On the Hofer-Zehnder conjecture for semipositive symplectic manifolds

The Hofer-Zehnder conjecture, which concerns the existence of infinitely many periodic points of Hamiltonian systems, originated in the study of celestial mechanics. In one of Poincaré’s remarkable contributions to the three-body problem, he proved the existence of infinitely many periodic solutions provided the planetary masses are sufficiently small. In attempting to generalize such a result for larger planetary masses, he was led to conjecture the existence of at least two fixed points of any time-one map of an area-preserving isotopy of the annulus satisfying a twist condition on the boundary [Poi12]. He proved some particular cases of the conjecture, while Birkhoff established it in general; this is the content of the celebrated Poincaré-Birkhoff theorem [Bir13, Bir26]. It was later shown [BN77] that a similar proof yields the existence of infinitely many periodic points¹¹1For $\phi:M\rightarrow M$, a point $x\in M$ is periodic if there is an integer $k\geq 1$ such that $\phi^{k}(x)=x$.. In a similar vein, Franks proved that, without the twisting condition, the existence of an interior fixed point is sufficient to guarantee the existence of infinitely many periodic points [Fra92, Fra96]. Furthermore, in what is one of the first results in the direction of the Hofer-Zehnder conjecture, and probably central to its motivation, Franks showed that any time-one map of an area-preserving isotopy of the sphere with at least three fixed points must have infinitely many periodic points. Note that having at least three fixed points is essential since any rotation of the sphere by an irrational fraction of $\pi$ has only two fixed points, which are also the only periodic points, namely, the North and South poles.

In the non-degenerate setting, a natural interpretation of Conjecture 1.1 leads to the statement that any Hamiltonian diffeomorphism $\phi$ of a closed symplectic manifold $(M,\omega)$ possessing more contractible fixed points $\mathrm{Fix}(\phi)$ than the total Betti number of $M$ must have infinitely many periodic points. Shelukhin suggested the following interpretation of Conjecture 1.1.

Here, $\operatorname{\mathrm{HF}}^{\operatorname{\mathrm{loc}}}(\phi,x)$ is a local version of Floer homology for a contractible fixed point $x$ of $\phi$. When $x$ is a non-degenerate fixed point, we have $\operatorname{\mathrm{HF}}^{\operatorname{\mathrm{loc}}}(\phi,x)\cong{\mathbb{K}}$, recovering the non-degenerate interpretation of Conjecture 1.1. In essence, one can think of $N(\phi,{\mathbb{K}})$ as a weighted sum of the number of contractible fixed points of $\phi$. In a striking result [She22, Theorem A], Shelukhin proved Conjecture 1.2 for spherically monotone symplectic manifolds with semisimple even quantum homology; a class of manifolds that includes complex projective spaces, complex Grassmannians, and their products. Furthermore, if ${\mathbb{K}}$ has characteristic $0$, then there is a simple $p$-periodic orbit for each sufficiently large prime $p$. This article aims to generalize Shelukhin’s theorem to the semipositive setting. We obtain the following result:

The Hofer-Zehnder conjecture is related to a conjecture by Conley [CZ83, CZ84], which postulates that for a broad class of symplectic manifolds, any Hamiltonian diffeomorphism must have infinitely many periodic points. Therefore, the Hofer-Zehnder conjecture is automatically satisfied when the Conley conjecture holds. Following plenty of previous works that confirmed the conjecture in several cases [SZ92, CZ84, CZ86, FH03, LC06, Hin09, Gin10, GG09, Hei12, GG12], Ginzburg and Gürel [GG19] proved the most general statement known to hold. They showed that the existence of a Hamiltonian diffeomorphism with finitely many periodic points implies there is a spherical homology class $A$ with ${\left<[\omega],A\right>}>0,{\left<c_{1}(M),A\right>}>0$, in particular, the Conley conjecture holds for closed symplectically aspherical, negative-monotone and Calabi-Yau symplectic manifolds. However, the simple example of irrational rotation of the two-sphere shows that the Conley conjecture does not hold in general, and those are the cases where the Hofer-Zehnder conjecture is interesting. Furthermore, the Conley conjecture also fails for ${\mathbb{C}}\mathrm{P}^{n}$ and complex Grassmannians, all of which have semisimple quantum homology [EP03].

Lastly, we point to other possible interpretations of the meaning of a fixed point not being “necessarily required by the V. Arnold conjecture” in Conjecture 1.1. More precisely, the presence of a hyperbolic fixed point [GG14, GG18], and the existence of a non-contractible fixed point [Gür13, GG16, Ori17, Ori19], are considered “unnecessary” from this viewpoint and were shown to imply the existence of infinitely many periodic points for a large class of symplectic manifolds.

