2.1 Admissible families of functions

Denoting the $\mathbb{R}$ -coordinate of $\mathbb{R}\times M$ by $\unicode[STIX]{x1D70F}$ , we consider the symplectization

$$\begin{eqnarray}(\mathbb{R}\times M,d(e^{\unicode[STIX]{x1D70F}}\unicode[STIX]{x1D706}_{0})).\end{eqnarray}$$

Definition 2.1. A smooth function $H:\mathbb{R}\times M\rightarrow \mathbb{R}$ is an admissible Hamiltonian if it is nonnegative, satisfies

(8) $$\begin{eqnarray}H(\unicode[STIX]{x1D70F},p)=0\quad \text{for all }\unicode[STIX]{x1D70F}\leqslant 0\end{eqnarray}$$

and if there is a $\mathbf{T}>0$ such that

(9) $$\begin{eqnarray}dH_{(\unicode[STIX]{x1D70F},p)}=0\quad \text{for all }\unicode[STIX]{x1D70F}\geqslant \mathbf{T}.\end{eqnarray}$$

The Hamiltonian vector field, $V_{H}$ , of $H$ is defined by the equation

$$\begin{eqnarray}i_{V_{H}}\unicode[STIX]{x1D714}=-dH.\end{eqnarray}$$

Conditions (8) and (9) imply that the support of $dH$ is compact and so the flow of $V_{H}$ is defined for all times.

Let $x(t)$ be a $1$ -periodic orbit of $H$ (i.e. $V_{H}$ ). We define the action of $x(t)$ by

$$\begin{eqnarray}\displaystyle {\mathcal{A}}_{H}(x) & = & \displaystyle -\int _{\mathbb{R}/\mathbb{Z}}x^{\ast }(e^{\unicode[STIX]{x1D70F}}\unicode[STIX]{x1D706}_{0})+\int _{\mathbb{R}/\mathbb{Z}}H(x(t))\,dt.\nonumber\end{eqnarray}$$

Let ${\mathcal{P}}^{-}(H)$ be the set of all $1$ -periodic orbits of $H$ which have negative action. Since $H$ is nonnegative, every $x$ in ${\mathcal{P}}^{-}(H)$ is nonconstant. We say that $H$ is nondegenerate if every $x$ in ${\mathcal{P}}^{-}(H)$ is also transversally nondegenerate. Strict nondegeneracy is impossible since $H$ is autonomous. In particular, every nonconstant $1$ -periodic orbit $x(t)$ of $H$ belongs to an $\mathbb{R}/\mathbb{Z}$ -family of such orbits, which we will denote by $X$ . The elements of $X$ all have the same action and so the notation ${\mathcal{A}}_{H}(X)$ can and will be used unambiguously. The collection of all $\mathbb{R}/\mathbb{Z}$ -families of $1$ -periodic orbits of $H$ with negative actions will be denoted by ${\mathcal{P}}_{\mathbb{R}/\mathbb{Z}}^{-}(H)$ .

The action gap of a nondegenerate admissible Hamiltonian $H$ is defined to be

$$\begin{eqnarray}\unicode[STIX]{x1D6E5}(H)=\sup _{X,Y\in {\mathcal{P}}_{\mathbb{R}/\mathbb{Z}}^{-}(H)}\{{\mathcal{A}}_{H}(X)-{\mathcal{A}}_{H}(Y):{\mathcal{A}}_{H}(X)\neq {\mathcal{A}}_{H}(Y)\},\end{eqnarray}$$

where we set $\unicode[STIX]{x1D6E5}(H)=0$ if there are no families $X$ and $Y$ meeting the required conditions (for example if $dH$ is sufficiently $C^{1}$ -small). Given two nondegenerate admissible Hamiltonians $H^{0}$ and $H^{1}$ , we set

$$\begin{eqnarray}\unicode[STIX]{x1D6E5}(H^{0},H^{1})=\sup \{{\mathcal{A}}_{H^{0}}(X^{0})-{\mathcal{A}}_{H^{1}}(X^{1}):{\mathcal{A}}_{H^{0}}(X^{0})\neq {\mathcal{A}}_{H^{1}}(X^{1})\},\end{eqnarray}$$

where here the supremum is over pairs $(X^{0},X^{1})\in {\mathcal{P}}_{\mathbb{R}/\mathbb{Z}}^{-}(H^{0})\times {\mathcal{P}}_{\mathbb{R}/\mathbb{Z}}^{-}(H^{1})$ and we again set $\unicode[STIX]{x1D6E5}(H^{0},H^{1})=0$ if no relevant pairs exist.

An admissible homotopy from $H^{0}$ to $H^{1}$ is a smooth family of functions $H^{s}$ for $s\in \mathbb{R}$ such that for some $\mathbf{S}>0$ and $\mathbf{T}>0$ we have:

( $\text{H}^{s}1$ )

$H^{s}=H^{0}$ for all $s\leqslant -\mathbf{S}$ ;

( $\text{H}^{s}2$ )

$H^{s}=H^{1}$ for all $s\geqslant \mathbf{S}$ ;

( $\text{H}^{s}3$ )

for all $s\in \mathbb{R}$ , the support of $d(H^{s})$ is contained in $[0,\mathbf{T}]\times M$ .

