Symplectomorphisms of surfaces preserving a smooth function, I

TL;DR

This paper studies the topology of symplectomorphisms of surfaces that preserve a smooth Morse function, revealing they are either contractible or homotopy equivalent to a circle, depending on the presence of saddle points.

Contribution

It constructs a canonical homeomorphism or infinite cyclic cover relating functions constant along Hamiltonian orbits to the symplectomorphism group component, extending to certain singular maps.

Findings

The group of symplectomorphisms preserving the function is either contractible or homotopy equivalent to a circle.

A canonical homeomorphism exists when the Morse function has at least one saddle point.

The results extend to maps with singularities similar to homogeneous polynomials without multiple factors.

Abstract

Let $M$ be a compact orientable surface equipped with a volume form $\omega$, $P$ be either $\mathbb{R}$ or $S^1$, $f:M\to P$ be a $C^{\infty}$ Morse map, and $H$ be the Hamiltonian vector field of $f$ with respect to $\omega$. Let also $\mathcal{Z}_{\omega}(f) \subset C^{\infty}(M,\mathbb{R})$ be set of all functions taking constant values along orbits of $H$, and $\mathcal{S}_{\mathrm{id}}(f,\omega)$ be the identity path component of the group of diffeomorphisms of $M$ mutually preserving $\omega$ and $f$. We construct a canonical map $\varphi: \mathcal{Z}_{\omega}(f) \to \mathcal{S}_{\mathrm{id}}(f,\omega)$ being a homeomorphism whenever $f$ has at least one saddle point, and an infinite cyclic covering otherwise. In particular, we obtain that $\mathcal{S}_{\mathrm{id}}(f,\omega)$ is either contractible or homotopy equivalent to the circle. Similar results hold in fact for a larger class of maps $M\to P$ whose singularities are equivalent to homogeneous polynomials without multiple factors.

Figures

Equations

$$\{{\omega }_t{\}}_{t\in D^n}\subset \mathrm{Vol}(M,1)$$

$$\{h_t{\}}_{t\in D^n}\subset {\mathrm{Diff}}_0(M)$$

$$p:{\mathrm{Diff}}_0(M)\to \mathrm{Vol}(M,1),$$

$${\mathrm{Symp}}_0(M,\omega )\subset {\mathrm{Diff}}_0(M)$$

$$\mathrm{Stab}(f)=\{h\in \mathrm{Diff}(M)\mid f\circ h=f\}$$

$$\mathcal{O}(f)=\{f\circ h\mid h\in \mathrm{Diff}(M)\}$$

$$\mathrm{Stab}(f,\omega )=\mathrm{Stab}(f)\cap \mathrm{Symp}(M,\omega )$$

$${\mathbf{H}}_{\alpha }:M\to M,$$

$${\mathrm{Stab}}_0(f,\omega )\subset {\mathrm{Stab}}_0(f)$$

$$j:\mathrm{Stab}(f,\omega )\subset {\mathrm{Stab}}^{+}(f)\equiv \mathrm{Stab}(f)\cap {\mathrm{Diff}}^{+}(M)$$

$${\mathbf{H}}_{\alpha }(x)=\mathbf{H}(x,\alpha (x)),$$

$${\phi }_H:C^{\infty }(M,\mathbb{R})\to C^{\infty }(M,M),$$

$${\Gamma }_H=\{\alpha \in C^{\infty }(M,\mathbb{R})\mid 1+H(\alpha )>0\},$$

$$\phi ({\Gamma }_H)\subset {\mathrm{Diff}}_0(H),$$

$$df(z)(u)=\omega (u,H(z)),$$

$$H(x,y)=\frac{1}{\gamma(x,y)}\bigl{(}-\hat{f}^{\prime}_{y}\tfrac{\partial}{\partial x}+\hat{f}^{\prime}_{x}\tfrac{\partial}{\partial y}\bigr{)}.$$

$$TM\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$s$$

$$TM\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$H(z)(f)=\omega (H(z),H(z))=0,$$

$${\mathcal{L}}_H\omega$$

$$\{\cdot ,\cdot \}:C^{\infty }(M,P)\times C^{\infty }(M,Q)⟶C^{\infty }(M,\mathbb{R})$$

$$\{f,g\}:=\omega (H_f,H_g)=\psi (H_f)(H_g)=H_f(g)=-H_g(f),$$

$${\mathcal{Z}}_{\omega }^Q(f)=\{g\in C^{\infty }(M,Q)\mid H_f(g)=\{f,g\}=0\}.$$

$$\Gamma =\{\alpha \in C^{\infty }(M,\mathbb{R})\mid 1+H(\alpha )>0\},$$

$$\phi :\Gamma \stackrel{p}{\to }\Gamma /\mathbb{Z}\underset{\cong }{\overset{r}{\to }}{\mathrm{Stab}}_0(f),$$

$${\phi }_F,{\phi }_H:C^{\infty }(M,\mathbb{R})\to C^{\infty }(M,M)$$

$$\sigma :M\times \mathbb{R}\to \mathbb{R},$$

$$\mathbf{H}(x,\alpha (x))=\mathbf{F}(x,\sigma (x,\alpha (x))).$$

$$\gamma :C^{\infty }(M,\mathbb{R})\to C^{\infty }(M,\mathbb{R}),$$

Full text

We construct a canonical map $\varphi:\mathcal{Z}_{\omega}(f)\to\mathcal{S}_{\mathrm{id}}(f,\omega)$ being a homeomorphism whenever $f$ has at least one saddle point, and an infinite cyclic covering otherwise. In particular, we obtain that $\mathcal{S}_{\mathrm{id}}(f,\omega)$ is either contractible or homotopy equivalent to the circle.

Similar results hold in fact for a larger class of maps $M\to P$ whose singularities are equivalent to homogeneous polynomials without multiple factors.

The author is indebted to Bogdan Feshchenko for useful discussions.

1. Introduction

Let $M$ be a closed oriented surface, $\mathrm{Diff}(M)$ be the group of all $C^{\infty}$ diffeomorphisms of $M$, and $\mathrm{Diff}_{0}(M)$ be the identity path component of $\mathrm{Diff}(M)$ consisting of all diffeomorphisms isotopic to the identity.

Let also $\mathrm{Vol}(M,1)$ be the space of all volume forms on $M$ having volume $1$ and $\omega\in\mathrm{Vol}(M,1)$. Since $\dim M=2$, $\omega$ is a closed non-degenerate $2$-form and so it defines a symplectic structure on $M$. Denote by $\mathrm{Symp}(M,\omega)$ the group of all $\omega$-preserving $C^{\infty}$ diffeomorphisms, and let $\mathrm{Symp}_{0}(M,\omega)$ be its identity path component.