We recall some notions about Hamiltonian diffeomorphisms and symplectic topology required to follow an overview of the main result.

From this point forward, $(M,\omega)$ denotes a closed semipositive symplectic manifold of dimension $2n$. Let $\phi=\phi_{H}$ be a Hamiltonian diffeomorphism generated by a Hamiltonian function⁵⁵5We usually consider normalized Hamiltonians $H$, i.e. $H(t,-)$ has zero $\omega^{n}$ mean for all $t\in[0,1]$ $H\in C^{\infty}({\mathbb{R}}/{\mathbb{Z}}\times M,{\mathbb{R}})$. We say that a fixed point $x$ of $\phi$ is contractible when the loop $\{\phi_{H}^{t}(x)\}_{t\in[0,1]}$ is contractible in $M$. It is a deep fact of symplectic topology that the class this loop represents in $\pi_{1}(M)$ is independent of the choice of $H$ generating $\phi$. We denote by $\mathrm{Fix}(\phi)$ the collection of contractible fixed points. Observe that $\mathrm{Fix}(\phi)$ includes naturally in $\mathrm{Fix}(\phi^{k})$ for all $k\geq 1$. For a fixed point $x$, we denote by $x^{(k)}$ its image under this inclusion. We call a fixed point $x$ of $\phi^{k}$ simple if it is not a fixed point of $\phi^{l}$ for any proper divisor $l$ of $k$.

We summarize the proof of Theorem 1.3 while pointing to the technical developments required to obtain the result in the semipositive setting.

There are two main components to the proof. One is a generalization to the semipositive setting, stated as Theorem 5.1, of the Smith type inequality

(1.2)

$$p\cdot\beta_{tot}(\phi,{\mathbb{F}}_{p})\leq\beta_{tot}(\phi^{p},{\mathbb{F}}_{p}),$$

shown to hold by Shelukhin for spherically monotone symplectic manifolds under the assumption that $\phi^{p}$ has finitely many fixed points. The main technical difficulty can be overcome using the recent work of Sugimoto [Sug21], generalizing the ${\mathbb{Z}}/(p)$-equivariant product-isomorphism to the semipositive setting. The Smith inequality is then obtained by following Shelukhin’s proof [She22] in the spherically monotone setting. The equivariant Floer theory required to obtain Inequality (1.2) is detailed in Section 5.

The other component of the proof, proven as Theorem 4.11, is the main technical achievement of this paper. We uniformly bound the boundary-depth

(1.3)

$$\beta(\phi^{k},{\mathbb{F}}_{p})\leq C$$

for sufficiently large iterations of $\phi$. In the monotone case, Shelukhin [She22] uses the relationship between indices and actions of contractible fixed points to obtain an upper bound for the boundary-depth that only depends on the dimension of the manifold. In the semipositive case, there is no such uniform relation between the actions and indices, therefore we carefully choose the coefficients of the relevant filtered Floer homology groups to have a good relation, before and after reducing coefficients, between the idempotents generating the even quantum homology of $M$. We are then able to bound the boundary-depth by a constant, independent of $p$, depending on the idempotents of the even quantum homology whose coefficient has characteristic $0$.

With the Smith-type inequality and the uniform bound on the boundary-depth established, we now summarize Shelukhin’s argument to prove Theorem 1.3. First, we consider the case where ${\mathbb{K}}$ has characteristic $0$ and the Hamiltonian diffeomorphism $\phi$ and all of its iterates are non-degenerate. By Inequality (1.2) and the simple observation that $\beta_{tot}(\psi,{\mathbb{K}})\leq K(\psi,{\mathbb{K}})\cdot\beta(\psi,{\mathbb{K}})$ for any Hamiltonian diffeomorphism $\psi$ and base field ${\mathbb{K}}$, we have

$$p\cdot\beta_{\mathrm{tot}}(\phi,{\mathbb{F}}_{p})\leq\beta_{\mathrm{tot}}(\phi^{p},{\mathbb{F}}_{p})\leq K(\phi^{p},{\mathbb{F}}_{p})\cdot\beta(\phi^{p},{\mathbb{F}}_{p}).$$

