On the generic behavior of the spectral norm

The main result of the paper is that for any closed symplectic manifold the spectral norm of the iterates of a Hamiltonian diffeomorphism is locally uniformly bounded away from zero $C^\infty$-generically.


Introduction
We show that for a Hamiltonian diffeomorphism ϕ of a closed symplectic manifold M, the spectral norm over ‫ޑ‬ of the iterates ϕ k is locally uniformly bounded away from zero C ∞ -generically in ϕ, without any additional assumptions on M.
The question of the behavior of the sequence γ (ϕ k ) of spectral norms goes back to the work of Polterovich [2002].Recently, there has been renewed interest in the problem whether and when this sequence is bounded away from zero.There are several reasons for this question, amounting roughly speaking to the fact that one can obtain pretty strong results on the symplectic dynamics of ϕ when the sequence is not bounded away from zero: (1-1) γ (ϕ) := lim inf of Hamiltonian diffeomorphisms when M is symplectic, etc. Assume that this class is equipped with some norm ∥ • ∥, e.g., the C 0 -or C 1 -norm or the γ -or Hofer-norm in the Hamiltonian case.A map ϕ is said to be ∥ • ∥-rigid if ϕ k i → id with respect to ∥ • ∥, i.e., ∥ϕ k i ∥ → 0, for some sequence k i → ∞.The term "rigid" is somewhat overused in dynamics and also frequently confused with structural stability, and in [Ginzburg and Gürel 2019a] we proposed to call such a map ϕ a ∥ • ∥-approximate identity, or a ∥ • ∥-a.i. for the sake of brevity.We refer the reader to, e.g., [Bramham 2015;Ginzburg and Gürel 2019a;Cineli and Seyfaddini 2022] for a further discussion of approximate identities, aka rigid maps, in different contexts.Here we only mention that C r -a.i. is obviously C s -a.i. for any s ≤ r and, when M is aspherical or M = ‫ސރ‬ n , a C 0 -a.i. is also a γ -a.i.; see [Buhovsky et al. 2021;Shelukhin 2022b].
Zeroing in on γ -a.i.'s we note that there are rather few examples of such maps.The most dynamically interesting examples are Hamiltonian pseudorotations.This class of maps has been extensively studied in a variety of settings by dynamical systems methods and more recently from the perspective of symplectic topology and Floer theory; see, for example, [Anosov and Katok 1970;Avila et al. 2020;Bramham 2015;Fayad and Katok 2004;Ginzburg and Gürel 2018;Joksimović and Seyfaddini 2023;Le Roux and Seyfaddini 2022].
While the official definitions of Hamiltonian pseudorotations vary, these are, roughly speaking, Hamiltonian diffeomorphisms with a finite and minimal possible number of periodic points (in the sense of Arnold's conjecture); see [Ginzburg and Gürel 2018;Shelukhin 2020;2021].For instance, when M = ‫ސރ‬ n this number is n + 1.Most likely, for many symplectic manifolds this condition can be relaxed.Namely, in all examples of Hamiltonian diffeomorphisms ϕ with finitely many periodic points, all periodic points are fixed points and their number is minimal possible.Thus ϕ is a pseudorotation.For a certain class of manifolds M, including ‫ސރ‬ n , this has been established rigorously under a minor nondegeneracy assumption; see [Shelukhin 2022a] and also [Çineli et al. 2022].In all examples to date of Hamiltonian diffeomorphisms ϕ with finitely many periodic points, ϕ is nondegenerate.