The cost of the homotopy $H^{s}$ is defined to be

$$\begin{eqnarray}\text{cost}(H^{s})=\int _{\mathbb{ R}}\max _{(\unicode[STIX]{x1D70F},p)}(\unicode[STIX]{x2202}_{s}(H^{s}(\unicode[STIX]{x1D70F},p)))\,ds.\end{eqnarray}$$

Finally, an admissible homotopy of homotopies between $H^{0}$ and $H^{1}$ is a smooth $\mathbb{R}^{2}$ -family of functions $H^{r,s}$ such that for each $r\in \mathbb{R}$ , $H^{r,s}$ is an admissible homotopy from $H^{0}$ to $H^{1}$ and the supports of all the $d(H^{r,s})$ are contained in a single neighborhood of the form $[0,\mathbf{T}]\times M$ . The cost of $H^{r,s}$ is defined as

$$\begin{eqnarray}\text{cost}(H^{r,s})=\int _{\mathbb{ R}}\max _{\substack{ (\unicode[STIX]{x1D70F},p,r)}}(\unicode[STIX]{x2202}_{s}(H^{r,s}(\unicode[STIX]{x1D70F},p)))\,ds.\end{eqnarray}$$

2.2 Dividing and tuned Hamiltonians

Let $H$ be an admissible Hamiltonian of the form

$$\begin{eqnarray}H(\unicode[STIX]{x1D70F},p)=h(e^{\unicode[STIX]{x1D70F}}).\end{eqnarray}$$

We will refer to $h$ as the profile of $H$ . The Hamiltonian vector field of $H$ is

$$\begin{eqnarray}V_{H}(\unicode[STIX]{x1D70F},p)=h^{\prime }(e^{\unicode[STIX]{x1D70F}})R_{\unicode[STIX]{x1D706}_{0}}(p).\end{eqnarray}$$

Thus, every nonconstant $1$ -periodic orbit $x(t)$ of $V_{H}$ corresponds to a unique closed Reeb orbit $\unicode[STIX]{x1D6FE}_{x}(t)$ of $\unicode[STIX]{x1D706}_{0}$ such that

(10) $$\begin{eqnarray}x(t)=(\unicode[STIX]{x1D70F}_{x},\unicode[STIX]{x1D6FE}_{x}(h^{\prime }(e^{\unicode[STIX]{x1D70F}_{x}})t)).\end{eqnarray}$$

In particular, $h^{\prime }(e^{\unicode[STIX]{x1D70F}_{x}})$ is the period of the orbit $\unicode[STIX]{x1D6FE}_{x}$ , which we will also denote by $T_{\unicode[STIX]{x1D6FE}_{x}}$ . The action of $x(t)$ is

(11) $$\begin{eqnarray}{\mathcal{A}}_{H}(x)=-e^{\unicode[STIX]{x1D70F}_{x}}h^{\prime }(e^{\unicode[STIX]{x1D70F}_{x}})+h(e^{\unicode[STIX]{x1D70F}_{x}}).\end{eqnarray}$$

Note also that the $\mathbb{R}/\mathbb{Z}$ -family of $1$ -periodic orbits of $V_{H}$ containing $x$ , $X$ corresponds to the unique $\mathbb{R}/\mathbb{Z}$ -family of closed Reeb orbits of $\unicode[STIX]{x1D706}_{0}$ containing $\unicode[STIX]{x1D6FE}_{x}$ . Denoting this family of Reeb orbits by $\unicode[STIX]{x1D6E4}_{X}$ , we have

$$\begin{eqnarray}\unicode[STIX]{x1D6E4}_{X}=\mathop{\bigcup }_{x\in X}\unicode[STIX]{x1D6FE}_{x}.\end{eqnarray}$$

We now define a class of Hamiltonians which have useful collections of $1$ -periodic orbits (related to useful collections of closed Reeb orbits) that can be identified simply by the fact that their actions are negative. For positive constants $a,b>0$ and $c>1+a$ , let $\mathfrak{h}_{a,b,c}$ be the space of smooth profile functions $h:\mathbb{R}\rightarrow \mathbb{R}$ with the following properties:

1. (h1) $h(s)=0\text{ for }s\leqslant 1$ ;

2. (h2) $h^{\prime \prime }(s)>0$ for $s\in (1,1+a)$ ;

3. (h3) $h(1+a)=a^{2}$ ;

4. (h4) $h^{\prime }(s)=b$ for $s\in [1+a,c]$ ;

5. (h5) $h^{\prime \prime }(s)<0$ for $s\in (c,c+a)$ ;

6. (h6) $h(s)=b(c-1-a)+a^{2}$ for $s\geqslant c+a$ .

Let ${\mathcal{R}}^{b}(\unicode[STIX]{x1D706}_{0})$ be the set of closed Reeb orbits of $\unicode[STIX]{x1D706}_{0}$ with period less than $b$ .

Lemma 2.2. Suppose that $H(\unicode[STIX]{x1D70F},p)=h(e^{\unicode[STIX]{x1D70F}})$ for some profile $h$ in $\mathfrak{h}_{a,b,c}$ . If $b$ is not the period of a closed Reeb orbit of $\unicode[STIX]{x1D706}_{0}$ and $c$ is sufficiently large, then every $x\in {\mathcal{P}}^{-}(H)$ is nonconstant and of the form

$$\begin{eqnarray}x=(\unicode[STIX]{x1D70F}_{x},\unicode[STIX]{x1D6FE}_{x}(h^{\prime }(e^{\unicode[STIX]{x1D70F}_{x}})t))\end{eqnarray}$$

for some $e^{\unicode[STIX]{x1D70F}_{x}}$ in $(1,1+a)$ and some $\unicode[STIX]{x1D6FE}_{x}$ in ${\mathcal{R}}^{b}(\unicode[STIX]{x1D706}_{0})$ . Moreover, the correspondence $x\rightarrow \unicode[STIX]{x1D6FE}_{x}$ defines a bijection between ${\mathcal{P}}^{-}(H)$ and ${\mathcal{R}}^{b}(\unicode[STIX]{x1D706}_{0})$ .

