Periodic points of rational area-preserving homeomorphisms

Questions on the existence and multiplicity of periodic points of area-preserving surface homeomorphisms have a long history, dating back to Poincaré and Birkhoff’s work on annulus twist maps and the restricted planar three-body problem, and have attracted significant attention since then. The existence question asks whether any periodic points exist, and the multiplicity question asks how many there are. On a closed surface of genus $g\geq 1$, area-preserving maps with any finite number $N\geq 2g-2$ of periodic points may be constructed by adding $N-2g+2$ singularities to an irrational translation flow ($g=1$) or a translation flow in any minimal direction ($g\geq 2$). The resulting time-one maps have finitely many periodic points, all of which are fixed, and are isotopic to the identity relative to their fixed point set. The current state-of-the-art, to our knowledge, is due to Le Calvez [LC22]. He shows, barring some edge cases when $g\leq 1$, that any area-preserving homeomorphism of a closed surface is either periodic, has periodic points of unbounded minimal period, or after passing to an iterate only has finitely many fixed points and is isotopic to the identity relative to the fixed point set. The last case is the one modeled by the example above.

In this note, we discuss a simple and dense homological condition which forces an area-preserving map isotopic to the identity to have infinitely many periodic points. Fix a compact surface $\Sigma$ and a smooth area form $\omega$. Any area-preserving map $\phi\in\operatorname{\operatorname{Homeo}}_{0}(\Sigma,\omega)$ isotopic to the identity has a rotation vector

where $\Gamma_{\omega}\subseteq H_{1}(\Sigma;\mathbb{Z})$ is a discrete subgroup. The map $\phi$ is said to have rational rotation direction if $\mathcal{F}(\phi)$ is a real multiple of a rational class, i.e. $c\cdot\mathcal{F}(\phi)\in H_{1}(\Sigma;\mathbb{Q})\,/\,\Gamma_{\omega}$ for some real number $c>0$. Any area-preserving map can be perturbed to an area-preserving map with rational rotation direction by a $C^{\infty}$-small perturbation, so a dense subset of maps have rational rotation direction. We now state our main result.

In addition to the historical backdrop presented above, Theorem A fits into an active stream of research in symplectic dynamics centered around the Conley conjecture. The original formulation of the conjecture asserts that any Hamiltonian diffeomorphism of a closed aspherical symplectic manifold has infinitely many periodic points. The Conley conjecture was resolved for surfaces by Franks–Handel [FH03] and extended to Hamiltonian homeomorphisms by Le Calvez [LC06] before being resolved in full generality by breakthrough work of Hingston [Hin09] for higher-dimensional tori and Ginzburg [Gin10] for the general case. The search for extensions of the Conley conjecture to Hamiltonian diffeomorphisms/homeomorphisms of more general symplectic manifolds has attracted a great deal of ongoing activity and progress [GG09b, GG09a, GG10, GG12, G1̈3, GG14, GG15, GG16, Ç18, GG19]. However, there has been much less progress in establishing Conley conjecture-type results for non-Hamiltonian symplectic maps (some notable results include [Bat15, Bat17, Bat18]). Moreover, to our knowledge, there is no agreed upon formulation of the Conley conjecture for non-Hamiltonian symplectic maps. Theorem A can be viewed not only as establishing such a “non-Hamiltonian Conley conjecture” in dimension $2$, but also as a guidepost towards formulating a “non-Hamiltonian Conley conjecture” for higher dimensional symplectic manifolds. We pose the following question.

The idea that a Conley conjecture-type result may hold for area-preserving homeomorphisms with rational rotation direction was motivated by recent work on the $C^{\infty}$-closing lemma [CGPZ21, EH21] for area-preserving surface diffeomorphisms. Herman famously showed [HZ94, Chapter $4.5$] that a version of the closing lemma using only Hamiltonian perturbations cannot hold for certain irrational maps (Diophantine torus rotations). This issue was avoided by proving a Hamiltonian $C^{\infty}$-closing lemma for many rational maps¹¹1Any map with rational asymptotic cycle (see Section 1.2). and then observing that such maps form a $C^{\infty}$-dense subset of $\operatorname{\operatorname{Diff}}(\Sigma,\omega)$. It seems reasonable to suspect that, given these marked differences in behavior, the rationality condition has dynamical significance. Further evidence that rationality may force infinitely many periodic points is furnished by the fact that the area-preserving maps with finitely many periodic points discussed above all have irrational rotation vector. We also learned while preparing the first version of this paper that Theorem A was also posed as a question, with the same motivation, by Sobhan Seyfaddini.