Then Moser’s stability theorem [20] implies that for any $C^{\infty}$ family

[TABLE]

of volume forms parameterized by points of a closed $n$-dimensional disk $D^{n}$, there exists a $C^{\infty}$ family of diffeomorphisms

[TABLE]

such that $\omega_{t}=h_{t}^{*}\omega$ for all $t\in D^{n}$. In particular, this implies that the map

[TABLE]

is a Serre fibration with fiber $\mathrm{Symp}_{0}(M,\omega)$, see e.g.​ [19, §3.2], [2], or [21, §7.2].

Since $\mathrm{Vol}(M,1)$ is convex and therefore contractible, it follows from exact sequence of homotopy groups of the Serre fibration $p$ that $p$ yields isomorphisms of the corresponding homotopy groups $\pi_{k}\mathrm{Symp}_{0}(M,\omega)\cong\pi_{k}\mathrm{Diff}_{0}(M)$, $k\geq 0$. Hence the inclusion

[TABLE]

turns out to be a weak homotopy equivalence. See also [18] for discussions of the inclusion (1.1) for non-compact manifolds.

Moreover, let $\mathrm{Diff}^{+}(M)$ be the group of orientation preserving diffeomorphisms. Then we have an inclusion $i:\mathrm{Symp}(M,\omega)\subset\mathrm{Diff}^{+}(M)$. Indeed, if $h$ preserves $\omega$, then it fixes the corresponding cohomology class $[\omega]\in H^{2}(M,{\mathbb{R}})\cong{\mathbb{R}}$, and so yields the identity on $H^{2}(M,{\mathbb{R}})$. In particular, $h$ preserves orientation of $M$. Hence (1.1) also implies that $i$ yields a monomorphism $i_{0}:\pi_{0}\mathrm{Symp}(M,\omega)\to\pi_{0}\mathrm{Diff}^{+}(M)$ on the set of isotopy classes.

It is well known that $\pi_{0}\mathrm{Diff}^{+}(M)$ is generated by isotopy classes of Dehn twists, [4], [9], and one easily shows that each Dehn twist can be realized by $\omega$-preserving diffeomorphism. This implies that $i_{0}$ is also surjective, and so $i$ is a weak homotopy equivalence as well.

On the other hand, let $f:M\to{\mathbb{R}}$ be a Morse function,

[TABLE]

be the group of $f$-preserving diffeomorphisms, i.e. the stabilizer of $f$ with respect to the right action of $\mathrm{Diff}(M)$ on $C^{\infty}(M,{\mathbb{R}})$, and $\mathrm{Stab}_{0}(f)$ be its identity path component. Let also

[TABLE]

be the corresponding orbit of $f$,

[TABLE]

be the group of diffeomorphisms mutually preserving $f$ and $\omega$, and $\mathrm{Stab}_{0}(f,\omega)$ be its identity path component.

In a series of papers the author proved that $\mathrm{Stab}_{0}(f)$ is either contractible or homotopy equivalent to the circle and computed the higher homotopy groups of $\mathcal{O}(f)$, [11], [15]; showed that $\mathcal{O}(f)$ is homotopy equivalent to a finite-dimensional CW-complex, [12]; and recently described precise algebraic structure of the fundamental group $\pi_{1}\mathcal{O}(f)$, [17]. E. Kudryavtseva, [7], [8], studied the homotopy type of the space of Morse maps on compact surfaces and using similar ideas as in [11], [15] proved that $\mathcal{O}(f)$ has the homotopy type of a quotient of a torus by a free action of a certain finite group.

The present paper is former in a series subsequent ones devoted to extension of the above results to the case of $\omega$-preserving diffeomorphisms. We will describe here the homotopy type of $\mathrm{Stab}_{0}(f,\omega)$. In next papers will study the homotopy type of the subgroup of $\mathrm{Stab}(f,\omega)$ trivially acting on the Kronrod-Reeb graph of $f$, see §3.2, and describe the precise algebraic structure of $\pi_{0}\mathrm{Stab}(f,\omega)$.

Notice that if $H$ is the Hamiltonian vector field of $f$ and $\mathbf{H}:M\times{\mathbb{R}}\to M$ is the corresponding Hamiltonian flow, then $\mathbf{H}_{t}\in\mathrm{Stab}(f,\omega)$ for all $t\in{\mathbb{R}}$.

More generally, given a $C^{\infty}$ function $\alpha:M\to{\mathbb{R}}$, one can define the map

[TABLE]

being in general just a $C^{\infty}$ map leaving invariant each orbit of $H$, and so preserving $f$. However, $\mathbf{H}_{\alpha}$ is not necessarily a diffeomorphism.

Let $\mathcal{Z}(f)=\{\alpha\in C^{\infty}(M,{\mathbb{R}})\mid H(\alpha)=0\}$ be the algebra of all smooth functions taking constant values along orbits of $\mathbf{H}$. Equivalently, $\mathcal{Z}(f)$ is the centralizer of $f$ with respect to the Poisson bracket induced by $\omega$, see §2.3. In Lemma 3.2.1 we also identify $\mathcal{Z}(f)$ with a certain subset of continuous functions on the Kronrod-Reeb graph of $f$. In particular, $\mathcal{Z}(f)$ contains all constant functions.

We will prove in Theorem 3.0.1 that $\mathbf{H}_{\alpha}\in\mathrm{Stab}_{0}(f,\omega)$ if and only if $\alpha\in\mathcal{Z}(f)$. Moreover if $f$ has at least one saddle critical point, then the correspondence $\alpha\mapsto\mathbf{H}_{\alpha}$ is a homeomorphism $\mathcal{Z}(f)\cong\mathrm{Stab}_{0}(f,\omega)$ with respect to $C^{\infty}$ topologies, and so $\mathrm{Stab}_{0}(f,\omega)$ is contractible. Otherwise, that correspondence is an infinite cyclic covering map and $\mathrm{Stab}_{0}(f,\omega)$ is homotopy equivalent to the circle. It will also follow that the inclusion

[TABLE]

is a homotopy equivalence. This statement can be regarded as an analogue of (1.1) for $f$-preserving diffeomorphisms.