Proof. Every point $(\unicode[STIX]{x1D70F},p)$ in $\mathbb{R}\times M$ with $e^{\unicode[STIX]{x1D70F}}\leqslant 1$ or $e^{\unicode[STIX]{x1D70F}}\geqslant c+a$ corresponds to a constant periodic orbit of $H$ . The constant orbits in the first region have action ( $H$ -value) equal to zero by (h1). The orbits in the second region have strictly positive action ( $H$ -value) by conditions (h3)–(h6). Thus, neither set contributes to ${\mathcal{P}}^{-}(H)$ .

It follows from our choice of $b$ that the nonconstant periodic orbits of $H$ are of the form

$$\begin{eqnarray}x(t)=(\unicode[STIX]{x1D70F}_{x},\unicode[STIX]{x1D6FE}_{x}(h^{\prime }(e^{\unicode[STIX]{x1D70F}_{x}})t))\end{eqnarray}$$

for some $e^{\unicode[STIX]{x1D70F}_{x}}$ in either $(1,1+a)$ or $(c,c+a)$ . As mentioned above, the action of such an orbit is

(12) $$\begin{eqnarray}{\mathcal{A}}_{H}(x)=-e^{\unicode[STIX]{x1D70F}_{x}}h^{\prime }(e^{\unicode[STIX]{x1D70F}_{x}})+h(e^{\unicode[STIX]{x1D70F}_{x}}).\end{eqnarray}$$

These action values correspond to values of the function

$$\begin{eqnarray}F(s)=-sh^{\prime }(s)+h(s).\end{eqnarray}$$

By (h1), we have $F(1)=0$ and by (h2) we have $F^{\prime }(s)=-sh^{\prime \prime }(s)<0$ in $(1,1+a)$ . Thus, all the nonconstant periodic orbits $x$ with $e^{\unicode[STIX]{x1D70F}_{x}}$ in $(1,1+a)$ have negative action and so appear in ${\mathcal{P}}^{-}(H)$ .

On the other hand, given a closed Reeb orbit $\unicode[STIX]{x1D6FE}$ of $\unicode[STIX]{x1D706}_{0}$ with period $T_{\unicode[STIX]{x1D6FE}}<b$ , it follows from (h2) and (h4) that there is a unique solution $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FE}}$ of

$$\begin{eqnarray}h^{\prime }(e^{\unicode[STIX]{x1D70F}})=T_{\unicode[STIX]{x1D6FE}}\end{eqnarray}$$

contained in the interval $(1,1+a)$ . Then

$$\begin{eqnarray}x(t)=(\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FE}},\unicode[STIX]{x1D6FE}(T_{\unicode[STIX]{x1D6FE}}t))\end{eqnarray}$$

belongs to ${\mathcal{P}}^{-}(H)$ .

To complete the proof it just remains to show that a nonconstant periodic orbit $x$ as above with $e^{\unicode[STIX]{x1D70F}_{x}}$ in $(c,c+a)$ has positive action for all large enough values of $c$ . For such an $x$ , we have

$$\begin{eqnarray}{\mathcal{A}}_{H}(x)>-(c+a)h^{\prime }(e^{\unicode[STIX]{x1D70F}_{x}})+(c-1-a)b\end{eqnarray}$$

and thus

(13) $$\begin{eqnarray}{\mathcal{A}}_{H}(x)>c(b-T_{\unicode[STIX]{x1D6FE}_{x}})-b(2a+1).\end{eqnarray}$$

Since the period spectrum, ${\mathcal{T}}(\unicode[STIX]{x1D706}_{0})$ , of $\unicode[STIX]{x1D706}_{0}$ is closed and $b$ lies outside it, the quantity

$$\begin{eqnarray}b-\max \{t\in {\mathcal{T}}(\unicode[STIX]{x1D706}_{0}):t<b\}\end{eqnarray}$$

is positive. From (13), we then get

$$\begin{eqnarray}{\mathcal{A}}_{H}(x)>c(b-\max \{t\in {\mathcal{T}}(\unicode[STIX]{x1D706}_{0}):t<b\})-b(2a+1).\end{eqnarray}$$

Hence, for all sufficiently large $c>0$ , we have ${\mathcal{A}}_{H}(x)>0$ for all $x\in \text{Crit}({\mathcal{A}}_{H})$ with $e^{\unicode[STIX]{x1D70F}_{x}}\in (c,c+a)$ , as desired.◻

Let ${\mathcal{H}}_{a,b,c,\unicode[STIX]{x1D705}}$ be the space of smooth functions of the form

$$\begin{eqnarray}H(\unicode[STIX]{x1D70F},p)=h(e^{\unicode[STIX]{x1D70F}-\unicode[STIX]{x1D705}}),\end{eqnarray}$$

where $\unicode[STIX]{x1D705}\geqslant 0$ and $h$ is in $\mathfrak{h}_{a,b,c}$ . For an $H$ in ${\mathcal{H}}_{a,b,c,\unicode[STIX]{x1D705}}$ , we have

$$\begin{eqnarray}V_{H}(\unicode[STIX]{x1D70F},p)=h^{\prime }(e^{\unicode[STIX]{x1D70F}-\unicode[STIX]{x1D705}})e^{-\unicode[STIX]{x1D705}}R_{\unicode[STIX]{x1D706}_{0}}(p)\end{eqnarray}$$

and for every nonconstant $1$ -periodic orbit $x(t)$ of $V_{H}$ there is a unique closed Reeb orbit $\unicode[STIX]{x1D6FE}_{x}(t)$ of $\unicode[STIX]{x1D706}_{0}$ such that

(14) $$\begin{eqnarray}x(t)=(\unicode[STIX]{x1D70F}_{x},\unicode[STIX]{x1D6FE}_{x}(h^{\prime }(e^{\unicode[STIX]{x1D70F}_{x}-\unicode[STIX]{x1D705}})e^{-\unicode[STIX]{x1D705}}t))\end{eqnarray}$$

and

(15) $$\begin{eqnarray}{\mathcal{A}}_{H}(x)=-e^{\unicode[STIX]{x1D70F}_{x}}T_{\unicode[STIX]{x1D6FE}_{x}}+h(e^{\unicode[STIX]{x1D70F}_{x}-\unicode[STIX]{x1D705}}).\end{eqnarray}$$

Arguing as above, we then get the following.