In general, the relation between pseudorotations and γ -a.i.'s is not obvious.All known Hamiltonian pseudorotations are γ -a.i.'s and for M = ‫ސރ‬ n this is proved in [Ginzburg and Gürel 2018] by using the results from [Ginzburg and Gürel 2009a].The converse is not true: for instance any element ϕ of a Hamiltonian torus action is a γ -a.i., although ϕ need not have isolated fixed points.(It is conceivable that for a strongly nondegenerate γ -a.i., the periodic points are necessarily the fixed points: in the obvious notation, Per(ϕ) = Fix(ϕ).However, a map ϕ with the latter property need not be a γ -a.i.For instance, γ (ϕ k ) can grow linearly for such a map; see Remark 4.10.) Most closed symplectic manifolds (M, ω) admit no pseudorotations, that is, every Hamiltonian diffeomorphism of M has infinitely many periodic points.This statement (for a specific manifold M) is usually referred to as the Conley conjecture.To date, the Conley conjecture has been shown to hold unless there exists A ∈ π 2 (M) such that ⟨[ω], A⟩ > 0 and ⟨c 1 (T M), A⟩ > 0; see [Çineli 2018;Ginzburg and Gürel 2015;2019b].In particular, the Conley conjecture holds when M is symplectically aspherical or negative monotone.For a broad class of closed symplectic manifolds, ϕ has infinitely many periodic points C ∞ -generically; see [Ginzburg and Gürel 2009b;Sugimoto 2021] and Section 4B.
Although the classes of Hamiltonian pseudorotations and γ -a.i.'s are certainly different, there is a clear parallel between these two classes and their existence conditions on M.
Conjecture.Let M be closed symplectic manifold.
(i) The manifold M admits no γ -a.i.'s unless there exists A ∈ π 2 (M) such that and only if all iterates ϕ k are Morse-Bott nondegenerate and dim H * (Fix(ϕ k ); ‫)ކ‬ = n + 1 for all k ∈ ‫ގ‬ and any ground field ‫.ކ‬This conjecture is supported by some evidence.For instance, M does not admit periodic Hamiltonian diffeomorphisms ϕ (i.e., ϕ N = id for some N > 1) when M satisfies the conditions of (i); see [Atallah and Shelukhin 2023;Polterovich 2002].In addition, Fix(ϕ k ) is Morse-Bott nondegenerate whenever ϕ is periodic.This is a consequence of the equivariant Darboux lemma; see, e.g., [Guillemin and Sternberg 1984, Theorem 22.2].Aspherical or negative monotone symplectic manifolds do not admit C 1 -a.i.'s; see [Polterovich 2002] and[Sugimoto 2023].Further results along these lines can be found in [Atallah and Shelukhin 2023].In [Cineli et al. 2022] both assertions are proved in dimension two for strongly nondegenerate Hamiltonian diffeomorphisms; see Corollary 3.4.In the setting of (i) the sequence of the spectral norms γ (ϕ p ) over ‫/ޚ‬ p‫,ޚ‬ where p ranges through all primes, is separated away from zero [Shelukhin 2023].As we have already mentioned the "if" part of (ii) is established in [Ginzburg and Gürel 2018] without any nondegeneracy assumption when |Per(ϕ)| = n + 1.With this in mind, part (ii) of the conjecture asserts, in particular, that every pseudorotation of ‫ސރ‬ n is strongly nondegenerate.
Remark 1.1.While part (ii) of the conjecture might extend to some other ambient symplectic manifolds M, some restriction on M is necessary.For instance, the torus ‫ޔ‬ 2n equipped with an irrational symplectic structure admits a Hamiltonian diffeomorphism ϕ such that the conditions of (ii) are satisfied but γ (ϕ k ) → ∞; see [Zehnder 1987] and also [Cineli 2023] for further constructions of this type with complicated dynamics.
In a similar vein, the main result of this paper can be thought of as the γ -a.i.analogue of the aforementioned theorem on the C ∞ -generic Conley conjecture, although at this moment the proof of the latter requires some additional conditions on the underlying manifold; see Section 4B.
