cases $\mathbb{N}\times\mathbb{Z}$ and $\mathbb{Z}\times\mathbb{Z}$ were left for future work.

September 28, 2026 claimed radial-$\ell^2$ progress

Chen, Wang, and Zhou report characterizing transitivity and hypercyclicity criteria in a radial weighted $\ell^2$ setting. This is a substantial restricted advance, not a solution of the full infinite-lattice problem, and the characterization is unverified here.

Current status (as of September 2026): the full necessary-and-sufficient criterion remains open; a radial weighted $\ell^2$ advance has been claimed but is unverified.