Again it implies that the inclusion

[TABLE]

yields an injection $j_{0}:\pi_{0}\mathrm{Stab}(f,\omega)\to\pi_{0}\mathrm{Stab}^{+}(f)$ on the sets of isotopy classes. However, now $j_{0}$ is not necessarily surjective, see §3.3. The reason is that $\mathrm{Stab}^{+}(f)$ has many invariant subsets, e.g.​ the sets of the from $M_{a}=f^{-1}(-\infty,a]$, $a\in{\mathbb{R}}$, and so if $h\in\mathrm{Stab}(f,\omega)$ interchanges connected components of $M_{a}$, then they must have the same $\omega$-volume.

In fact, our results hold for a larger class of smooth maps $f$ from $M$ into ${\mathbb{R}}$ and $S^{1}$, see §2.4. On the other hand, we also provide in §3.1 an example of a function with isolated critical points for which the above correspondence $\alpha\mapsto\mathbf{H}_{\alpha}$ is not surjective.

The author is indebted to Bogdan Feshchenko for useful discussions.

2. Preliminaries

2.1. Shift map

Let $M$ be a connected $n$-dimensional $C^{\infty}$ manifold, $H$ be a $C^{\infty}$ vector field tangent to $\partial M$ and generating a flow $\mathbf{H}:M\times{\mathbb{R}}\to M$. For each $\alpha\in C^{\infty}(M,{\mathbb{R}})$ define the following $C^{\infty}$ map $\mathbf{H}_{\alpha}:M\to M$ by

[TABLE]

for $x\in M$. Evidently, $\mathbf{H}_{\alpha}$ leaves invariant each orbit of $\mathbf{H}$ and is homotopic to $\mathrm{id}_{M}$ by the homotopy $\{\mathbf{H}_{t\alpha}\}_{t\in[0,1]}$. Also notice that if $\alpha\equiv t$ is a constant function, then $\mathbf{H}_{\alpha}=\mathbf{H}_{t}$ is a diffeomorphism belonging to the flow $\mathbf{H}$.

For $\alpha\in C^{\infty}(M,{\mathbb{R}})$ we will denote by $H(\alpha)$ the Lie derivative of $\alpha$ along $H$.

2.1.1 Lemma**.**

[10, Theorem 19]* Let $\alpha\in C^{\infty}(M,{\mathbb{R}})$, $y\in M$, and $z=\mathbf{H}_{\alpha}(y)$. Then the tangent map $T_{y}\mathbf{H}_{\alpha}:T_{y}M\to T_{z}M$ is an isomorphism if and only if $1+H(\alpha)(y)\not=0$.*

2.1.2 Remark**.**

In fact, [10, Lemma 20], if $\alpha(y)=0$, so $z=\mathbf{H}_{\alpha}(y)=\mathbf{H}(y,0)=y$ is a fixed point of $\mathbf{H}_{\alpha}$, then the determinant of $T_{y}\mathbf{H}_{\alpha}:T_{y}M\to T_{y}M$ does not depend on a particular choice of local coordinates at $z$ and equals $1+H(\alpha)(y)$. The general case $\alpha(y)=a\not=0$ reduces to $a=0$ by observation that $\mathbf{H}_{\alpha}=\mathbf{H}_{\alpha-a}\circ\mathbf{H}_{a}$.

To get a global variant of Lemma 2.1.1 notice that the correspondence $\alpha\mapsto\mathbf{H}_{\alpha}$ can also be regarded as the following mapping

[TABLE]

It will be called the shift map along orbits of $\mathbf{H}$, [10], [16]. Consider the following subset of $C^{\infty}(M,{\mathbb{R}})$:

[TABLE]

and let $\mathrm{Diff}_{0}(H)$ be the group of all diffeomorphisms of $M$ which leave invariant each orbit of $H$ and isotopic to the identity via an orbit preserving isotopy.

2.1.3 Lemma**.**

[10, Theorem 19]* If $M$ is compact, then*

[TABLE]

In other words, suppose $\alpha\in C^{\infty}(M,{\mathbb{R}})$. Then $\alpha\in\Gamma$ if and only if $\mathbf{H}_{\alpha}\in\mathrm{Diff}_{0}(H)$.

2.2. Hamiltonian vector field

Let $M$ be a compact orientable surface equipped with a volume form $\omega$ and $P$ be either ${\mathbb{R}}$ or $S^{1}$. Since $\dim M=2$, $\omega$ is a closed $2$-form, and therefore it defines a symplectic structure on $M$. Then for each $C^{1}$ map $f:M\to P$ there exists a unique vector field $H$ on $M$ satisfying

[TABLE]

for each point $z\in M$ and a tangent vector $u\in T_{z}M$. This vector field is called the Hamiltonian vector field of $f$ with respect to $\omega$. For the convenience of the reader we recall its construction as it is usually defined for functions $f:M\to{\mathbb{R}}$ only.

Let $z\in M$. Fix local charts $h:U\to M$ and $q:J\to P$ at $z$ and $f(z)$ respectively, where $U$ is an open subset of the upper half-plane ${\mathbb{R}}_{+}^{2}=\{(x,y)\mid y\geq 0\}$ and $J$ is an open interval in ${\mathbb{R}}$. Decreasing $U$ one can assume that $f(h(U))\subset q(J)$. Then the map $\hat{f}=q^{-1}\circ f\circ h:U\to J$ is called a local representation of $f$ at $z$.

Now if in coordinates $(x,y)$ on $U$ we have that $\omega(x,y)=\gamma(x,y)dx\wedge dy$ for some non-zero $C^{\infty}$ function $\gamma:U\to{\mathbb{R}}\setminus\{0\}$, then

[TABLE]

A definition of $H$ that does not use local coordinates can be given as follows. Since the restriction of $\omega$ to each tangent space $T_{x}M$ is a non-degenerate skew-symmetric form, it follows that $\omega$ yields a bundle isomorphism

[TABLE]

defined by the formula $\psi(u)(v)=\omega(u,v)$ for all $u,v\in T_{x}M$ and $x\in M$.

Further notice, that the tangent bundle of $P$ is trivial, so we have the unit section

[TABLE]

Now for a $C^{1}$ map $f:M\to P$ its differential $df:M\to T^{*}M$ and the Hamiltonian vector field $H:M\to TM$ are unique maps for which the following diagram is commutative:

[TABLE]

Thus $df=T^{*}f\circ s\circ f$, and $H=\psi^{-1}\circ df$. It follows that

[TABLE]

as $\omega$ is skew-symmetric, and so $H$* is tangent to level curves of $f$*.