Remark 1.2.Overall, rather little is known about the behavior of the γ -norm under iterations.For a certain class of manifolds, including ‫ސރ‬ n , the spectral norm is a priori bounded from above [Entov and Polterovich 2003;Kislev and Shelukhin 2021].However, such manifolds appear to be rare; see Remark 4.10.Also, the sequence γ (ϕ k ) is bounded from above when supp ϕ is displaceable in M, but not much beyond these facts and the results of this paper is known about the behavior of this sequence.For instance, when M is a surface of positive genus, it is not known if γ (ϕ k ) necessarily grows linearly or can be bounded from above when ϕ is strongly nondegenerate or, at the opposite extreme, autonomous and supp ϕ is not displaceable.
Remark 1.3.It is worth keeping in mind that in contrast with some other dynamics concepts, in most if not all settings a.i.'s are sensitive to reparametrization.To be more specific, let an a.i.ϕ be the time-one map of the flow of a vector field X and let ψ be the time-one map of f X for some function f > 0.Then, in general, ψ need not be an a.i.For instance, assume that X is a solid rotation vector field on M = S 2 and f is not constant.Then one can show that ψ is not a C 0 -a.i., and hence not a C r -a.i. for any r ≥ 0. Apparently, the same is true for the γ -norm, but this fact is yet to be proved rigorously; cf.item (ii) of the Conjecture.

Preliminaries and notation
We very briefly set our notation and conventions which are quite standard and spelled out in more detail in, for example, [Cineli and Seyfaddini 2022].The reader may find it convenient to jump to Section 3 and consult this section only as needed.
Throughout the paper, all manifolds, functions and maps are assumed to be C ∞ -smooth unless specifically stated otherwise. Let We say that ϕ is nondegenerate if all fixed points of ϕ are nondegenerate, and strongly nondegenerate if all periodic points of ϕ are nondegenerate.We will denote by Ham(M, ω) the group of Hamiltonian diffeomorphisms of (M, ω).
Recall that the spectral norm, also known as the γ -norm, of ϕ is defined as where is the spectral invariant associated with the fundamental class [M] ∈ H 2n (M).(Here we can take as H inv any Hamiltonian generating this flow with the same time/space average as H .) The infimum is taken over all 1-periodic in time Hamiltonians H generating ϕ, i.e., ϕ = ϕ H .The Hofer norm of ϕ is defined as where the infimum is again taken over all 1-periodic in time Hamiltonians H generating ϕ.Then γ (ϕ) ≤ ∥ϕ∥ H .
Here we are interested in the behavior of γ (ϕ k ), k ∈ ‫,ގ‬ and in particular in the question when this sequence is bounded away from zero.As in the introduction, set These definitions implicitly depend on the construction of the filtered Floer homology HF a (H ) for the action window (−∞, a).In this paper we do not in general assume that the class [ω] is rational or that ϕ is nondegenerate.Hence, we feel, a word is due on the specifics of the definitions.
Assume first that H is nondegenerate.Then we utilize Pardon's VFC package [2016], to define the filtered Floer homology HF a (H ) over ‫ޑ‬ and spectral invariants; see, for example, [Cineli and Seyfaddini 2022;Usher 2008].To be more specific, HF a (H ) is the homology of the subcomplex CF a (H ) of the Floer complex CF(H ) generated by Floer chains with action below a. Virtually any choice of the Novikov field can be used here.We take the standard Novikov field Alternatively, we could have used the universal Novikov field.Then, for any α ∈ H * (M)⊗ , the spectral invariant c α (H ) is defined as is the natural inclusion-induced map and the identification on the right is the PSSisomorphism.We note that all spectral invariants necessarily belong to the action spectrum S(H ) of H when H is nondegenerate [Usher 2008].When H is not necessarily nondegenerate, we set where H is nondegenerate and the convergence H → H is taken to be C 0 .The second and third equalities and the existence of the limit follow from that c α is monotone and c α ( and then used (2-1) and (2-2) to get the same result.Defined in this way, spectral invariants c α can be easily shown to have all the standard properties: c α (H ) is monotone and Lipschitz continuous in H with Lipschitz constant one; c α (H + k) = c α (H ) + k for any constant function k; etc. (We refer the reader to, e.g., [Cineli and Seyfaddini 2022] for more details.)The exception is that c α (H ) has been proven to be spectral, i.e., an element of S(H ), only when [ω] is rational or H is nondegenerate; see [Entov and Polterovich 2003;Oh 2005a;Usher 2008].