Theorem A can also be extended, with slightly weaker conclusions, to maps preserving arbitrary full-support Borel probability measures. Given an isotopy $\Phi$ from the identity to $\phi$ and a $\phi$-invariant Borel probability measure $\mu$, a rotation vector $\mathcal{F}(\Phi,\mu)\in H_{1}(\Sigma;\operatorname{\mathbb{R}})$ can be defined. When $\Sigma$ has genus $\geq 2$, the rotation vector does not depend on the isotopy $\Phi$, and we write it as $\mathcal{F}(\phi,\mu)$.

Theorem B follows from a short argument combining Theorem A and a theorem of Oxtoby–Ulam [OU41], which was suggested to the author by Patrice Le Calvez. Theorem A follows from Theorem C, which may be of independent interest, and the results in [LC06, LC22].

A recent result from symplectic geometry is central to the proof of Theorem C. In [CGPZ21], a precise non-vanishing theorem is proved for the periodic Floer homology (PFH) of area-preserving diffeomorphisms of closed surfaces with rational rotation vector $(\mathcal{F}(\phi)\in H_{1}(\Sigma;\mathbb{Q}))$. PFH is a homology theory for area-preserving surface maps built out of their periodic orbits. The non-vanishing theorem relies on a deep result of Lee–Taubes [LT12] showing PFH is isomorphic to monopole Floer homology. We explain more about PFH and state the non-vanishing theorem at the beginning of Section 4.

There are two significant issues to overcome in the proof of Theorem C. The first issue is that PFH is only well-defined for diffeomorphisms, not homeomorphisms. To get around this, we observe that a quantitative version (Proposition 4.2) of Theorem C holds; there exists a non-contractible periodic point with an upper bound on its minimal period depending only on the rotation vector. This allows us to extend Theorem C to homeomorphisms by an approximation argument.

The second issue is that the non-vanishing of PFH is only known for maps with rotation vector in $H_{1}(\Sigma;\mathbb{Q})$ and not all maps with rational rotation direction. We cannot hope for too much when $\mathcal{F}(\phi)\not\in H_{1}(\Sigma;\mathbb{Q})$, in this case there are many examples (irrational torus rotations, translation flows in minimal directions) where PFH essentially vanishes²²2More precisely, it vanishes in nontrivial homological gradings, so it only detects null-homologous periodic orbit sets, in which every periodic orbit could be contractible.. We deal with this by introducing a blow-up argument to reduce from the case of rational rotation direction to the case where $\mathcal{F}(\phi)\in H_{1}(\Sigma;\mathbb{Q})$. When $\Sigma$ has genus $\geq 2$, $\phi$ has a fixed point, and we can assume without loss of generality that it is contractible since otherwise we would already be done. We blow up this fixed point and attach a disk of a specified area to the boundary circle, and then extend the isotopy over the new surface. The extended map has a rotation vector in the same direction, but scaled by a constant in $(0,1)$, which depends on the area of the attached disk. We are free to attach a disk of any area, so we scale the rotation vector down to a rational vector, and note that the only new periodic points introduced by blowing up are contractible, so the original map must have a non-contractible periodic point. This argument reveals more generally that, in genus $\geq 2$, the dynamics are essentially identical for maps with $\mathcal{F}(\phi)\in H_{1}(\Sigma;\mathbb{Q})$ and those with rational rotation direction.

Assume that $\Sigma$ is closed and has genus $\geq 2$. It is known [LC22] that either $\phi$ has periodic points of unbounded minimal period or it has periodic Nielsen–Thurston class. Therefore, if $\phi$ has periodic Nielsen–Thurston class and some iterate $\phi^{q}\in\operatorname{\operatorname{Homeo}}_{0}(\Sigma,\omega)$ has rational rotation direction, Theorem A implies $\phi$ is either periodic or has periodic points of unbounded minimal period. We will now show that maps with periodic Nielsen–Thurston class and rational asymptotic cycle have an iterate with rational rotation direction.

