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