Relations among Hamiltonian, area-preserving, and non-wandering flows on surfaces

Denote by $\overline{A}$ the closure of a subset $A$ of a topological space, by $\mathrm{int}A$ the interior of $A$, and by $\partial A:=\overline{A}-\mathrm{int}A$ the boundary of $A$, where $B-C$ is used instead of the set difference $B\setminus C$ when $B\subseteq C$. We define the border $\partial^{-}A$ by $A-\mathrm{int}A$ of $A$. A boundary component of a subset $A$ is a connected component of the boundary of $A$. A subset is locally dense if its closure has a nonempty interior. A boundary component of a subset $A$ is a connected component of the boundary of $A$. By a surface, we mean a two dimensional paracompact manifold, that does not need to be orientable.

We recall some basic notions. A good reference for most of what we describe is a book by S. Aranson, G. Belitsky, and E. Zhuzhoma [aranson1996introduction].

By a flow, we mean a continuous $\mathbb{R}$-action on a surface. Let $v\colon\mathbb{R}\times S\to S$ be a flow on a compact surface $S$. Then $v_{t}:=v(t,\cdot)$ is a homeomorphism on $S$. For $t\in\mathbb{R}$, define $v_{t}:S\to S$ by $v_{t}:=v(t,\cdot)$. For a point $x$ of $S$, we denote by $O(x)$ the orbit of $x$ (i.e. $O(x):=\{v_{t}(x)\mid t\in\mathbb{R}\}$). An orbit arc is a non-degenerate arc contained in an orbit. A subset of $S$ is said to be invariant (or saturated) if it is a union of orbits. The saturation $\mathrm{Sat}_{v}(A)=v(A)$ of a subset $A\subseteq S$ is the union of orbits intersecting $A$. A point $x$ of $S$ is singular if $x=v_{t}(x)$ for any $t\in\mathbb{R}$ and is periodic if there is a positive number $T>0$ such that $x=v_{T}(x)$ and $x\neq v_{t}(x)$ for any $t\in(0,T)$. A point is closed if it is either singular or periodic. Denote by $\mathop{\mathrm{Sing}}(v)$ (resp. $\mathop{\mathrm{Per}}(v)$) the set of singular (resp. periodic) points. A point is wandering if there are its neighborhood $U$ and a positive number $N$ such that $v_{t}(U)\cap U=\emptyset$ for any $t>N$. Then such a neighborhood is called a wandering domain. A point is non-wandering if it is not wandering (i.e. for any its neighborhood $U$ and for any positive number $N$, there is a number $t\in\mathbb{R}$ with $|t|>N$ such that $v_{t}(U)\cap U\neq\emptyset$).

For a point $x\in S$, define the $\omega$-limit set $\omega(x)$ and the $\alpha$-limit set $\alpha(x)$ of $x$ as follows: $\omega(x):=\bigcap_{n\in\mathbb{R}}\overline{\{v_{t}(x)\mid t>n\}}$, $\alpha(x):=\bigcap_{n\in\mathbb{R}}\overline{\{v_{t}(x)\mid t<n\}}$. For an orbit $O$, define $\omega(O):=\omega(x)$ and $\alpha(O):=\alpha(x)$ for some point $x\in O$. Note that an $\omega$-limit (resp. $\alpha$-limit) set of an orbit is independent of the choice of a point in the orbit. A separatrix is a non-singular orbit whose $\alpha$-limit or $\omega$-limit set is a singular point.

A point $x$ of $S$ is Poisson stable (or strongly recurrent) if $x\in\omega(x)\cap\alpha(x)$. A point $x$ of $S$ is recurrent if $x\in\omega(x)\cup\alpha(x)$. An orbit is singular (resp. periodic, closed, non-wandering, recurrent, Poisson stable) if it consists of singular (resp. periodic, closed, non-wandering, recurrent, Poisson stable) points. A flow is non-wandering (resp. recurrent) if each point is non-wandering (resp. recurrent). A quasi-minimal set is an orbit closure of a non-closed recurrent orbit. The closure of a non-closed recurrent orbit is called a Q-set.

Recall a following characterization of non-wandering flow which is essentially based on [yokoyama2016topological, Theorem 2.5].