Let $M_{\phi}$ denote the mapping torus of $\phi$. The asymptotic cycle $\operatorname{\mathcal{C}}(\phi)\in H_{1}(M_{\phi};\operatorname{\mathbb{R}})$ is an analogue of the rotation vector for area-preserving maps not isotopic to the identity; by rational asymptotic cycle we mean $\operatorname{\mathcal{C}}(\phi)\in H_{1}(M_{\phi};\mathbb{Q})$. It is defined by applying Schwartzman’s construction [Sch57] to the suspension flow. When $\phi$ is isotopic to the identity, a choice of identity isotopy $\Phi$ defines a diffeomorphism $M_{\phi}\simeq\mathbb{T}\times\Sigma$ where $\mathbb{T}:=\mathbb{R}/\mathbb{Z}$ denotes the circle. Lemma 3.1 proves that, under this identification,

If $\phi\in\operatorname{\operatorname{Homeo}}_{0}(\Sigma,\omega)$ has rational asymptotic cycle, this computation shows that $\phi$ has rational rotation direction. The property $\operatorname{\mathcal{C}}(\phi)\in H_{1}(M_{\phi};\mathbb{Q})$ also behaves well under iteration; Lemma 3.3 below shows that if $\operatorname{\mathcal{C}}(\phi)$ is rational, then so is $\operatorname{\mathcal{C}}(\phi^{k})$ for each $k>1$. Putting these facts together implies our claim above, that if $\phi$ has periodic Nielsen–Thurston class and $\operatorname{\mathcal{C}}(\phi)\in H_{1}(M_{\phi};\mathbb{Q})$, then it has an iterate $\phi^{q}\in\operatorname{\operatorname{Homeo}}_{0}(\Sigma,\omega)$ with rational rotation direction. Combining this with Theorem A proves the following theorem.

There are also asymptotic cycles $\operatorname{\mathcal{C}}(\phi,\mu)\in H_{1}(M_{\phi};\operatorname{\mathbb{R}})$ for each $\phi$-invariant Borel probability measure $\mu$. The same proof as Theorem D, replacing Theorem A with Theorem B, implies the following result.

The analogues of the above theorems when $\Sigma$ has genus $0$ are already known. Collapsing the boundary components and appealing to [FH03, LC06] gives a sharp characterization of existence and multiplicity of periodic points. A genus $1$ version of Theorem A follows from work of Le Calvez.

The assumptions are stronger than in Theorem A, but the result is sharp. Any translation $(x,y)\mapsto(x+a,y+b)$ on $\mathbb{T}^{2}:=\operatorname{\mathbb{R}}^{2}/\mathbb{Z}^{2}$, where at least one of $a$ and $b$ is not rational, has irrational rotation vector and also has no periodic points. The proof of Proposition 1.1 is straightforward. Any map satisfying the conditions of Proposition 1.1 has a Hamiltonian iterate, and the main result of [LC06] shows Hamiltonian torus homeomorphisms are either the identity or have periodic points of unbounded minimal period. A sharp version of Theorem C for smooth torus maps³³3An explanation for why we need smoothness is provided below the statement of Proposition 4.3. is also true.

The proof is given in Section 4. We also remark that if $\mathcal{F}(\phi)\in H_{1}(\mathbb{T}^{2};\mathbb{Q})$, then $\phi$ has a Hamiltonian iterate, so it has a periodic point by Conley–Zehnder’s fixed point theorem [CZ83]. There is also a version of Theorem D for the torus.