Suppose, in addition, that $f$ takes constant values at boundary components of $M$. Then, due to (2.5), $H$ is tangent to $\partial M$, and therefore it yields a flow $\mathbf{H}:M\times{\mathbb{R}}\to M$. It also follows from (2.5) that each diffeomorphism $\mathbf{H}_{t}:M\to M$ preserves $f$, in the sense that $f\circ\mathbf{H}_{t}=f$. Moreover, the well known Liouville’s theorem claims that each diffeomorphism $\mathbf{H}_{t}$ also preserves $\omega$. In fact, that theorem is a simple consequence of Cartan’s identity:

[TABLE]

since $\iota_{H}\omega=\omega(H,\cdot)=df$ by (2.3), and $d\omega=0$ as $\dim\omega=\dim M$.

2.3. Poisson multiplication

Let $Q$ be another one-dimensional manifold without boundary, so $Q$ is either ${\mathbb{R}}$ or $S^{1}$ as well as $P$. Then $\omega$ yields a Poisson multiplication

[TABLE]

defined by one of the following equivalent formulas:

[TABLE]

where $H_{f}$ and $H_{g}$ are Hamiltonian vector fields of $f\in C^{\infty}(M,P)$ and $g\in C^{\infty}(M,Q)$ respectively.

In particular, for each $f\in C^{\infty}(M,P)$ one can define its annulator with respect to (2.8) by

[TABLE]

Thus $\mathcal{Z}_{\omega}^{Q}(f)$ consists of all maps $g\in C^{\infty}(M,Q)$ taking constant values along orbits of the Hamiltonian vector field $H_{f}$. It follows from (2.8) that $g\in\mathcal{Z}_{\omega}^{Q}(f)$ iff $f\in\mathcal{Z}_{\omega}^{P}(g)$.

When $P=Q={\mathbb{R}}$, this multiplication is the usual Poisson bracket, and $\mathcal{Z}_{\omega}^{{\mathbb{R}}}(f)$ is the centralizer of $f$, see [19, §3].

2.4. Class ${\mathcal{F}}(M,P)$

Let ${\mathcal{F}}(M,P)$ be the subspace of $C^{\infty}(M,P)$ consisting of maps $f$ satisfying the following two axioms:

Axiom (B)*

The map $f$ takes a constant value at each connected component of $\partial M$ and has no critical points on $\partial M$. *

Axiom (L)*

For every critical point $z$ of $f$ there is a local presentation $\hat{f}:{\mathbb{R}}^{2}\to{\mathbb{R}}$ of $f$ near $z$ in which $\hat{f}$ is a homogeneous polynomial ${\mathbb{R}}^{2}\to{\mathbb{R}}$ without multiple factors. *

In particular, since the polynomial $\pm x^{2}\pm y^{2}$ (a non-degenerate singularity) is homogeneous and has no multiple factors, we see that ${\mathcal{F}}(M,P)$ contains an open and everywhere dense subset $\mathrm{Morse}(M,P)$ consisting of maps satisfying Axiom (B) and having non-degenerate critical points only.

Figure 2.1 describes possible singularities satisfying Axiom (L).

2.4.1 Definition**.**

We will say that a vector field $F$ on $M$ is Hamiltonian like for $f\in{\mathcal{F}}(M,P)$ if

1. $(a)$

$F(f)=0$, and, in particular, $F$ is tangent to $\partial M$ and generates a flow on $M$; 2. $(b)$

$F(z)=0$* if and only if $z$ is a critical point of $f$;* 3. $(c)$

for each $z$ critical point of $f$ there exists a local representation $\hat{f}:{\mathbb{R}}^{2}\to{\mathbb{R}}$ of $f$ as a homogeneous polynomial without multiple factors such that in these coordinates $F(x,y)=-\hat{f}^{\prime}_{y}\tfrac{\partial}{\partial x}+\hat{f}^{\prime}_{x}\tfrac{\partial}{\partial y}$.

One can easily prove that for each $f\in{\mathcal{F}}(M,P)$ there exists a Hamiltonian like vector field, [11, Lemma 5.1].

Notice also that every Hamiltonian vector field $H$ of $f$ has properties $(a)$ and $(b)$ of Definition 2.4.1. Moreover, if $H$ is also a Hamiltonian like, then due to (2.4) in the corresponding coordinates satisfying property $(c)$ of Definition 2.4.1 we have that $\omega=dx\wedge dy$.

2.4.2 Lemma**.**

Let $F$ be any Hamiltonian like vector field for $f\in{\mathcal{F}}(M,P)$, and $H$ be the Hamiltonian vector field for $f$ with respect to $\omega$. Then there exists an everywhere non-zero $C^{\infty}$ function $\lambda:M\to{\mathbb{R}}\setminus\{0\}$ such that $H=\lambda F$.

Proof.

Denote by $\Sigma_{f}$ the set of critical point of $f$, being also the set of zeros of $H$ as well as of $F$. Since $F$ and $H$ are parallel and non-zero on $M\setminus\Sigma_{f}$, it follows that there exists a $C^{\infty}$ non-zero function $\lambda:M\setminus\Sigma_{f}\to{\mathbb{R}}$ such that $H=\lambda F$. It remains to show that $\lambda$ can be defined by non-zero values on $\Sigma_{f}$ to give a $C^{\infty}$ function on all of $M$.

Let $z$ be a critical point of $f$. Then by definition of Hamiltonian like vector field there exists a local representation $\hat{f}:{\mathbb{R}}^{2}\to{\mathbb{R}}$ of $f$ such that $z=(0,0)\in{\mathbb{R}}^{2}$, $\hat{f}$ is a homogeneous polynomial without multiple factors, and $F=-\hat{f}^{\prime}_{y}\tfrac{\partial}{\partial x}+\hat{f}^{\prime}_{x}\tfrac{\partial}{\partial y}$.

Then $\omega(x,y)=\gamma(x,y)dx\wedge dy$ for some non-zero $C^{\infty}$ function $\gamma$, and by formula (2.4), we have $H=\tfrac{1}{\gamma}F$ on $U\setminus z$. Hence $\lambda=1/\gamma$, and so $\lambda$ smoothly extends to all of $U$ by $\lambda(z)=1/\gamma(z)$. ∎

The following statement is a particular case of results of [13] on parameter rigidity.