A $C^{r}$ vector field $X$ for any $r\in\mathbb{Z}_{\geq 0}$ on an orientable surface $S$ is Hamiltonian if there is a $C^{r+1}$ function $H\colon S\to\mathbb{R}$ such that $dH=\omega(X,\cdot)$ as a one-form, where $\omega$ is a volume form of $S$. In other words, locally the Hamiltonian vector field $X$ is defined by $X=(\partial H/\partial x_{2},-\partial H/\partial x_{1})$ for any local coordinate system $(x_{1},x_{2})$ of a point $p\in S$. A flow is Hamiltonian if it is topologically equivalent to a flow generated by a smooth Hamiltonian vector field. Note that a volume form on an orientable surface is a symplectic form. It is known that a $C^{r}$ ($r\geq 1$) Hamiltonian vector field on a compact surface is structurally stable with respect to the set of $C^{r}$ Hamiltonian vector fields if and only if both each singular point is non-degenerate and each separatrix of a saddle is homoclinic, and each separatrix on a $\partial$-saddle connects a boundary component (see [ma2005geometric, Theorem 2.3.8, p. 74]).

By [cobo2010flows, Theorem 3], any singular points of a Hamiltonian vector field $X$ with finitely many singular points on a compact surface $S$ is either a center or a multi-saddle, and the Reeb graph of the Hamiltonian $H$ generating $X$ is a directed topological graph which is also a quotient space of the orbit space of the generating flow. In other words, the Reeb graph is obtained from the orbit space by collapsing multi-saddle connections into singletons. Note that the orbit space of a Hamiltonian flow is not a directed topological graph.

In the analogy of the collapse, we define the extended orbit to describe “reduced” orbit spaces as follows. By an extended orbit of a flow, we mean a multi-saddle connection or an orbit that is not contained in any multi-saddle connection. Denote by $O_{\mathrm{ex}}(x)$ the extended orbit containing $x$. The quotient space by extended orbits of a flow $v$ on a surface $S$ is called the extended orbit space and denoted by ${S}/{v_{\mathrm{ex}}}$.

Suppose that $v$ is Hamiltonian. Let $H\colon S\to\mathbb{R}$ be the Hamiltonian of $v$ and $\pi\colon S\to S/v_{\mathrm{ex}}$. Then there is a quotient map $p\colon S/v_{\mathrm{ex}}\to\mathbb{R}$ with $H=p\circ\pi$. This means that there are no directed cycles in $S/v_{\mathrm{ex}}$. Lemma 3.2 implies the converse.

Let $v$ be a flow with finitely many singular points on a compact connected surface $S$. Lemma 2.2 implies the equivalence between area-preserving property and divergence-free property for $v$. Suppose that $v$ is divergence-free. [cobo2010flows, Theorem 3] implies that each singular point is either a center or a multi-saddle. Since any divergence-free vector fields on compact surfaces have no wandering domains, the flow $v$ is non-wandering.

Conversely, suppose that $v$ is non-wandering. Lemma 2.1 implies the non-existence of exceptional orbits of $v$. The smoothing theorem [gutierrez1986smoothing] implies that $v$ is topologically equivalent to a $C^{\infty}$-flow. It is known that a total number of Q-sets for $v$ cannot exceed $g$ if $S$ is an orientable surface of genus $g$ [mayer1943trajectories], and $\frac{p-1}{2}$ if $S$ is a non-orientable surface of genus $p$ [markley1970number] (cf. Remark 2 [aranson1996maier]). There are finitely many distinct Q-sets. By induction of the number of Q-sets, we will show the assertion. If there are no Q-sets, then Lemma 3.2 implies that $v$ is divergence-free. Suppose that there are Q-sets. Fix a Q-set $Q$. [cherry1937topological, Theorem VI] implies that $Q$ contains infinitely many Poisson stable orbits. Fix a Poisson stable orbit $O$ in $Q$. By Lemma 4.1, there is a closed transversal $\gamma\subset Q$ intersecting $O$. Then $\gamma=\overline{\gamma\cap O}$. Let $Z$ be a smooth vector field whose generating flow is topological equivalent to $v$.

We claim that the first return map $P_{Z}\colon\gamma\to\gamma$ of $Z$ is topologically conjugate to a minimal interval exchange transformation $E\colon\mathbb{\mathbb{missing}}{S}^{1}\to\mathbb{\mathbb{missing}}{S}^{1}$ on the circle $\mathbb{\mathbb{missing}}{S}^{1}$. Indeed, by the structure theorem of Gutierrez [gutierrez1986smoothing] (see also [gutierrez1986smoothing, Lemma 3.9] and [nikolaev1999flows, Theorem 2.5.1]), for the closed transversal $\gamma$ for the Q-set $Q$, the first return map $P_{Z}\colon\gamma\to\gamma$ is well-defined and topologically semi-conjugate to a minimal interval exchange transformation $E\colon\mathbb{\mathbb{missing}}{S}^{1}\to\mathbb{\mathbb{missing}}{S}^{1}$. Since $Q$ is locally dense and contains $\gamma$, the first return map $P_{Z}\colon\gamma\to\gamma$ is topologically conjugate to the minimal interval exchange transformation $E$.