Main results
The key to bounding γ from below is the following fact connecting the behavior of γ (ϕ k ) with the dynamics of ϕ and, in particular, its hyperbolic points.
Without the "Also" part, this theorem was originally proved in [Cineli et al. 2022].We give a complete proof in Section 4. Let us emphasize that in Theorem 3.1 we impose no nondegeneracy requirements on ϕ, and also that the property of ϕ to have more than dim H * (M) hyperbolic periodic points, or more than any fixed number of hyperbolic periodic points, is open in C 1 -topology.
The requirement of the theorem that the number of hyperbolic points is greater than dim H * (M) can be further relaxed by looking only at the odd/even-degree homology of M, depending on whether n = dim M/2 is odd or even; see Remark 4.2.
The main result of the paper is the following theorem relying on Theorem 3.1 and the strong closing lemma from [Cineli and Seyfaddini 2022].
Theorem 3.3.Let M be a closed symplectic manifold.The function γ is locally uniformly bounded away from zero on a C ∞ -open and dense set of Hamiltonian diffeomorphisms ϕ : M → M, i.e., for every ϕ in this set there exists δ > 0, possibly depending on ϕ but not on ψ, such that We note that we do not assert here that in general the set of Hamiltonian diffeomorphisms ϕ with γ (ϕ) > 0 is itself C ∞ -open, but rather that this set contains a set which is C ∞ -open and dense.Nor do we impose any restrictions on the (symplectic) topology of M or require any of the iterates ϕ k to be nondegenerate.The proof of Theorem 3.3 given in Section 4A is based on a variant of the Birkhoff-Lewis-Moser theorem.The key new ingredient of the proof is the strong closing lemma from [Cineli and Seyfaddini 2022].It is also worth pointing out that if we replaced the statement that the set is C ∞ -dense by that it is C 1 -dense, the theorem would turn into an easy consequence of already known facts; see Remark 4.5.
In several situations, Theorem 3.3 can be made slightly more precise.For instance, we have the following result, also originally proved in [Cineli et al. 2022] without the "Also" part.
Corollary 3.4.Assume that M is a surface and ϕ is strongly nondegenerate.Then γ (ϕ) > 0 when M has positive genus.When M is the two-sphere, γ (ϕ) = 0 if and only if ϕ is a pseudorotation.Also, γ is locally uniformly bounded from 0 on the set of all strongly nondegenerate Hamiltonian diffeomorphisms ϕ when M has positive genus and on the set of such ϕ with at least three fixed points when M = S 2 .
Proof.When M has positive genus, a Conley conjecture type argument guarantees that ϕ has infinitely many hyperbolic periodic points; see [Franks and Handel 2003;Ginzburg and Gürel 2015;Salamon and Zehnder 1992] or [Le Calvez and Sambarino 2022].Thus, in this case, the statement follows directly from Theorem 3.1.
Concentrating on M = S 2 , first note that for all, not necessarily nondegenerate, pseudorotations of ‫ސރ‬ n , the sequence γ (ϕ k ) contains a subsequence converging to zero, and hence γ (ϕ) = 0; see [Ginzburg and Gürel 2018].In the opposite direction, when M = S 2 , the existence of one positive hyperbolic periodic point is enough to ensure that γ (ϕ) > 0 and, moreover, γ is locally uniformly bounded away from zero; see Remark 4.2.Hence, more generally, without any nondegeneracy assumption, if γ (ϕ) = 0, then all periodic points of ϕ are elliptic.For strongly nondegenerate Hamiltonian diffeomorphisms ϕ, this forces ϕ to be a pseudorotation.□ Since the Hofer norm is bounded from below by the spectral norm, we have the following.
Corollary 3.5.In all results from this section, we can replace the spectral norm by the Hofer norm.