Lemma 2.3. Let $H$ be in ${\mathcal{H}}_{a,b,c,\unicode[STIX]{x1D705}}$ . If $be^{-\unicode[STIX]{x1D705}}$ is not in ${\mathcal{T}}(\unicode[STIX]{x1D706}_{0})$ and $c$ is sufficiently large, then every $x\in {\mathcal{P}}^{-}(H)$ is nonconstant and of the form

$$\begin{eqnarray}x(t)=(\unicode[STIX]{x1D70F}_{x},\unicode[STIX]{x1D6FE}_{x}(h^{\prime }(e^{\unicode[STIX]{x1D70F}_{x}-\unicode[STIX]{x1D705}})e^{-\unicode[STIX]{x1D705}}t))\end{eqnarray}$$

for some $e^{\unicode[STIX]{x1D70F}_{x}}\in (e^{\unicode[STIX]{x1D705}},(1+a)e^{\unicode[STIX]{x1D705}})$ and some $\unicode[STIX]{x1D6FE}_{x}\in {\mathcal{R}}^{be^{-\unicode[STIX]{x1D705}}}(\unicode[STIX]{x1D706}_{0})$ . Moreover, the correspondence $x\rightarrow \unicode[STIX]{x1D6FE}_{x}$ defines a bijection between ${\mathcal{P}}^{-}(H)$ and ${\mathcal{R}}^{be^{-\unicode[STIX]{x1D705}}}(\unicode[STIX]{x1D706}_{0})$ .

Every $H$ tuned to ${\mathcal{C}}_{\unicode[STIX]{x1D706}_{0},\unicode[STIX]{x1D6FC}}(T)$ is dividing. So, the sets ${\mathcal{P}}^{-}(H)$ and ${\mathcal{R}}^{be^{-\unicode[STIX]{x1D705}}}(\unicode[STIX]{x1D706}_{0})$ are in bijection. The class $\unicode[STIX]{x1D6FC}\in [\mathbb{S}^{1},M]$ determines a unique class in $[\mathbb{R}/\mathbb{Z},\mathbb{R}\times M]$ , which we will also denote by $\unicode[STIX]{x1D6FC}$ . Let ${\mathcal{P}}_{\unicode[STIX]{x1D6FC}}^{-}(H)$ be the subset ${\mathcal{P}}^{-}(H)$ consisting of $1$ -periodic orbits of $H$ which have negative action and which represent the class $\unicode[STIX]{x1D6FC}$ . The point of Definition 2.6 is that, if $H$ is tuned to ${\mathcal{C}}_{\unicode[STIX]{x1D706}_{0},\unicode[STIX]{x1D6FC}}(T)$ , then the elements of ${\mathcal{P}}_{\unicode[STIX]{x1D6FC}}^{-}(H)$ correspond precisely to the elements of ${\mathcal{C}}_{\unicode[STIX]{x1D706}_{0},\unicode[STIX]{x1D6FC}}(T)$ , i.e.

(16) $$\begin{eqnarray}x\in {\mathcal{P}}_{\unicode[STIX]{x1D6FC}}^{-}(H)\longleftrightarrow \unicode[STIX]{x1D6FE}_{x}\in {\mathcal{C}}_{\unicode[STIX]{x1D706}_{0},\unicode[STIX]{x1D6FC}}(T).\end{eqnarray}$$

To identify tuned Hamiltonians that are suitable for defining Floer theory on the symplectization of $(M,\unicode[STIX]{x1D706}_{0})$ , we must refine this notion further. It is clear from (15) that for a tuned $H$ in ${\mathcal{H}}_{a,b,c,\unicode[STIX]{x1D705}}$ with a small value of $a$ , the actions of the orbits in ${\mathcal{P}}_{\unicode[STIX]{x1D6FC}}^{-}(H)$ are close to $-e^{\unicode[STIX]{x1D705}}$ times the period of an element of ${\mathcal{C}}_{\unicode[STIX]{x1D706}_{0},\unicode[STIX]{x1D6FC}}(T)$ . Developing this further, we get the following useful set of inequalities.