Therefore we may assume that the first return map $P_{v}\colon\gamma\to\gamma$ by $v$ is a minimal interval exchange transformation. Fix an interval exchange transformation $E^{\prime}\colon\mathbb{\mathbb{missing}}{S}^{1}\to\mathbb{\mathbb{missing}}{S}^{1}$ such that $E^{\prime}\circ E$ has periodic orbits. Denote by $I_{1},I_{2},\ldots,I_{k}$ the intervals of interval exchange transformation $E^{\prime}\circ E$. Let $d_{i}$ be the length of $I_{i}\subset\mathbb{S}^{1}$ with $d_{1}+d_{2}+\cdots+d_{k}=1$. Fix an open transverse annulus $A\subset Q$ which is a neighborhood of $\gamma$ and contains no singular points. Then we also may assume that the restriction of $Z$ to $A$ can be identified with a vector field $(1,0)$ on $[0,1]\times\gamma$ by a homeomorphism. Consider a smooth divergence-free vector field $Z_{A}$ on $A=[0,1]\times\gamma$ which depends only on $\gamma$ and which is a periodic annulus such that the first return map on $\gamma$ by the flow $v_{Z^{\prime}}$ generated by the resulting vector field $Z^{\prime}:=Z+Z_{A}$ is topological equivalent to the interval exchange transformation $E^{\prime}\circ E\colon\mathbb{\mathbb{missing}}{S}^{1}\to\mathbb{\mathbb{missing}}{S}^{1}$. Then $v_{Z^{\prime}}$ has a periodic orbit $O_{A}$ intersecting $\gamma$ and $O$. Fix the corresponding periodic orbit $O_{1}$ of $Z$ intersecting $\gamma$. Denote by $I^{\prime}_{i}\subset\gamma$ the intervals of $P_{Z^{\prime}}$ corresponding to $I_{i}$. Then the saturations of any intervals $I^{\prime}_{i}$ of $P_{Z^{\prime}}$ are invariant open periodic annuli or Möbius bands $A^{\prime}_{i}$. Since any interval exchange transformation preserves the Lebesgue measure and so non-wandering, the flow $v_{Z^{\prime}}$ generated by $Z^{\prime}$ is non-wandering. By the inductive hypothesis, the flow $v_{Z^{\prime}}$ is divergence-free. By construction, $v_{Z}|_{S-v(\gamma)}=v_{Z^{\prime}}|_{S-v(\gamma)}$. Fix a divergence-free vector field $W$ whose generating flow $v_{W}$ is topologically equivalent to $v_{Z^{\prime}}$ and a closed transversal $\gamma_{W}$ with respect to $v_{W}$ which corresponds to $\gamma$. Then the first return map $P_{W}\colon\gamma_{W}\to\gamma_{W}$ is topologically conjugate to $P_{Z^{\prime}}$. If $A_{i}$ is an open Möbius band, then the orbit space $A_{i}/v_{W}$ for the flow $v_{W}$ generated by $W$ is homeomorphic to a half open interval $[0,1)$ such that the periodic orbit corresponding to $0$ is one-sided and denote by $O_{A_{i}}$. Denote by $D_{i}$ be the differences of the values the boundary components of the open invariant annulus $A_{i}$ with respect to the multi-valued Hamiltonians $H_{i}\colon A_{i}\to\mathbb{R}$ if $A_{i}$ is an annulus, and by $D_{i}$ the twice of the differences of the values the boundary components of the open invariant annulus $A_{i}-O_{A_{i}}$ with respect to the multi-valued Hamiltonians $H_{i}\colon A_{i}-O_{A_{i}}\to\mathbb{R}$. Put $D:=D_{1}+D_{2}+\cdots+D_{k}$. Write $\mathbb{A}_{i}:=A_{i}$ if $A_{i}$ is an annulus, and $\mathbb{A}_{i}:=A_{i}-O_{A_{i}}$ if $A_{i}$ is Möbius band.