We refer the reader to Section 4 for further refinements of Theorems 3.1 and 3.3.
Remark 3.6.Throughout the paper all homology groups are taken over ‫.ޑ‬This choice of the background coefficient field is necessitated by the use of Floer theory for an arbitrary closed symplectic manifold M. When M is weakly monotone, ‫ޑ‬ can be replaced by any coefficient field.

Proofs and refinements
In Section 4A, we prove Theorems 3.1 and 3.3.In Section 4B, we refine the latter result under certain additional assumptions on M and further comment on the class of γ -a.i.'s.
Proof of Theorem 3.1.By the conditions of the theorem, for some N ∈ ‫,ގ‬ the Hamiltonian diffeomorphism ϕ has more than dim H * (M) hyperbolic N -periodic points.We denote the set of these points by K. Thus |K| > dim H * (M) and clearly K is a locally maximal hyperbolic set.Furthermore, every point in K is also ℓNperiodic for all ℓ ∈ ‫.ގ‬For ϵ > 0, denote by b ϵ (ϕ) the number of bars in the barcode of ϕ of length greater than ϵ including infinite bars; see, for example, [Cineli et al. 2021].Then, we claim that, for a sufficiently small ϵ > 0 and any ℓ ∈ ‫,ގ‬ In particular, ϕ ℓN has at least one finite bar of length greater than ϵ > 0. This inequality is essentially a consequence of [Cineli et al. 2021, Proposition 3.8 and 6.2].To prove (4-1), first note that the number of infinite bars in the barcode of any Hamiltonian diffeomorphism is equal to dim H * (M).Secondly, it follows from [Cineli et al. 2021, Proposition 6.2] and the proof of [Cineli et al. 2021, Proposition 3.8] that every periodic point in K appears as an "end point" of a bar of length greater than ϵ > 0. Combining these two facts, we conclude that ϕ ℓN has at least ⌈(|K| − dim H * (M))/2⌉ finite bars of length greater than ϵ > 0, and (4-1) follows.
Furthermore, since the crossing energy lower bound in [Cineli et al. 2021, Theorem 6.1] is stable under C ∞ -small perturbations of the Hamiltonian, for every positive η < ϵ the same is true for any Hamiltonian diffeomorphism which is and hence the barcode of ℓ has a finite bar of length greater than η.
Also, recall that as is proved in [Kislev and Shelukhin 2021, Theorem A], for any ϕ, where the left-hand side is the boundary depth, i.e., the longest finite bar in the barcode of ϕ.Thus, for a sufficiently small η > 0, Next, set δ = η/2 and arguing by contradiction, assume that there exist ψ sufficiently C ∞ -close to ϕ and a sequence k i → ∞ such that γ (ψ k i ) < δ.
Since the sequence k i is infinite and there are only finitely many residues modulo N , there exists a pair k i < k j such that close to ϕ, and hence (4-2) holds.Then by the triangle inequality for γ , we have This contradiction concludes the proof of the theorem.□ Remark 4.1.It might be worth a second to examine how the invariants of ϕ involved in the proof depend on the isotopy ϕ t H in Ham(M, ω) generated by H and its lift to the universal covering of the group.Namely, γ (ϕ) is a priori independent of the isotopy only on the universal covering.On Ham(M, ω) it is defined by passing to the infimum over often infinitely many elements.However, the boundary depth β max is well defined on Ham(M, ω).In the proof we bound β max (ϕ) from below (see, e.g., [Usher 2011]) and that bounds γ (ϕ) from below regardless of the lift [Kislev and Shelukhin 2021].
Remark 4.2.When n = dim M/2 is odd, it is sufficient to require in Theorem 3.1 that the number of hyperbolic periodic points is greater than b = dim H odd (M).For instance, this is the case when M is a surface.Indeed, in the proof of the theorem, by taking N even and sufficiently large, we can guarantee that the number of positive hyperbolic N -periodic points is greater than b.Such points necessarily have even Conley-Zehnder index, and hence contribute to the odd-degree homology of M under the isomorphism HF * (ϕ N ) ∼ = H * +n (M).Likewise, when n is even, it suffices to require the number of hyperbolic periodic points to be greater than dim H even (M).
