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Autonomous norm shell separation: distance from S(3) to S(2) in the Hofer metric

Determine whether there exist elements of the shell S(3) = { φ ∈ Ham(M, ω) | ||φ||_Aut = 3 } that are arbitrarily far (in Hofer distance) from the shell S(2) = { φ ∈ Ham(M, ω) | ||φ||_Aut = 2 } within Ham(M, ω), i.e., prove or refute the existence of sequences φ_k ∈ S(3) with d_Hof(φ_k, S(2)) → ∞.

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Background

The autonomous norm ||·||_Aut on Ham(M, ω) is defined as the word metric generated by autonomous Hamiltonian diffeomorphisms, and its geometry is only partially understood: it is known to be bounded in some settings (e.g., ℝ{2n}) and unbounded on surfaces.

Polterovich and Shelukhin proved that elements of S(2) can be arbitrarily far from S(1) in Hofer distance. The paper highlights that extending this to the next shell remains open, specifically whether elements of S(3) can be arbitrarily far from S(2), which connects to understanding how the shells S(n) fill Ham(M, ω).

References

However, already the question of whether elements of S(3) can be arbitrarily far from S(2) is wide open.

— Random Hamiltonians I: Probability measures and random walks on the Hamiltonian diffeomorphism group (2510.03190 - Dawid, 3 Oct 2025) in Random walks on the Hamiltonian diffeomorphism group; remark following Theorem \ref{thm:random-walk-filling}