The assumption that $N(\phi,{\mathbb{K}})>\dim_{{\mathbb{K}}}H_{*}(M;{\mathbb{K}})$ and Remark 1.7 imply that the total bar-length $\beta_{\mathrm{tot}}(\phi,{\mathbb{F}}_{p})$ is positive for a sufficiently large prime $p$. Furthermore, Inequality (1.3) yields

$$p\cdot\beta_{\mathrm{tot}}(\phi,{\mathbb{F}}_{p})\leq C\cdot K(\phi^{p},{\mathbb{F}}_{p}),$$

which means that $K(\phi^{p},{\mathbb{K}})$ grows at least linearly with respect to $p$. We now observe that in the non-degenerate setting Equation (1.1) yields

$$\mathrm{Fix}(\phi^{p})=\dim_{{\mathbb{K}}}\operatorname{\mathrm{H}}_{*}(M;{\mathbb{K}})+2K(\phi^{p},{\mathbb{K}}),$$

which implies that $\phi$ must have infinitely many contractible periodic points. In general, when the Hamiltonian diffeomorphism $\phi$ is possibly degenerate, we can again achieve Inequality (1.3) by the local equivariant Floer homology argument [She22, Section 7.4]. Furthermore, by the canonical complex whose properties are listed in Theorem 2.20, the upper bound for the boundary-depth, which is also independent of $p$, continues to hold. Therefore, we can use the same argument as in the non-degenerate case to obtain the linear growth of $K(\phi^{p},{\mathbb{K}})$ and, thus, of $N(\phi^{p},{\mathbb{K}})$. To conclude the argument, we assume that $p$ is large enough to guarantee $\phi^{p}$ is an admissible iteration in the sense of Definition 2.17, it then follows by Theorem 2.18, [GG10, Theorem 1.1],[She22, Theorem C], that $\operatorname{\mathrm{HF}}^{\operatorname{\mathrm{loc}}}(\phi^{p_{1}},x)\cong\operatorname{\mathrm{HF}}^{\operatorname{\mathrm{loc}}}(\phi^{p_{2}},x)$ for all $x\in\mathrm{Fix}(\phi^{p_{1}})$ for any two primes $p_{2}\geq p_{1}\geq p$. In particular, there must be a new simple $p^{\prime}$-periodic point for each prime $p^{\prime}>p$. In fact, if $\mathrm{Fix}(\phi^{p_{1}})=\mathrm{Fix}(\phi^{p_{2}})$ for $p_{2}\geq p_{1}\geq p$, then $N(\phi^{p_{1}},{\mathbb{K}})=N(\phi^{p_{2}},{\mathbb{K}})$ contradicting the linear growth of $N(\phi^{p^{\prime}},{\mathbb{K}})$ for $p^{\prime}\geq p$. A similar argument works when ${\mathbb{K}}$ has characteristic $p$ the details of which can be found in [She22, Section 8].

Floer theory was first developed by A. Floer in [Flo86, Flo87, Flo89] as a generalization of Morse-Novikov homology, to prove the non-degenerate Arnold conjecture. We refer to [HS95, MS17, Oh15] and references therein for the details of the construction and to [Abo15, Sei02, Zap] for in-depth discussions of canonical orientations. We shall consider the construction of Hamiltonian Floer homology in the semipositive setting.

Let $(M,\omega)$ be a closed symplectic manifold and ${\mathbb{K}}$ a choice of base field. Let $H$ be a non-degenerate Hamiltonian function on $M$ and $J=\{J_{t}\}$ be a $1$-periodic family of $\omega$-compatible almost complex structures. For $a\in{\mathbb{R}}\setminus\operatorname{\mathrm{Spec}}(H)$ the Floer chain complex at filtration level $a$ is defined by

$$\operatorname{\mathrm{CF}}_{*}(H;J)^{<a}=\Big{\{}\sum a_{i}\overline{x}_{i}\,\Big{|}\,a_{i}\in{\mathbb{K}},\,\overline{x}_{i}\in\widetilde{\mathcal{P}}(H),\,\mathcal{A}_{H}(\overline{x}_{i})<a\Big{\}},$$

where every summation satisfies the condition that the set $\{i\,|\,a_{i}\neq 0,\mathcal{A}_{H}(\overline{x}_{i})>c\}$ is finite for all $c\in{\mathbb{R}}$. The complex is graded by the Conley-Zehnder index, which assigns an integer $\operatorname{\mathrm{CZ}}(\overline{x})$ to each $\overline{x}\in\widetilde{\mathcal{P}}(H)$ of $\mathcal{A}_{H}$ and satisfies $\operatorname{\mathrm{CZ}}(\overline{x}\#A)=\operatorname{\mathrm{CZ}}(\overline{x})-{\left<c_{1}(M),A\right>}$. Note that $\operatorname{\mathrm{CF}}_{*}(H;J)=\operatorname{\mathrm{CF}}_{*}(H;J)^{+\infty}$ is naturally a finitely generated module over the Novikov ring

$$\Lambda_{\omega,{\mathbb{K}}}=\Bigg{\{}\sum_{A\in G_{\omega}}a_{A}T^{\omega(A)}\,\Big{|}\,a_{A}\in{\mathbb{K}},\,\#\{A\,|\,a_{A}\neq 0,\omega(A)<c\}<\infty,\,\forall c\in{\mathbb{R}}\Bigg{\}}.$$