Proof of Theorem 3.3.To prove the theorem, it suffices to show that every C ∞ -open set U in the group of Hamiltonian diffeomorphisms contains an open subset W such that γ (ϕ) > δ for all ϕ ∈ W and some δ = δ(W) > 0 independent of ϕ.Indeed, then fixing W for every U we can take the union of sets W for all U as the desired open and dense subset.
Let q = dim H * (M).For any U, there are two alternatives: (i) There exists ϕ ∈ U with more than q periodic points.
(ii) Every ϕ ∈ U has at most q periodic points.
Let us first focus on case (i).Pick ϕ ∈ U with more than q periodic points and fix q +1 of them.Denote these points by x 0 , . . ., x q , and note that arbitrarily C ∞ -close to ϕ there exists a Hamiltonian diffeomorphism ϕ ′ ∈ U such that x 0 , . . ., x q are nondegenerate periodic points of ϕ ′ .This is essentially a linear algebra fact and to construct ϕ ′ , it suffices to perturb ϕ near these points, changing Dϕ slightly.(Note that ϕ ′ may have many other periodic points, nondegenerate or not.We can ensure in addition that ϕ ′ is strongly nondegenerate, but we do not need this fact.)We replace ϕ by ϕ ′ , keeping the notation ϕ.
If all periodic points x 0 , . . ., x q are hyperbolic, we can take as W any C ∞ -small neighborhood of ϕ by Theorem 3.1.
If one of the points x 0 , . . ., x q is not hyperbolic, we argue by perturbing ϕ again.Namely, recall that by the Birkhoff-Lewis-Moser theorem (see [Moser 1977]), whenever ϕ has a nonhyperbolic, nondegenerate periodic point x, there exists an arbitrarily C ∞ -small perturbation ϕ ′ ∈ U of ϕ with infinitely many periodic points near x.Moreover, ϕ ′ can be chosen so that infinitely many of these periodic points are hyperbolic; see [Arnaud 1992, Proposition 8.2].(This follows from the proof of the Birkhoff-Lewis-Moser theorem.)Thus, again by Theorem 3.1, we can take a sufficiently C ∞ -small neighborhood of ϕ ′ as W.
To deal with case (ii), we need the following quantitative variant of the strong closing lemma: Lemma 4.3 [Cineli and Seyfaddini 2022].Let ψ be a Hamiltonian diffeomorphism of a closed symplectic manifold M. Assume that there is a closed ball V ⊂ M containing no periodic points of ψ, that is, V ∩ Per(ψ) = ∅.Let G ≥ 0 be a Hamiltonian supported in V and such that c(G) > γ (ψ).
Then the composition ψϕ G has a periodic orbit passing through V .
Pick a nondegenerate Hamiltonian diffeomorphism ϕ ∈ U, where U is as in case (ii).Such a map exists since U is C ∞ -open and the set of nondegenerate Hamiltonian diffeomorphisms is C ∞ -dense (and open).We will show that there exists δ > 0 such that γ (ψ) > δ for all ψ ∈ U which are C ∞ -close to ϕ.Hence, in this case, we can take a small C ∞ -neighborhood of ϕ as W.
Lemma 4.4.Let (M, ω) be a closed symplectic manifold.Suppose that there exists a C ∞ -open U ⊂ Ham(M, ω) such that all ϕ ∈ U have at most q = dim H * (M) periodic points.Then the function γ : U → [0, ∞) is locally uniformly bounded away from zero at every nondegenerate ϕ ∈ U.
Note that the proof of Theorem 3.3 will be completed once we prove Lemma 4.4.To prove the lemma, arguing by contradiction, fix a nondegenerate ϕ ∈ U and assume that there exists a sequence Here and below convergence of maps is always understood in the C ∞ -sense.