Addas Zanata–Tal [AZT07] proved that an area-preserving torus homeomorphism $\phi$ either has periodic points of unbounded minimal period, is isotopic to a Dehn twist with no periodic points and vertical rotation set reduced to an irrational number, or has an iterate isotopic to the identity. The assumption $\operatorname{\mathcal{C}}(\phi)\in H_{1}(M_{\phi};\mathbb{Q})$ rules out the second case, so we can assume $\phi$ has an iterate $\phi^{q}$ isotopic to the identity. The assumption $\operatorname{\mathcal{C}}(\phi)\in H_{1}(M_{\phi};\mathbb{Q})$ implies $\phi^{q}$ has rational rotation direction, and then by Proposition 1.1 $\phi^{q}$ is either the identity or has periodic points of unbounded minimal period.

Section 2 reviews some important preliminaries. Section 3 does some computations of asymptotic cycles which seem standard but which we could not find elsewhere. Section 4 presents a brief overview of PFH and the non-vanishing theorem from [CGPZ21], and then proves Theorems A–C and F. Theorems D and E were proved above using computations from Section 3 and Theorems A and B.

Thanks to Dan Cristofaro-Gardiner for discussions regarding this project. Similar results were proved independently and simultaneously by Guihéneuf–Le Calvez–Passeggi [GLCP23], using topological methods (homotopically transverse foliations, forcing theory). I would like to thank them for discussions regarding their work. This work was supported by the National Science Foundation under Award No. DGE-1656466 and the Miller Institute at the University of California Berkeley.

Fix any $\phi\in\operatorname{\operatorname{Homeo}}(\Sigma)$. The mapping torus of $\phi$ is the compact $3$-manifold $M_{\phi}$ defined by quotienting $\mathbb{R}_{t}\times\Sigma$ by the relation $(1,p)\sim(0,\phi(p))$. Translation in the $t$-direction with speed $1$ yields a continuous flow $\{\psi^{t}_{R}\}_{t\in\operatorname{\mathbb{R}}}$ on $M_{\phi}$ called the suspension flow. Its closed integral curves are in one-to-one correspondence with simple periodic orbits of $\phi$. If $\phi$ preserves a Borel measure $\mu$, the suspension flow preserves the measure $dt\otimes\mu$ on $M_{\phi}$.

Suppose $\phi\in\operatorname{\operatorname{Homeo}}_{0}(\Sigma)$. Choose an identity isotopy $\Phi=\{\phi_{t}\}_{t\in[0,1]}$. This choice defines a homeomorphism

by the map $[(t,p)]\mapsto[(t,\phi_{t}^{-1}(p))]$. This homeomorphism provides a useful method for recovering the rotation vector of a periodic orbit. Let $S=\{x_{1},\ldots,x_{k}\}\subset\Sigma$ be a simple periodic orbit, and let $\gamma\subset M_{\phi}$ be the associated closed integral curve of the suspension flow. Using $\eta$ to realize $\gamma$ as a loop in $\mathbb{T}\times\Sigma$, its homology class is easily computed:

We define $\operatorname{\mathcal{C}}(\phi,\mu)$ as a real-valued linear functional on $[M_{\phi},\mathbb{T}]$. Fix any continuous $f:M_{\phi}\to\mathbb{T}$. For each $s\in\operatorname{\mathbb{R}}$, write $f_{s}:=f\circ\psi^{s}_{R}$. The functions $f_{s}-f$ are a continuous family of null-homotopic circle-valued functions, so they lift to a unique continuous family $\{g_{s}\}_{s\in\operatorname{\mathbb{R}}}$ of functions $M_{\phi}\to\operatorname{\mathbb{R}}$ with $g_{0}\equiv 0$. Kingman’s subadditive ergodic theorem implies that $G:=\lim_{s\to\infty}g_{s}/s$ is a well-defined $(dt\otimes\mu)$-integrable function. We set $\langle\operatorname{\mathcal{C}}(\phi,\mu),f\rangle$ to be the integral of $G$. This is linear and homotopy-invariant in $f$ (see [Sch57]), so it defines a real-valued linear functional on $H^{1}(M_{\phi};\operatorname{\mathbb{Z}})$, and therefore defines a class in $H_{1}(M_{\phi};\operatorname{\mathbb{R}})$.

When $\phi\in\operatorname{\operatorname{Homeo}}_{0}(\Sigma)$, the rotation vector of $\mu$ can be recovered from $\operatorname{\mathcal{C}}(\phi,\mu)$.