Proof. By (15) and the properties of $h$ , we have

(20) $$\begin{eqnarray}-e^{\unicode[STIX]{x1D705}}(1+a)T<{\mathcal{A}}_{H}(x)<-e^{\unicode[STIX]{x1D705}}T_{\min }(\unicode[STIX]{x1D706}_{0},\unicode[STIX]{x1D6FC})+a^{2}\end{eqnarray}$$

for every $x\in {\mathcal{P}}_{\unicode[STIX]{x1D6FC}}^{-}(H)$ . The fact that inequality (17) holds for sufficiently small $a$ follows immediately from this and the fact that $\unicode[STIX]{x1D705}\geqslant 0$ . The inequalities of (20) also imply that

(21) $$\begin{eqnarray}\unicode[STIX]{x1D6E5}(H)<e^{\unicode[STIX]{x1D705}}(T-T_{\min }(\unicode[STIX]{x1D706}_{0},\unicode[STIX]{x1D6FC}))+ae^{\unicode[STIX]{x1D705}}(T-a).\end{eqnarray}$$

Since ${\mathcal{C}}_{\unicode[STIX]{x1D706}_{0},\unicode[STIX]{x1D6FC}}(T)$ is rigid, we have $T<T_{\min }(\unicode[STIX]{x1D706}_{0},\unicode[STIX]{x1D6FC})+T_{\min }(\unicode[STIX]{x1D706}_{0}).$ Together with condition (t1), this yields

(22) $$\begin{eqnarray}e^{\unicode[STIX]{x1D705}}<\frac{T_{\min }(\unicode[STIX]{x1D706}_{0})}{T-T_{\min }(\unicode[STIX]{x1D706}_{0},\unicode[STIX]{x1D6FC})}.\end{eqnarray}$$

The fact that inequality (18) holds for sufficiently small $a$ now follows immediately from (21) and (22). Finally, (20) also implies that

(23) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E5}(H^{0},H^{1}) & {\leqslant} & \displaystyle -e^{\unicode[STIX]{x1D705}^{0}}T_{\min }(\unicode[STIX]{x1D706}_{0},\unicode[STIX]{x1D6FC})+e^{\unicode[STIX]{x1D705}^{1}}T+(a^{0})^{2}+e^{\unicode[STIX]{x1D705}^{1}}a^{1}T\end{eqnarray}$$

(24) $$\begin{eqnarray}\displaystyle & {\leqslant} & \displaystyle e^{\unicode[STIX]{x1D705}^{1}}T-T_{\min }(\unicode[STIX]{x1D706}_{0},\unicode[STIX]{x1D6FC})+(a^{0})^{2}+e^{\unicode[STIX]{x1D705}^{1}}a^{1}T.\end{eqnarray}$$

This, together with (t1), implies that (19) holds when both $a^{0}$ and $a^{1}$ are sufficiently small.◻

2.3 Almost complex structures

For the next four subsections we will assume that $H$ is an admissible Hamiltonian and that each element of ${\mathcal{P}}^{-}(H)$ , that is, each $1$ -periodic orbit of $H$ with negative action, is transversally nondegenerate. Given such an $H$ , we now define a useful class of almost complex structures on $\mathbb{R}\times M$ . Recall first that an almost complex structure $J$ on the symplectization $(\mathbb{R}\times M,d(e^{\unicode[STIX]{x1D70F}}\unicode[STIX]{x1D706}))$ is said to be cylindrical if it is invariant under $\unicode[STIX]{x1D70F}$ -translations and satisfies $J(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}\unicode[STIX]{x1D70F})=R_{\unicode[STIX]{x1D706}}$ . The related notion of being cylindrical on subsets of the form $\{\unicode[STIX]{x1D70F}\leqslant \mathbf{T}\}$ or $\{\unicode[STIX]{x1D70F}\geqslant \mathbf{T}\}$ is defined in the obvious way.

Denote by ${\mathcal{J}}(H)$ the set of smooth almost complex structures on $\mathbb{R}\times M$ with the following properties:

1. (J1) $J$ is compatible with $d(e^{\unicode[STIX]{x1D70F}}\unicode[STIX]{x1D706}_{0})$ ;

2. (J2) $J=J_{0}$ on $\{\unicode[STIX]{x1D70F}\leqslant 0\}$ , where $J_{0}$ is fixed and cylindrical;

3. (J3) $J$ is cylindrical on $\{\unicode[STIX]{x1D70F}\geqslant \mathbf{T}\}$ for some $\mathbf{T}>0$ ;

4. (J4) for any point $z=(\unicode[STIX]{x1D70F},p)$ on the image of a family $X$ in ${\mathcal{P}}_{\mathbb{R}/\mathbb{Z}}^{-}(H)$ , we have

   $$\begin{eqnarray}[V_{H},JV_{H}](z)\neq 0\quad \text{and}\quad [V_{H},JV_{H}](z)\notin \text{Span}\{V_{H}(z),JV_{H}(z)\}.\end{eqnarray}$$

2.4 Floer trajectories: transversality

Consider a pair

$$\begin{eqnarray}F=(H,J)\end{eqnarray}$$

consisting of an admissible Hamiltonian $H$ and an almost complex structure $J$ in ${\mathcal{J}}(H)$ . We will refer to $F$ as a Floer data set and will denote the set of all Floer data sets by $\mathbf{F}$ .

Given two (nonconstant) $\mathbb{R}/\mathbb{Z}$ -families $X$ and $Y$ in ${\mathcal{P}}_{\mathbb{R}/\mathbb{Z}}^{-}(H)$ and an $F=(H,J)$ in $\mathbf{F}$ , we define

$$\begin{eqnarray}\widehat{{\mathcal{M}}}(X,Y;F)\end{eqnarray}$$

to be the space of solutions $u:\mathbb{R}\times \mathbb{R}/\mathbb{Z}\rightarrow \mathbb{R}\times M$ of

$$\begin{eqnarray}\unicode[STIX]{x2202}_{s}u+J(u)(\unicode[STIX]{x2202}_{t}u-V_{H}(u))=0\end{eqnarray}$$

which satisfy the asymptotic conditions

$$\begin{eqnarray}\lim _{s\rightarrow -\infty }u(s,t)\in X,\quad \lim _{s\rightarrow +\infty }u(s,t)\in Y\quad \text{and}\quad \lim _{s\rightarrow \pm \infty }\unicode[STIX]{x2202}_{s}u(s,t)=0,\end{eqnarray}$$

where the convergences are all uniform in $t$ . The following transversality statement for these spaces is established by Bourgeois and Oancea in [Reference Bourgeois and OanceaBO09].

