Hofer and spectral non-recurrence for zero-entropy sphere diffeomorphisms

Determine whether every Hamiltonian diffeomorphism \(\phi\in Ham(S^{2})\) that is not torsion, has at least three periodic points, and has zero topological entropy admits a constant \(\delta>0\) such that \(\gamma(\phi^{k})\geq\delta\) for every integer \(k\geq1\).

Background

The paper proves uniform lower bounds for the spectral norm, and hence for Hofer’s norm, along all positive iterations of every non-identity Hamiltonian diffeomorphism on a closed surface of genus at least one. On the sphere, however, rational rotations, irrational rotations, and more generally pseudo-rotations exhibit spectral recurrence along subsequences. The remaining case identified by the authors consists of non-torsion Hamiltonian diffeomorphisms with at least three periodic points and zero topological entropy.

The authors explain that their invariant-annulus methods do not directly apply to this spherical case. Resolving the question would therefore extend the paper’s two-dimensional non-recurrence theorem to the remaining zero-entropy, non-pseudo-rotation setting on S2S^{2}.

References

Therefore it would be interesting to investigate the following question:

Suppose $\phi\in Ham(S{2})$ is not torsion, has at least $3$ periodic points and $h_{top}(\phi) = 0$. Does there exist a constant $\delta>0$ so that $\gamma(\phi{k}) \geq \delta$ for all $k\geq 1$ ?

— Invariant sets and spectral rigidity of Hamiltonian diffeomorphisms  (2610.12279 - Alizadeh et al., 8 Oct 2026) in Question in the subsection “Surfaces,” Section 1 (Introduction)