On the generic behavior of the spectral norm

Our main result is that for any closed symplectic manifold, the spectral norm of the iterates of a Hamiltonian diffeomorphism is locally uniformly bounded away from zero C ∞ -generically.k→∞ γ (ϕ k ) = 0.Among these are, for instance, Lagrangian Poincaré recurrence Ginzburg and Gürel 2018; Joksimović and Seyfaddini 2023, and the variant of the strong closing lemma from Cineli and Seyfaddini 2022.Simultaneously, fairly explicit criteria for this sequence to be bounded away from zero have been established, based on the crossing energy theorem from Ginzburg and Gürel 2014; 2018; see, e.g., Cineli et al. 2022 and Theorem 3.1.Let us now provide some more context for the question.First, note that the condition (1-1) can be interpreted as that ϕ is γ -rigid or, in other words, a γ -approximate identity.This notion is a particular case of a much more general concept.Namely, consider a class of diffeomorphisms ϕ or even homeomorphisms of a manifold M, which we assume here to be closed.For instance, this can be the class of all diffeomorphisms or