The assumption that either $X$ or $Y$ is simple implies that the elements of $\widehat{{\mathcal{M}}}(X,Y;F)$ are all somewhere injective. Having thus avoided the fundamental difficulty of dealing with multiply covered maps, transversality can then be established in the manner of [Reference Hofer and SalamonHS95]. The subtle point, observed and overcome in [Reference Bourgeois and OanceaBO09], is that condition (J4) can be used to prove that the set of injective points in the domain of an element of $\widehat{{\mathcal{M}}}(X,Y;F)$ constitutes an open and dense subset of some neighborhood of an end asymptotic to a simple orbit. Let $\mathbf{F}_{\text{reg}}$ be the subset of $\mathbf{F}$ consisting of Floer data sets $F=(H,J)$ with $J\in {\mathcal{J}}_{\text{reg}}(H)$ .

Next we consider spaces of Floer continuation trajectories. Let $H^{s}$ be an admissible homotopy between admissible Hamiltonians $H^{0}$ and $H^{1}$ whose nonconstant $1$ -periodic orbits with negative action are transversally nondegenerate. Let $J^{s}$ be a smooth family of $d(e^{\unicode[STIX]{x1D70F}}\unicode[STIX]{x1D706}_{0})$ -compatible almost complex structures such that for some $\mathbf{S}>0$ and $\mathbf{T}>0$ , we have:

( $\text{J}^{s}1$ )

$J^{s}=J^{0}\in {\mathcal{J}}(H^{0})$ for all $s\leqslant -\mathbf{S}$ ;

( $\text{J}^{s}2$ )

$J^{s}=J^{1}\in {\mathcal{J}}(H^{1})$ for all $s\geqslant \mathbf{S}$ ;

( $\text{J}^{s}3$ )

$J^{s}=J_{0}$ on $(-\infty ,0]\times M$ for all $s\in \mathbb{R}$ ;

( $\text{J}^{s}4$ )

$J^{s}$ is cylindrical (for $\unicode[STIX]{x1D706}_{0}$ ) on $[\mathbf{T},+\infty )\times M$ for all $s\in \mathbb{R}$ .

We refer to the pair $F^{s}=(H^{s},J^{s})$ as Floer continuation data (connecting $F^{0}=(H^{0},J^{0})$ to $F^{1}=(H,J^{1})$ ) and will denote the set of all such triples as $\mathbf{F}^{s}=\mathbf{F}^{s}(F^{0},F^{1})$ .

For an $F^{s}=(H^{s},J^{s})$ in $\mathbf{F}^{s}$ and families $X^{0}$ in ${\mathcal{P}}_{\mathbb{R}/\mathbb{Z}}^{-}(H^{0})$ and $X^{1}$ in ${\mathcal{P}}_{\mathbb{R}/\mathbb{Z}}^{-}(H^{1})$ , let

$$\begin{eqnarray}\widehat{{\mathcal{M}}}_{s}(X^{0},X^{1};F^{s})\end{eqnarray}$$

be the space of solutions $u:\mathbb{R}\times \mathbb{R}/\mathbb{Z}\rightarrow \mathbb{R}\times M$ of

$$\begin{eqnarray}\unicode[STIX]{x2202}_{s}u+J^{s}(u)(\unicode[STIX]{x2202}_{t}u-V_{H^{s}}(u))=0\end{eqnarray}$$

which satisfy the asymptotic conditions

$$\begin{eqnarray}\lim _{s\rightarrow -\infty }u(s,t)\in X^{0},\quad \lim _{s\rightarrow +\infty }u(s,t)\in X^{1}\quad \text{and}\quad \lim _{s\rightarrow \pm \infty }\unicode[STIX]{x2202}_{s}u(s,t)=0.\end{eqnarray}$$

Arguing as above, one gets the following basic transversality statement.

More generally, we have the moduli space of Floer trajectories corresponding to an admissible homotopy of homotopies, $H^{r,s}$ , from $H^{0}$ to $H^{1}$ . Let $J^{r,s}$ be a smooth $\mathbb{R}^{2}$ -family of $d(e^{\unicode[STIX]{x1D70F}}\unicode[STIX]{x1D706}_{0})$ -compatible almost complex structures such that for some smooth positive function $S(r)$ and for some $\mathbf{T}>0$ , the following condition holds:

( $\text{J}^{r,s}1$ )

for each $r$ , the $\mathbb{R}$ -family $J^{r,s}$ satisfies ( $\text{J}^{s}1$ )–( $\text{J}^{s}4$ ) for $S=S(r)$ .

Following the pattern above, we refer to the triple $F^{r,s}=(H^{r,s},J^{r,s})$ as Floer homotopy data and will denote the set of all such triples by $\mathbf{F}^{r,s}$ . For an $F^{r,s}=(H^{r,s},J^{r,s})$ in $\mathbf{F}^{r,s}$ and two families $X^{0}$ in ${\mathcal{P}}_{\mathbb{R}/\mathbb{Z}}^{-}(H)$ , $X^{1}$ in ${\mathcal{P}}_{\mathbb{R}/\mathbb{Z}}^{-}(H^{1})$ , let

$$\begin{eqnarray}\widehat{{\mathcal{M}}}_{r,s}(X^{0},X^{1};F^{r,s})=\{(r,u):r\in \mathbb{R},u\in \widehat{{\mathcal{M}}}_{s}(X^{0},X^{1};F^{r,s})\}.\end{eqnarray}$$

2.5 Floer trajectories: $C^{0}$ -bounds

With transversality in hand, we now turn to compactness. Since the manifold $\mathbb{R}\times M$ is open, we must first establish $C^{0}$ -bounds. The positive end of this manifold ( $\unicode[STIX]{x1D70F}\rightarrow +\infty$ ) is never a possible source of noncompactness. Since every (family of) Hamiltonian(s) we consider is constant for $\unicode[STIX]{x1D70F}\gg 0$ and every (family of) almost complex structure(s) we consider is cylindrical for $\unicode[STIX]{x1D70F}\gg 0$ , the maximal principle forbids our curves from entering these regions. It remains for us to deal with the negative end of $\mathbb{R}\times M$ .