2.4.3 Corollary**.**

c.f. [13, §4 & Theorem 11.1] For any two Hamiltonian like vector fields $F_{1}$ and $F_{2}$ there exists an everywhere non-zero $C^{\infty}$ function $\mu:M\to{\mathbb{R}}\setminus\{0\}$ such that $F_{1}=\mu F_{2}$.

Proof.

It follows from Lemma 2.4.2 that $H=\lambda_{1}F_{1}=\lambda_{2}F_{2}$ for some everywhere non-zero $C^{\infty}$ functions $\lambda_{1},\lambda_{2}:M\to{\mathbb{R}}\setminus\{0\}$. Hence $\mu=\lambda_{2}/\lambda_{1}$. ∎

2.5. Topological type of $\mathrm{Stab}_{0}(f)$

Let $f\in{\mathcal{F}}(M,P)$, $H$ be a Hamiltonian like vector field for $f$, and $\mathbf{H}:M\times{\mathbb{R}}\to M$ be the corresponding Hamiltonian flow.

2.5.1 Theorem**.**

[14], [15], [16]. Let $\varphi:C^{\infty}(M,{\mathbb{R}})\to C^{\infty}(M,M)$ be the shift map along orbits of $H$ and

[TABLE]

see (2.1).

1. (1)

$\varphi(\Gamma)=\mathrm{Stab}_{0}(f)$*  and  \Gamma=\varphi^{-1}\bigl{(}\mathrm{Stab}_{0}(f)\bigr{)}.* 2. (2)

Suppose all critical points of $f$ are non-degenerate local extremes, so, in particular, $f\in\mathrm{Morse}(M,P)$. Then the restriction map $\varphi|_{\Gamma}:\Gamma\to\mathrm{Stab}_{0}(f)$ is an infinite cyclic covering, and so $\mathrm{Stab}_{0}(f)$ is homotopy equivalent to the circle. More precisely, in this case there exists $\theta\in\Gamma$ such that

1. (i)

$\theta>0$* on all of $M$;* 2. (ii)

each non-constant orbit $\gamma$ of $F$ is periodic, and $\theta$ takes a constant value on $\gamma$ being an positive integral multiple of the period $\mathrm{Per}(\gamma)$ of $\gamma$; 3. (iii)

there exists a free action of ${\mathbb{Z}}$ on $\Gamma$ defined by $n*\alpha=\alpha+n\theta$, for $n\in{\mathbb{Z}}$ and $\alpha\in\Gamma$, such that the map $\varphi$ is a composite

[TABLE]

where $p$ is a projection onto the factor space $\Gamma/{\mathbb{Z}}$ endowed with the corresponding final topology, and $r$ is a homeomorphism. 3. (3)

Suppose $f$ has a critical point being not a non-degenerate local extreme. Then $\varphi|_{\Gamma}:\Gamma\to\mathrm{Stab}_{0}(f)$ is a homeomorphism, and so $\mathrm{Stab}_{0}(f)$ is contractible.

Proof.

In fact, Theorem 2.5.1 is stated and proved in [15] for any Hamiltonian like vector field $F$ of $f$. The advantage of using Hamiltonian like vector fields is that we have precise formulas for $F$ near critical points of $f$.

Let $\lambda:M\to{\mathbb{R}}$ be every where non-zero $C^{\infty}$ function and $H=\lambda F$. We will deduce from results of [14] that Theorem 2.5.1 also holds for $H$. Due to Lemma 2.4.2 this includes the case when $H$ is Hamiltonian.

Let $\mathbf{F},\mathbf{H}:M\times{\mathbb{R}}\to M$ be the flows of $F$ and $H=\lambda F$ respectively,

[TABLE]

be the corresponding shift maps, $\mathrm{image}(\varphi_{H})$, $\mathrm{image}(\varphi_{F})$ be their images in $C^{\infty}(M,M)$, and $\Gamma_{F}$, $\Gamma_{H}$ be corresponding the subsets of $C^{\infty}(M,{\mathbb{R}})$ defined by (2.1). Define the following $C^{\infty}$ function

[TABLE]

Then it is well known and easy to see, e.g. [14], that for each $\alpha\in C^{\infty}(M,{\mathbb{R}})$ we have that

[TABLE]

Consider the map

[TABLE]

Evidently, $\gamma$ is continuous with respect to $C^{\infty}$ topologies. Moreover, (2.11) means that $\mathbf{H}_{\alpha}=\mathbf{F}_{\gamma(\alpha)}$ for all $\alpha\in C^{\infty}(M,{\mathbb{R}})$. Hence $\varphi_{H}=\varphi_{F}\circ\gamma$, $\mathrm{image}(\varphi_{H})\subset\mathrm{image}(\varphi_{F})$, and we get the following commutative diagram:

[TABLE]

Since $\lambda\not=0$ everywhere, one can interchange $F=\tfrac{1}{\lambda}H$ and $H$. Hence by the same arguments as above we get that $\mathrm{image}(\varphi_{H})=\mathrm{image}(\varphi_{F})$ and $\gamma$ is a homeomorphism. Also notice that the orbit structures of $F$ and $H$ coincide. Hence $\mathrm{Diff}_{0}(F)=\mathrm{Diff}_{0}(H)$, and so

[TABLE]

Thus $\gamma$ yields a homeomorphism of $\Gamma_{H}$ onto $\Gamma_{F}$. Since Theorem 2.5.1 holds for $F$, we get the following commutative diagram

[TABLE]

implying that $\varphi_{H}|_{\Gamma_{H}}$ has the same topological properties (1)-(3) as $\varphi_{F}|_{\Gamma_{F}}$, and so Theorem 2.5.1 holds for $H$ as well. ∎

2.5.2 Remark**.**

Let us discus the case (2) of Theorem 2.5.1 which is realized precisely for the following four types of Morse maps, see [11, Theorem 1.9]:

1. (A)

$M=S^{2}$ is a $2$-sphere and $f:S^{2}\to P$ has exactly two critical points: non-degenerate local minimum and maximum; 2. (B)

$M=D^{2}$ is a $2$-disk and $f:D^{2}\to P$ has exactly one critical point being a non-degenerate local extreme; 3. (C)

$M=S^{1}\times[0,1]$ is a cylinder and $f:S^{1}\times[0,1]\to P$ has no critical points; 4. (D)

$M=T^{2}$ is a $2$-torus, $P=S^{1}$ is a circle, and $f:T^{2}\to P$ has no critical points.