When $\phi\in\operatorname{\operatorname{Diff}}(\Sigma,\omega)$, the area form $\omega$ defines a closed $2$-form $\omega_{\phi}$ on the mapping torus $M_{\phi}$, which restricts to $0$ on the boundary. Its cohomology class is Poincaré dual to the asymptotic cycle.

We show that rationality of $\operatorname{\mathcal{C}}(\phi,\mu)$ is preserved under iteration.

Fix a closed surface $\Sigma$ and a smooth area form $\omega$. Fix an area-preserving diffeomorphism $\phi\in\operatorname{\operatorname{Diff}}(\Sigma,\omega)$. The area form $\omega$ defines a closed 2-form on the mapping torus $M_{\phi}$, denoted by $\omega_{\phi}$. Let $t$ be the coordinate for the interval component of $[0,1]\times\Sigma$. Then $dt$ pushes forward to a smooth 1-form on $M_{\phi}$. The pair $(dt,\omega_{\phi})$ forms a stable Hamiltonian structure on $M_{\phi}$, and the Reeb vector field is a smooth vector field $R$ generating the suspension flow $\{\psi^{t}_{R}\}_{t\in\operatorname{\mathbb{R}}}$. The associated two-plane bundle $\ker(dt)$ is equal to the vertical tangent bundle of the fibration $M_{\phi}\to\mathbb{T}$, which we denote by $V$.

Fix some nonzero homology class $\Gamma\in H_{1}(M_{\phi};\operatorname{\mathbb{Z}})$. The PFH generators are finite sets $\Theta=\{(\gamma_{i},m_{i})\}$ of pairs of embedded Reeb orbits $\gamma_{i}$ and multiplicities $m_{i}\in\mathbb{N}$ which satisfy the following three conditions: (1) the orbits $\gamma_{i}$ are distinct, (2) the multiplicity $m_{i}$ is $1$ whenever $\gamma_{i}$ is a hyperbolic orbit, (3) $\sum_{i}m_{i}[\gamma_{i}]=\Gamma$. The chain complex $\operatorname{\text{PFC}}_{*}(\phi,\Gamma)$ is the free module over a commutative coefficient ring⁴⁴4This can be anything when $\Gamma$ solves (8) for some $d$, but in general $\Lambda$ must be a Novikov ring. $\Lambda$ generated by the set of all PFH generators.

The differential on $\operatorname{\text{PFC}}_{*}(\phi,\Gamma)$ counts “ECH index 1” $J$-holomorphic currents between orbit sets. The homology of this chain complex is the periodic Floer homology $\operatorname{\text{PFH}}_{*}(\phi,\Gamma)$. This homology theory was constructed by Hutchings [Hut02]; see [Hut14] for a detailed exposition of the closely related theory of embedded contact homology. The PFH group depends only on the Hamiltonian isotopy class of $\phi$; this allows us to define PFH for a degenerate map as the PFH of any sufficiently close nondegenerate Hamiltonian perturbation. We now precisely state the non-vanishing theorem for PFH.

The result as stated in [CGPZ21] only asserts non-vanishing for $d$ sufficiently large, but the explicit lower bound is not difficult to extract once the details are understood. We only need $d$ large enough to ensure that $\operatorname{\text{PFH}}$ is isomorphic to the “bar” version $\overline{\text{HM}}$ of monopole Floer homology. Lee–Taubes [LT12, Theorem $1.2$, Corollary $1.5$] prove this isomorphism assuming $d>\max(2g-2,0)$. The non-vanishing theorem is a key ingredient in the following technical result.

The same argument proves an analogue for smooth torus maps, but we need to rule out maps without fixed points in the statement.

Theorem C and Theorem F from the introduction respectively follow from Proposition 4.2 and Proposition 4.3. We now outline the plan for the rest of the section. Section 4.2 proves Theorems A and B assuming Theorem C. Section 4.3 proves Proposition 4.2. Section 4.4 proves Proposition 4.3.

We prove Theorems A and B using Theorem C. We assume that $\Sigma$ is a closed surface of genus $\geq 2$, since we can reduce to this case by collapsing the boundary components.