Before proceeding, we recall the relevant notions of energy in this context and some useful equalities and inequalities involving them. For $F=(H,J)$ , the $L^{2}$ -energy of each $u\in \widehat{{\mathcal{M}}}(X,Y;F)$ is defined to be

(25) $$\begin{eqnarray}E(u)=\int _{\mathbb{R}\times \mathbb{R}/\mathbb{Z}}d(e^{\unicode[STIX]{x1D70F}}\unicode[STIX]{x1D706}_{0})(\unicode[STIX]{x2202}_{s}u,J(u)\unicode[STIX]{x2202}_{s}u)\,ds\,dt.\end{eqnarray}$$

The following well-known identity then follows from Stokes’ theorem and the definition of $\widehat{{\mathcal{M}}}(X,Y;F)$ :

(26) $$\begin{eqnarray}E(u)={\mathcal{A}}_{H}(X)-{\mathcal{A}}_{H}(Y).\end{eqnarray}$$

Hence, we have

(27) $$\begin{eqnarray}E(u)\leqslant \unicode[STIX]{x1D6E5}(H)\end{eqnarray}$$

for any $u$ in any $\widehat{{\mathcal{M}}}(X,Y;F)$ .

Similarly, for $F^{s}=(H^{s},J^{s})$ , the $L^{2}$ -energy of $u$ in $\widehat{{\mathcal{M}}}_{s}(X^{0},X^{1};F^{s})$ is defined to be

(28) $$\begin{eqnarray}E_{s}(u)=\int _{\mathbb{R}\times \mathbb{R}/\mathbb{Z}}d(e^{\unicode[STIX]{x1D70F}}\unicode[STIX]{x1D706}_{0})(\unicode[STIX]{x2202}_{s}u,J^{s}(u)\unicode[STIX]{x2202}_{s}u)\,ds\,dt.\end{eqnarray}$$

In this case, Stokes’ theorem yields

(29) $$\begin{eqnarray}E_{s}(u)={\mathcal{A}}_{H^{0}}(X^{0})-{\mathcal{A}}_{H^{1}}(X^{1})+\int _{\mathbb{ R}\times \mathbb{R}/\mathbb{Z}}(\unicode[STIX]{x2202}_{s}H^{s})(u(s,t))\,ds\,dt\end{eqnarray}$$

and

(30) $$\begin{eqnarray}E_{s}(u)\leqslant \unicode[STIX]{x1D6E5}(H^{0},H^{1})+\text{cost}(H^{s}).\end{eqnarray}$$

Finally, for $F^{r,s}=(H^{r,s},J^{r,s})$ and $(r,u)$ in $\widehat{{\mathcal{M}}}_{r,s}(X^{0},X^{1};F^{r,s})$ , we have

(31) $$\begin{eqnarray}E_{r,s}((r,u))=\int _{\mathbb{R}\times \mathbb{R}/\mathbb{Z}}d(e^{\unicode[STIX]{x1D70F}}\unicode[STIX]{x1D706}_{0})(\unicode[STIX]{x2202}_{s}u,J^{r,s}(u)\unicode[STIX]{x2202}_{s}u)\,ds\,dt\leqslant \unicode[STIX]{x1D6E5}(H^{0},H^{1})+\text{cost}(H^{r,s}).\end{eqnarray}$$

We now prove that for a fixed $H$ we have uniform $C^{0}$ -bounds for the elements of the spaces $\widehat{{\mathcal{M}}}(X,Y;F)$ . Recall that $T_{\min }(\unicode[STIX]{x1D706}_{0})$ is the smallest period of any closed Reeb orbit of $\unicode[STIX]{x1D706}_{0}$ .

Proposition 2.13. Suppose that $H$ is an admissible Hamiltonian and that $X$ and $Y$ are transversally nondegenerate families in ${\mathcal{P}}_{\mathbb{R}/\mathbb{Z}}^{-}(H)$ . If

$$\begin{eqnarray}{\mathcal{A}}_{H}(X)-{\mathcal{A}}_{H}(Y)<T_{\min }(\unicode[STIX]{x1D706}_{0}),\end{eqnarray}$$

then there is a $K>0$ such that for any choice of Floer data of the form $F=(H,J)$ the image of every $u\in \widehat{{\mathcal{M}}}(X,Y;F)$ is contained in $[-K,+\infty )\times M\subset \mathbb{R}\times M$ .