Due to (2)(i) and (2)(ii) each regular point $x\in M$ of $f$ is periodic of some period $\mathrm{Per}(x)$, and there exists $k_{x}\in{\mathbb{N}}$ depending on $x$ such that $\theta(x)=k_{x}\mathrm{Per}(x)$. Hence

[TABLE]

and so $\mathbf{H}_{\theta}=\mathrm{id}_{M}$. Moreover, if $\alpha\in\Gamma$, then

[TABLE]

This implies correctness of the ${\mathbb{Z}}$-action from (2)(iii) of Theorem 2.5.1 and existence of decomposition (2.10) with continuous $p$ and $r$. The principal difficulty was to prove that $r$ is a homeomorphism.

The aim of the present paper is to deduce from Theorem 2.5.1 a description of the homotopy type of $\mathrm{Stab}_{0}(f,\omega)$, see Theorem 3.0.1 below.

3. Main result

Let $M$ be a compact orientable surface equipped with a volume form $\omega$, $f\in{\mathcal{F}}(M,P)$, $H$ be the Hamiltonian vector field of $f$ with respect to $\omega$, $\mathbf{H}:M\times{\mathbb{R}}\to M$ be the corresponding Hamiltonian flow, and

[TABLE]

be the shift map along orbits of $\mathbf{H}$. Let also

[TABLE]

be the space of functions taking constant values along orbits of $H$, see (2.9). Then $\mathcal{Z}$ is a linear subspace of $C^{\infty}(M,{\mathbb{R}})$ and is contained in $\Gamma$, see (2.1). In particular, $\mathcal{Z}$ is contractible as well as $\Gamma$.

3.0.1 Theorem**.**

The following statements hold true.

1. $(1)$

$\varphi(\mathcal{Z})=\mathrm{Stab}_{0}(f,\omega)=\mathrm{Stab}_{0}(f)\cap\mathrm{Stab}(f,\omega)$* and \mathcal{Z}=\varphi^{-1}\bigl{(}\mathrm{Stab}_{0}(f,\omega)\bigr{)}.* 2. $(2)$

If all critical points of $f$ are non-degenerate local extremes, then the restriction $\varphi|_{\mathcal{Z}}:\mathcal{Z}\to\mathrm{Stab}_{0}(f,\omega)$ is an infinite cyclic covering, and $\mathrm{Stab}_{0}(f,\omega)$ is homotopy equivalent to the circle. 3. $(3)$

Otherwise, $\varphi|_{\mathcal{Z}}:\mathcal{Z}\to\mathrm{Stab}_{0}(f,\omega)$ is a homeomorphism, and so $\mathrm{Stab}_{0}(f,\omega)$ is contractible. 4. $(4)$

The inclusion $\mathrm{Stab}_{0}(f,\omega)\subset\mathrm{Stab}_{0}(f)$ is a homotopy equivalence. 5. $(5)$

The inclusion map $j:\mathrm{Stab}(f,\omega)\subset\mathrm{Stab}(f)$ induces an injection $j_{0}:\pi_{0}\mathrm{Stab}(f,\omega)\to\pi_{0}\mathrm{Stab}(f)$.

Proof.

First we need the following lemma.

3.0.2 Lemma**.**

Let $\alpha\in\mathcal{C}^{\infty}(M,{\mathbb{R}})$. Then the action of $\mathbf{H}_{\alpha}$ on $\omega$ is given by

[TABLE]

Proof.

Since the set of critical points is finite and so nowhere dense, it suffices to check this relation at regular points of $f$ only.

So let $p$ be a regular point of $f$. Then $H(p)\not=0$, whence there are local coordinates $(x,y)$ at $p$ in which $p=(0,0)$, $H(x,y)=\tfrac{\partial}{\partial x}$, and $\mathbf{H}(x,y,t)=(x+t,y)$ for sufficiently small $x,y,t$. In particular, $H(\alpha)=\tfrac{\partial\alpha}{\partial x}$. We also have that $\omega(x,y)=\gamma(x,y)dx\wedge dy$ for some $C^{\infty}$ function $\gamma$.

Notice that one may also assume that $\alpha(p)=0$. Indeed, let $b=\alpha(p)$. Then $\mathbf{H}_{\alpha}=\mathbf{H}_{\alpha-b}\circ\mathbf{H}_{b}$. Since $\mathbf{H}_{b}$ preserves $\omega$, see (2.6), it follows that

[TABLE]

Thus suppose $\alpha(p)=0$, whence $\mathbf{H}_{\alpha}(p)=p$. Then

[TABLE]

In particular, at $p$ we have that

[TABLE]

which proves (3.1). ∎

Now we can complete Theorem 3.0.1.

$(1)$ Let us check that

[TABLE]

Let $\alpha\in\mathcal{Z}$. As $\mathbf{H}_{\alpha}$ leaves invariant each orbit of $H$, and therefore it preserves $f$, we have that $\mathbf{H}_{\alpha}\in\mathrm{Stab}(f)$.

Moreover, by formula (3.1), $\mathbf{H}_{\alpha}^{*}\omega=\omega$, so $\mathbf{H}_{\alpha}\in\mathrm{Stab}(f,\omega)$.

Now notice that $t\alpha\in\mathcal{Z}$ for all $t\in{\mathbb{R}}$, and so $\mathbf{H}_{t\alpha}\in\mathrm{Stab}(f,\omega)$ as well. Thus the homotopy $\mathbf{H}_{t\alpha}:M\to M$, $t\in[0,1]$, is in fact an isotopy in $\mathrm{Stab}(f,\omega)$ between $\mathrm{id}_{M}=\mathbf{H}_{0}$ and $\mathbf{H}_{\alpha}$. Hence $\mathbf{H}_{\alpha}\in\mathrm{Stab}_{0}(f,\omega)$.

Further we claim that

[TABLE]

Indeed, let $h\in\mathrm{Stab}_{0}(f)\cap\mathrm{Stab}(f,\omega)$. Then by (1) of Theorem 2.5.1 $h=\mathbf{H}_{\alpha}$ for some $\alpha\in\Gamma$. As $\omega$ is everywhere non-zero on $M$, it follows from formula (3.1) that $H(\alpha)=0$ on all of $M$, that is $\alpha\in\mathcal{Z}$.

Hence

[TABLE]

[TABLE]

This proves $(1)$.

$(2)$, $(4)$ Suppose $\varphi:\Gamma\to\mathrm{Stab}_{0}(f)$ is an infinite cyclic covering map, and let $\theta\in\Gamma$ be the function from (2) of Theorem 2.5.1.