We claim that we may assume that $D_{1}/d_{1}=D_{2}/d_{2}=\cdots=D_{k}/d_{k}$ by deformations. Indeed, by the topological equivalence between $P_{w}$ and $E^{\prime}\circ E$, the saturations of any intervals of the interval exchange transformation $P_{W}$ are also invariant open periodic annuli $\mathbb{A}_{i}$. Identify annuli $\mathbb{A}_{i}$ with an embedded annuli $\{(x,H_{i}(x))\mid x\in\mathbb{A}_{i}\}$ of $\mathbb{A}_{i}\times\mathbb{R}$. Let $\mathbb{A}^{\prime}_{i}$ be an invariant open annulus with $\overline{\mathbb{A}^{\prime}_{i}}\subset\mathbb{A}_{i}$ and $\mathbb{A}^{\prime}_{i+},\mathbb{A}^{\prime}_{i-}$ the connected components of $\mathbb{A}_{i}-\mathbb{A}^{\prime}_{i}$. Then the set difference $\mathbb{A}_{i}-\mathbb{A}^{\prime}_{i}$ is a disjoint union of two invariant annuli. Removing $\mathbb{A}^{\prime}_{i}$, taking translation $\mathbb{A}^{\prime\prime}_{i+}$ of $\mathbb{A}^{\prime}_{i+}$ with respect to the heights of $H_{i}$, and interpolating between $\mathbb{A}^{\prime\prime}_{i+}$ and $\mathbb{A}^{\prime}_{i-}$ as in Figure 5, the resulting annuli $\mathbb{A}^{\prime\prime}_{i}$ have the arbitrarily differences of the values the boundary components of $\mathbb{A}^{\prime\prime}_{i}$ with respect to the height.

Therefore we can choose the differences of the values the boundary components of $\mathbb{A}^{\prime\prime}_{i}$ such that $D_{1}/d_{1}=D_{2}/d_{2}=\cdots=D_{k}/d_{k}$. Since the constructions preserve topological equivalence, the claim is completed.

We claim that we may assume that there are a closed transversal $\mu$ with $v_{Z^{\prime}}(\mu)=v_{Z^{\prime}}(\gamma)$ and an open annular neighborhood $V_{0}$ of $\mu$ which can be isometrically embedded into $(-1,1)\times\mathbb{R}/\mathbb{Z}\times\mathbb{R}/D\mathbb{Z}$ such that $\overline{V_{0}}\subset v(\mu)$ and that $Z^{\prime}|_{V_{0}}=(d,0,0)$ for some positive number $d\in\mathbb{R}$, by deformation. Indeed, fix a small open annular neighborhood $V$ of $\gamma_{W}$ which is a closed transverse annulus such that $\overline{V}\subset v(\gamma_{W})$. For a point $t\in\gamma_{W}$, denote by $C_{t}$ the orbit arc in $V$ containing $t$. Identify $V$ with $\{(x,H(x))\mid x\in V\}$, where $H\colon V\to\mathbb{R}/D\mathbb{Z}$ is a function defined by $H_{i}\colon\mathbb{A}_{i}\to\mathbb{R}$. Cutting $V$ along the closed transversal $\gamma_{W}$, inserting a closed annulus $V_{0}:=[-0.1,0,1]\times\{(t,Dt)\mid t\in\mathbb{R}/\mathbb{Z}\}$ into the resulting two closed annuli $V_{-}$ and $V_{+}$, and interpolate between $V_{-}$ and $V_{0}$ and between $V_{+}$ and $V_{0}$ as in Figure 6, we can obtain the resulting annuli $U$.

Then $\mu:=\{0\}\times\{(t,Dt)\mid t\in\mathbb{R}/\mathbb{Z}\}\subset W\subset U$ is a closed transversal. This completes the claim.

By construction, there is a divergence-free vector field $W_{A}$ whose support is contained in $V_{0}$ and that any non-singular orbit is a periodic orbit of form $\{x_{1}\}\times\{(t,Dt)\mid t\in\mathbb{R}/\mathbb{Z}\}\subset V_{0}$ and $W_{A}$ is depended only on the $x_{1}$-coordinate in $V_{0}$. Multiplying $W_{A}$ by a scalar if necessary, we may assume that the first return map on $\mu$ by $W+W_{A}$ is topological conjugate to $E\colon\mathbb{\mathbb{missing}}{S}^{1}\to\mathbb{\mathbb{missing}}{S}^{1}$ and so to $P_{Z}\colon\gamma\to\gamma$. This means that the flow generated by $W_{A}+W$ is topological equivalent to $v$. Recall that, for any divergence-free vector field $Y$, a vector field $X$ is divergence-free if and only if so is the sum $X+Y$. This implies that the vector field $W_{A}+W$ is divergence-free and so is the flow $v$.

From the non-existence of non-closed recurrent orbit, Lemma 3.1 implies the quotient space $\mathbb{S}^{2}/v_{\mathrm{ex}}$ is a finite directed topological graph. Since the quotient space of a sphere is simply connected, the graph $\mathbb{S}^{2}/v_{\mathrm{ex}}$ has no directed circuit. Theorem A and Theorem B imply Corollary C.