We claim that when i is large enough, all periodic points of ψ i are close to periodic points of ϕ, and hence there exists a closed ball V ⊂ M containing no periodic points of any of these maps.Indeed, since ϕ is nondegenerate and by the Arnold conjecture (see [Fukaya and Ono 1999;Liu and Tian 1998]  Furthermore, when i is large enough, ψ i ∈ U is also nondegenerate since ψ i → ϕ.Therefore, again by the Arnold conjecture, It follows that Per(ψ i ) converges to Per(ϕ).
Next, take G ≥ 0 as in Lemma 4.3, which is supported in V and small enough so that ϕϕ G ∈ U. Hence, ψ i ϕ G ∈ U when i is large; for ψ i → ϕ and thus ψ i ϕ G → ϕϕ G .On the other hand, due to the assumption that γ (ψ i ) → 0, we have c(G) > γ (ψ i ), when again i is sufficiently large.By the strong closing lemma, the composition ψ i ϕ G has a periodic orbit passing through V .On the other hand, the fixed points of ψ i (or equivalently the periodic points) are among the fixed points of ψ i ϕ because supp G ⊂ V .It follows that |Per(ψ i ϕ G )| ≥ q + 1, when i is large enough, which is impossible since ψ i ϕ G ∈ U.This contradiction completes the proof of Lemma 4.4 and hence of Theorem 3.3.□ Remark 4.5.If in Theorem 3.3 we were to find a C 1 -dense (and open) set of Hamiltonian diffeomorphisms rather than C ∞ -dense, the argument would be considerably simpler.Namely, in this case it would be enough to first construct a map ϕ with just one hyperbolic periodic point.Once this is done, we could apply the results from [Hayashi 1997;Xia 1996] to create nontrivial transverse homoclinic intersections, and hence a horseshoe (see [Katok and Hasselblatt 1995]) by a C 1small perturbation.As a consequence, the perturbed map ψ would have infinitely many hyperbolic periodic points.For any m ∈ ‫,ގ‬ having at least m such points is a C 1 -open property.Now we can take any m > dim H * (M).
4B. Sugimoto manifolds and further remarks.As is shown in [Sugimoto 2021], a strongly nondegenerate Hamiltonian diffeomorphism ϕ of a closed symplectic manifold M 2n has either a nonhyperbolic periodic point or infinitely many hyperbolic periodic points when M meets one of the following requirements: (i) n is odd.
(iii) the minimal Chern number of M is greater than 1.
Below we refer to a closed symplectic manifold meeting at least one of these requirements as a Sugimoto manifold.For this class of manifolds Theorem 3.3 has a more direct proof and can be slightly refined.We do this in two steps.
Denote by V m , m ∈ ‫,ގ‬ the set of Hamiltonian diffeomorphisms with at least m hyperbolic points.Note that we do not require the elements of V m to be strongly nondegenerate.The key difference with [Sugimoto 2021] is that the periodic points of ϕ here are specified to be hyperbolic.The residual set in this corollary is, of course, We note that this set is not C 1 -and even C ∞ -open.However, one can require in addition ϕ to be strongly nondegenerate.Indeed, the set of such maps is residual and its intersection with V is still a residual set.
Closer to the immediate subject of the paper we have the following refinement of Theorem 3.Here we can take any V m with m > dim H * (M) as a C 1 -open and C ∞ -dense set, where γ is locally uniformly bounded away from zero.Note also that in this corollary we can again replace the spectral norm by the Hofer norm.