Then due to property (2)(ii) in Theorem 2.5.1 $\theta$ takes constant values along orbits of $H$, and therefore $\theta\in\mathcal{Z}$. Since, in addition, $\mathcal{Z}$ is a group, it follows that $\mathcal{Z}$ is invariant with respect to the ${\mathbb{Z}}$-action on $\Gamma$, i.e. $\alpha+n\theta\in\mathcal{Z}$ for all $\alpha\in\mathcal{Z}$. Therefore $\mathcal{Z}=\varphi^{-1}(\mathrm{Stab}_{0}(f,\omega))$. Hence $\varphi|_{\mathcal{Z}}:\mathcal{Z}\to\mathrm{Stab}_{0}(f,\omega)$ is a ${\mathbb{Z}}$-covering as well as $\varphi|_{\Gamma}$. As $\mathcal{Z}$ is contractible, we obtain that the quotient $\mathrm{Stab}_{0}(f,\omega)$ is homotopy equivalent to the circle.

Consider the following path $\tau:[0,1]\to\mathcal{Z}\subset\Gamma$, $\tau(t)=t\theta$. Then $\varphi\circ\tau$ is a loop in $\mathrm{Stab}_{0}(f,\omega)\subset\mathrm{Stab}_{0}(f)$, since

[TABLE]

This loop is a generator of $\pi_{1}\mathrm{Stab}_{0}(f,\omega)\cong{\mathbb{Z}}$ as well as a generator of $\pi_{1}\mathrm{Stab}_{0}(f)\cong{\mathbb{Z}}$. Hence the inclusion $j:\mathrm{Stab}_{0}(f,\omega)\subset\mathrm{Stab}_{0}(f)$ yields an isomorphism of fundamental groups. Since these spaces homotopy equivalent to the circle, we obtain that $j$ is a homotopy equivalence.

$(3)$, $(4)$ If $\varphi:\Gamma\to\mathrm{Stab}_{0}(f)$ is a homeomorphism, then due to $(1)$ it yields a homeomorphism of $\mathcal{Z}$ onto $\mathrm{Stab}_{0}(f,\omega)$. In particular, both $\mathrm{Stab}_{0}(f,\omega)$ and $\mathrm{Stab}_{0}(f)$ are contractible, and so the inclusion $\mathrm{Stab}_{0}(f,\omega)\subset\mathrm{Stab}_{0}(f)$ is a homotopy equivalence.

$(5)$ Injectivity of $j_{0}$ follows from the relation $\mathrm{Stab}_{0}(f,\omega)=\mathrm{Stab}_{0}(f)\cap\mathrm{Stab}(f,\omega)$. Theorem 3.0.1 is completed. ∎

3.0.3 Remark**.**

Though the inclusion $\mathrm{Stab}_{0}(f,\omega)\subset\mathrm{Stab}_{0}(f)$ is a homotopy equivalence, it seems to be a highly non-trivial task to find precise formulas for a strong deformation retraction of $\mathrm{Stab}_{0}(f)$ onto $\mathrm{Stab}_{0}(f,\omega)$. For the case $(3)$ of Theorem 3.0.1 this is equivalent to a construction of a strong deformation retraction of $\Gamma$ onto $\mathcal{Z}$. In fact, it suffices to find a retraction $r:\Gamma\to\mathcal{Z}$, so to associate to each $\alpha\in\Gamma$ a function $r(\alpha)$ taking constant values along orbits of $H$ so that each $\beta\in\mathcal{Z}$ remains unchanged. Then a strong deformation $r_{t}:\Gamma\to\mathcal{Z}$, $t\in[0,1]$, of $\Gamma$ onto $\mathcal{Z}$ can be given by $r_{t}(\alpha)=(1-t)\alpha+tr(\alpha)$.

3.1. Counterexample for maps $g\not\in{\mathcal{F}}(M,P)$

Let $D^{2}=\{|z|\leq 1\}$ be the unit disk in the complex plane ${\mathbb{C}}$ and $\omega=dx\wedge dy$ be the standard symplectic form. Consider the following two functions $f,g:D^{2}\to[0,1]$ defined by

[TABLE]

Then the foliations by level sets of $f$ and $g$ coincide, whence

[TABLE]

However, $f\in{\mathcal{F}}(D^{2},{\mathbb{R}})$, while $g$ does not belong to ${\mathcal{F}}(D^{2},{\mathbb{R}})$ since it is a polynomial with multiple factors.

Notice that the Hamiltonian vector fields $F$ and $G$ of $f$ and $g$ are given by

[TABLE]

In particular, the Hamiltonian flow $\mathbf{F}:D^{2}\times{\mathbb{R}}\to D^{2}$ of $f$ is given by $\mathbf{F}(z,t)=e^{2it}z$, and so the tangent map $T_{0}\mathbf{F}_{t}:T_{0}D^{2}\to T_{0}D^{2}$ is not the identity for $t\not=\pi n$, $n\in{\mathbb{Z}}$.

On the other hand, the linear part of $G$ at [math] vanishes, whence for the Hamiltonian flow $\mathbf{G}:D^{2}\times{\mathbb{R}}\to D^{2}$ of $g$ the corresponding tangent map $T_{0}\mathbf{G}_{t}:T_{0}D^{2}\to T_{0}D^{2}$ is always the identity. Hence for any $C^{\infty}$ function $\alpha$ the tangent map at [math] of $\mathbf{G}_{\alpha}$ is the identity as well. Therefore for $t\not=\pi n$, $n\in{\mathbb{Z}}$, then $\mathbf{F}_{t}\not=\mathbf{G}_{\alpha}$ for any $\alpha\in C^{\infty}(M,{\mathbb{R}})$.

By Theorem 3.0.1 the shift map $\varphi_{F}:\mathcal{Z}_{\omega}^{{\mathbb{R}}}(f)\to\mathrm{Stab}_{0}(f,\omega)$ of $F$ is an infinite cyclic covering, while the shift map

[TABLE]

of $G$ turns out to be not surjective, since its image does not contain $\mathbf{F}_{t}$ for $t\not=\pi n$, $n\in{\mathbb{Z}}$.

Thus we see that the centralizer of $g$ does not “detect” all the diffeomorphisms from $\mathrm{Stab}_{0}(g)$, while the centralizer of $f$ does so. This shows that the assumption $f\in{\mathcal{F}}(M,P)$ in Theorem 3.0.1 is essential.

3.2. Kronrod-Reeb graph of $f$

Now we will give an interpretation of $\mathcal{Z}$ in terms of functions on the Kronrod-Reeb graph of $f$.