A Floer trajectory between capped orbits $\overline{x}_{-},\overline{x}_{+}$, is a smooth map $u:{\mathbb{R}}\times\mathrm{S}^{1}\rightarrow M$ satisfying the Floer equation

$$\frac{\partial u}{\partial s}+J_{t}(u)\Bigg{(}\frac{\partial u}{\partial t}-X_{H}^{t}(u)\Bigg{)}=0,$$

with asymptotics,

$$\lim_{s\rightarrow\pm\infty}u(s,t)=x_{\pm}(t),$$

such that the capping $u\#\overline{x}_{+}$ is equivalent to $\overline{x}_{-}$, and $\operatorname{\mathrm{CZ}}(\overline{x}_{-})-\operatorname{\mathrm{CZ}}(\overline{x}_{+})=1$. In the semipositive setting, the compactified moduli space $\mathcal{M}(\overline{x},\overline{y};J)$ of Floer-trajectories from $\overline{x}$ to $\overline{y}$ (modulo the natural ${\mathbb{R}}$-action) is a manifold of dimension $\operatorname{\mathrm{CZ}}(\overline{x})-\operatorname{\mathrm{CZ}}(\overline{y})-1$. The Floer differential

$$d_{F}:\operatorname{\mathrm{CF}}_{k}(H;J)^{<a}\rightarrow\operatorname{\mathrm{CF}}_{k-1}(H;J)^{<a}$$

is defined by

$$d_{F}(\overline{x})=\sum_{\operatorname{\mathrm{CZ}}(\overline{x})-\operatorname{\mathrm{CZ}}(\overline{y})=1}\big{(}\#\mathcal{M}(\overline{x},\overline{y};J)\big{)}\cdot\overline{y}.$$

It squares to zero and preserves the filtration induced by $\mathcal{A}_{H}$. For an interval $I=(a,b)$, $a<b$, where $a,b\in{\mathbb{R}}\setminus\operatorname{\mathrm{Spec}}(H)$ we define the Floer complex in the action window $I$ as the quotient complex

$$\operatorname{\mathrm{CF}}_{*}(H;J)^{I}=\operatorname{\mathrm{CF}}_{*}(H;J)^{<b}/\operatorname{\mathrm{CF}}_{*}(H;J)^{<a}.$$

We denote by $\operatorname{\mathrm{HF}}_{*}(H)^{I}$ the resulting homology of this complex with the differential induced by $d_{F}$ and call it the Floer homology of $H$ in the action window $I$. Note that it is independent of the generic choice of almost complex structure $J$. The (total) Floer homology $\operatorname{\mathrm{HF}}_{*}(H)$ of $H$ is obtained by setting $a=-\infty$ and $b=+\infty$ and does not depend on the choice of Hamiltonian (by a standard continuation argument). Furthermore, for all $H\in\mathcal{H}$, $\operatorname{\mathrm{HF}}_{*}(H)^{I}$ depends only on the homotopy class of $\{\phi_{H}^{t}\}_{t\in[0,1]}$ in the universal cover $\widetilde{\operatorname{\mathrm{Ham}}}(M,\omega)$ of the group of Hamiltonian diffeomorphisms $\operatorname{\mathrm{Ham}}(M,\omega)$.

When $H$ is degenerate, we have to consider perturbation data $\mathcal{D}=(K^{H},J^{H})$, where $K\in\mathcal{H}$ is such that $H^{\mathcal{D}}=H\#K^{H}$ is a non-degenerate Hamiltonian and $J^{H}$ is a choice of generic almost complex structure with respect to $H^{\mathcal{D}}$. We take the action functional to be $\mathcal{A}_{H;\mathcal{D}}=\mathcal{A}_{H^{\mathcal{D}}}$. When $(M,\omega)$ is rational, an admissible action window $I=(a,b)$ for $H$, i.e. $a,b\in{\mathbb{R}}\setminus\operatorname{\mathrm{Spec}}(H)$, will remain so for $H^{\mathcal{D}}$ for sufficiently $C^{2}$-small $K^{H}$, furthermore, the groups $\operatorname{\mathrm{HF}}(H;\mathcal{D})^{I}$ are canonically isomorphic. Therefore, $\operatorname{\mathrm{HF}}(H)^{I}$ is defined as the colimit of the induced directed system. The general case is dealt with by taking the colimit over partially ordered non-degenerate perturbations whose action spectrums do not include $a$ or $b$; refer to [Hei12] or [AS23, Section 2.2.2] for a detailed exposition.