Remark 4.9.In contrast with Theorem 3.3, C ∞ -generic existence of infinitely many periodic points is not known to hold without some additional assumptions on M. The class of Sugimoto manifolds is the broadest to date for which such existence has been proved [Sugimoto 2021].(See also [Ginzburg and Gürel 2009b] for the original result and a different approach.)Remark 4.10.Continuing the discussion from the introduction and Remark 1.2, we give some "textbook" examples where γ (ϕ k ) grows linearly, and hence γ (ϕ) = ∞, and at the same time all periodic points of ϕ are fixed points: Per(ϕ) = Fix(ϕ).Namely, let H : M → ‫ޒ‬ be a nonconstant autonomous Hamiltonian such that H has only finitely many critical values and all nonconstant periodic orbits of the flow of H are noncontractible.Set ϕ = ϕ H .Then, as is easy to see, γ (ϕ k ) grows linearly and the only periodic points of ϕ are the critical points of H .For instance, we can take H = sin 2πθ, where θ is the first angular coordinate θ on ‫ޔ‬ 2 = ‫ޒ‬ 2 ‫ޚ/‬ 2 .Alternatively, let ‫ޔ(‬ 4 , ω) be a Zehnder's torus, that is, a torus equipped with a sufficiently irrational translation invariant symplectic structure ω (see [Zehnder 1987]), and again let θ : ‫ޔ‬ 4 → ‫ޚ/ޒ‬ be a fixed angular coordinate.Then the flow of H given by the same formula has no periodic orbits at all, contractible or not, other than the critical points of H : the 3-dimensional tori θ = 1 2 and θ = 3 2 .In both cases, γ (ϕ k ) = 2k.More surprisingly, there exists a Hamiltonian diffeomorphism ϕ : S 2 × S 2 → S 2 × S 2 such that γ (ϕ k ) grows linearly; see [Shelukhin 2022a, Remark 8] and [Polterovich and Rosen 2014, Theorem 6.2.6], although the argument is quite indirect.
In all these examples, dim H * (Fix(ϕ)) = dim H * (M) over any field, in addition to the condition that Per(ϕ) = Fix(ϕ).Loosely following [Atallah and Shelukhin 2023], we call such a map ϕ a generalized pseudorotation.Generalized pseudorotations from the above examples have simple dynamics.However, this is not necessarily so in general.For instance, in dimension six and higher Morse-Bott nondegenerate, generalized pseudorotations ϕ with positive topological entropy have been recently constructed in [Cineli 2023].Such a generalized pseudorotation can be neither a C 0 -a.i.since h top (ϕ) > 0 (see [Avila et al. 2020]) nor a γ -a.i.In fact, γ (ϕ k ) also grows linearly since M is aspherical and Per(ϕ) = Fix(ϕ) has finitely many connected components.
Proposition 4.6.Let M be a Sugimoto manifold.Then for any m ∈ ‫ގ‬ the set V m is C 1 -open and C ∞ -dense in the space of all Hamiltonian diffeomorphisms.Proof.The statement that V m is C 1 -open is obvious.(It is essential here that m is finite.)To show that it is C ∞ -dense we argue as in [Sugimoto 2021] and the proof of Theorem 3.3.Let ϕ be a Hamiltonian diffeomorphism.To prove the proposition, we need to find ψ ∈ V m arbitrarily C ∞ -close to ϕ.Since the set of strongly nondegenerate Hamiltonian diffeomorphisms is C ∞ -dense, we can assume that ϕ is in this class.As shown in [Sugimoto 2021], ϕ has infinitely many hyperbolic periodic points or a (nondegenerate) nonhyperbolic point.In the former case, ϕ ∈ V m for all m ∈ ‫.ގ‬In the latter case, by [Arnaud 1992, Proposition 8.2], for any m ∈ ‫ގ‬ there exists ψ ∈ V m arbitrarily close to ϕ. □As an immediate consequence, we obtain a slightly more precise variant of the main result from [Sugimoto 2021]:Corollary 4.7.Assume that M is a Sugimoto manifold.Then C ∞ -generically a Hamiltonian diffeomorphism ϕ of M has infinitely many hyperbolic periodic points.
3 and Corollary 3.4: Corollary 4.8.Assume that M is a Sugimoto manifold.Then γ is locally uniformly bounded away from zero on a C 1 -open and C ∞ -dense set of Hamiltonian diffeomorphisms of M.