Let $f\in{\mathcal{F}}(M,P)$. Consider the partition $\Delta$ of $M$ into connected components of level-sets of $f$. Let $K:=M/\Delta$ be the corresponding quotient space and $p:M\to K$ be the factor map. Then we have a natural decomposition

[TABLE]

Endow $K$ with the final topology, so a subset $U\subset K$ is open if and only if $p^{-1}(U)$ is open in $M$. Then it is well known that $K$ has a natural structure of a one-dimensional CW-complex. It is called a Lyapunov or Kronrod-Reeb graph of $f$, [1], [22], [6], [5], [3].

We will briefly recall the correspondence between elements of $\Delta$ (i.e. points of $K$) and orbits of $H$. Let $\gamma\in\Delta$. If $\gamma$ contains at least one critical point of $f$, then it follows from Axiom (L) that $\gamma$ is a connected 1-dimensional CW-complex such that each of its vertices has even (possibly zero) degree, and $p(\gamma)$ is a vertex of $K$. In this case the vertices of $\gamma$ are critical points of $f$ being also zeros of $H$, while edges of $\gamma$ are non-closed orbits of $H$.

If $\gamma$ has no critical point of $f$, then $\gamma$ is a closed orbit of $H$.

3.2.1 Lemma**.**

Each $\alpha\in\mathcal{Z}_{\omega}^{Q}(f)$ yields a unique continuous function $\widehat{\alpha}:K\to Q$ such that $\alpha=\widehat{\alpha}\circ p$. Moreover, the correspondence $\alpha\mapsto\widehat{\alpha}$ is a continuous injective map $\eta:\mathcal{Z}_{\omega}^{Q}(f)\to C(K,Q)$ with respect to $C^{0}$ topology on $C(K,Q)$.

Proof.

Let $\alpha\in\mathcal{Z}_{\omega}^{Q}(f)$, so $\alpha$ takes constant values along orbits of $H$. First we should show that $\alpha$ takes constant value at each element of $\Delta$.

Consider any element $\gamma\in\Delta$. If $\gamma$ contains no critical point of $f$, then $\gamma$ is a closed orbit of $H$, and so $\alpha$ takes a constant value at $\gamma$, see Figure 3.1.

Otherwise, $\gamma$ is a connected $1$-dimensional CW-complex whose vertices and edges are orbits of $H$. Then $\alpha$ takes constant values along edges of $\gamma$, and it follows from continuity of $\alpha$ and connectedness of $\gamma$ that $\alpha$ is constant on all of $\gamma$.

Thus $\alpha$ yields a unique function $\widehat{\alpha}:K\to Q$ such that $\alpha=\widehat{\alpha}\circ p$.

Since $K$ has final topology with respect to $p$ and $\alpha$ is continuous, it easily follows and is well known that $\widehat{\alpha}$ is continuous. Continuity of the correspondence $\alpha\mapsto\widehat{\alpha}$ is left for the reader. ∎

Lemma 3.2.1 together with Theorem 3.0.1 implies that for $f\in{\mathcal{F}}(M,P)$ the elements of $\mathrm{Stab}_{0}(f,\omega)$ are parametrized by continuous functions on the Kronrod-Reeb graph $K$ of $f$, see Figure 3.1.

More precisely, due to Theorem 3.0.1 for each $h\in\mathrm{Stab}_{0}(f,\omega)$ there exists $\alpha\in\mathcal{Z}_{\omega}^{Q}(f)$ such that $h=\mathbf{H}_{\alpha}$. This function takes constant values on connected components of level-sets of $f$, and therefore induces a continuous function $\widehat{\alpha}:K\to{\mathbb{R}}$. Then the value of $\widehat{\alpha}$ at some point $v\in K$ equals to the common time shift induced by $h$ on all the orbits of $H$ constituting $p^{-1}(v)$.

3.3. Non-surjectivity of the map $j_{0}:\pi_{0}\mathrm{Stab}(f,\omega)\to\pi_{0}\mathrm{Stab}^{+}(f)$

Let $g:{\mathbb{R}}^{2}\to{\mathbb{R}}$ be the function defined by

[TABLE]

It has three critical points: one saddle $p_{0}=(0,0)$ and two local minimums $p_{1}=(-1,0)$ and $p_{2}=(1,0)$. Let $D=g^{-1}[0,2]$, and $f=g|_{D}:D\to{\mathbb{R}}$ be the restriction of $g$ to $D$. Then $D$ is a $2$-disk and $f$ belongs to the class ${\mathcal{F}}(D,{\mathbb{R}})$.

Consider the following subset $A=f^{-1}[0,0.5]\subset D$, see Figure 3.2. It consists of two connected components $A_{1}$ and $A_{2}$ containing the points $p_{1}$ and $p_{2}$ respectively.

Notice that $h(A)=A$ for each $h\in\mathrm{Stab}(f)$, whence $h$ either preserves both $A_{1}$ and $A_{2}$ or interchanges them. Also notice that if $h,k\in\mathrm{Stab}(f)$ and $h(A_{1})=A_{2}$, while $k(A_{1})=A_{1}$, then $h$ and $k$ belong to distinct path components of $\mathrm{Stab}(f)$.

Let $h:D\to D$ be a diffeomorphism defined by $D(x,y)=(-x,-y)$. Evidently, $h$ belongs to $\mathrm{Stab}^{+}(f)$ and interchanges $A_{1}$ and $A_{2}$.

3.3.1 Lemma**.**

Let $\omega$ be any volume form on $D$ such that $\mathrm{Vol}_{\omega}(A_{1})\not=\mathrm{Vol}_{\omega}(A_{2})$. Then the isotopy class $[h]\in\pi_{0}\mathrm{Stab}^{+}(f)$ of $h$ does not contain any $k\in\mathrm{Stab}(f,\omega)$. Hence for such an $\omega$ the map $j_{0}:\pi_{0}\mathrm{Stab}(f,\omega)\to\pi_{0}\mathrm{Stab}^{+}(f)$ is not surjective.

Proof.

Each $k\in\mathrm{Stab}(f,\omega)$ preserves $\omega$-volume. Since $\mathrm{Vol}_{\omega}(A_{1})\not=\mathrm{Vol}_{\omega}(A_{2})$, it follows that $k(A_{i})=A_{i}$ for $i=1,2$. But $h(A_{1})=A_{2}$, whence $h$ and $k$ are not isotopic in $\mathrm{Stab}(f)$. ∎