Proof. In terms of the product structure of $\mathbb{R}\times M$ , any $u\in \widehat{{\mathcal{M}}}(X,Y;F)$ can be written in the form

$$\begin{eqnarray}u(s,t)=(\unicode[STIX]{x1D70C}(s,t),\unicode[STIX]{x1D709}(s,t)).\end{eqnarray}$$

Arguing by contradiction, we assume that there are a sequence of almost complex structures $J_{k}$ in ${\mathcal{J}}(H)$ and a sequence of curves

$$\begin{eqnarray}u_{k}(s,t)=(\unicode[STIX]{x1D70C}_{k}(s,t),\unicode[STIX]{x1D709}_{k}(s,t))\end{eqnarray}$$

in $\widehat{{\mathcal{M}}}(X,Y;F_{k})$ for $F_{k}=(H,J_{k})$ such that

(32) $$\begin{eqnarray}\lim _{k\rightarrow \infty }\Bigl(\min _{(s,t)\in \mathbb{R}\times \mathbb{R}/\mathbb{Z}}\unicode[STIX]{x1D70C}_{k}(s,t)\Bigr)=-\infty .\end{eqnarray}$$

To obtain the desired contradiction, we will argue as in [Reference Albers, Fuchs and MerryAFM15] (see also [Reference Eliashberg, Hofer and SalamonEHS95]). We begin by isolating purely $J_{0}$ -holomorphic portions of the $u_{k}$ . Fix a decreasing sequence $\unicode[STIX]{x1D716}_{k}{\searrow}0$ such that $u_{k}$ is transverse to $\{-\unicode[STIX]{x1D716}_{k}\}\times M$ for all $k\in \mathbb{N}$ and set

$$\begin{eqnarray}V_{k}=u_{k}^{-1}((-\infty ,-\unicode[STIX]{x1D716}_{k}]\times M).\end{eqnarray}$$

By (32), we may assume, by passing to a subsequence if necessary, that each $V_{k}$ is nonempty. Let $v_{k}=u_{k}|_{V_{k}}$ . Since $H$ is admissible and $J_{k}$ belongs to ${\mathcal{J}}(H)$ , each $v_{k}$ is a $J_{0}$ -holomorphic curve with a possibly disconnected domain and image in $\{\unicode[STIX]{x1D70F}\leqslant 0\}$ . For these curves, we have

$$\begin{eqnarray}\int _{V_{k}}v_{k}^{\ast }d(e^{\unicode[STIX]{x1D70F}}\unicode[STIX]{x1D706}_{0})=\int _{V_{k}}d(e^{\unicode[STIX]{x1D70F}}\unicode[STIX]{x1D706}_{0})(\unicode[STIX]{x2202}_{s}u_{k},J_{k}(u_{k})\unicode[STIX]{x2202}_{s}u_{k})\,ds\,dt<E(u_{k})\end{eqnarray}$$

and so, by (26), we have

(33) $$\begin{eqnarray}\int _{V_{k}}v_{k}^{\ast }d(e^{\unicode[STIX]{x1D70F}}\unicode[STIX]{x1D706}_{0})<{\mathcal{A}}_{H}(X)-{\mathcal{A}}_{H}(Y).\end{eqnarray}$$

The Hofer energy of $v_{k}$ is

$$\begin{eqnarray}E_{\text{Hofer}}(v_{k})=\sup _{\unicode[STIX]{x1D719}}\int _{V_{k}}v_{k}^{\ast }d(\unicode[STIX]{x1D719}\unicode[STIX]{x1D706}_{0}),\end{eqnarray}$$

where the supremum is over all functions $\unicode[STIX]{x1D719}$ in $C^{\infty }(\mathbb{R},[0,1])$ that are nondecreasing.

Applying Stokes’ theorem twice, we get

$$\begin{eqnarray}\displaystyle E_{\text{Hofer}}(v_{k}) & = & \displaystyle \sup _{\unicode[STIX]{x1D719}}\int _{V_{k}}v_{k}^{\ast }d(\unicode[STIX]{x1D719}\unicode[STIX]{x1D706}_{0})\nonumber\\ \displaystyle & = & \displaystyle \sup _{\unicode[STIX]{x1D719}}\int _{\unicode[STIX]{x2202}V_{k}}v_{k}^{\ast }(\unicode[STIX]{x1D719}\unicode[STIX]{x1D706}_{0})\nonumber\\ \displaystyle & = & \displaystyle \int _{\unicode[STIX]{x2202}V_{k}}v_{k}^{\ast }\unicode[STIX]{x1D706}_{0}\nonumber\\ \displaystyle & = & \displaystyle e^{\unicode[STIX]{x1D716}_{k}}\int _{\unicode[STIX]{x2202}V_{k}}v_{k}^{\ast }(e^{\unicode[STIX]{x1D70F}}\unicode[STIX]{x1D706}_{0})\nonumber\\ \displaystyle & = & \displaystyle e^{\unicode[STIX]{x1D716}_{k}}\int _{V_{k}}v_{k}^{\ast }d(e^{\unicode[STIX]{x1D70F}}\unicode[STIX]{x1D706}_{0})\nonumber\\ \displaystyle & = & \displaystyle e^{\unicode[STIX]{x1D716}_{k}}\int _{V_{k}}v_{k}^{\ast }d(e^{\unicode[STIX]{x1D70F}}\unicode[STIX]{x1D706}_{0}).\nonumber\end{eqnarray}$$

Hence, by (33), we have

$$\begin{eqnarray}E_{\text{Hofer}}(v_{k})<e^{\unicode[STIX]{x1D716}_{k}}({\mathcal{A}}_{H}(X)-{\mathcal{A}}_{H}(Y)).\end{eqnarray}$$

So, we have a sequence, $v_{k}$ , of $J_{0}$ -holomorphic curves in the symplectization $(\mathbb{R}\times M,d(e^{\unicode[STIX]{x1D70F}}\unicode[STIX]{x1D706}_{0}))$ whose Hofer energies are uniformly bounded from above, which all intersect the hypersurface $\{-\unicode[STIX]{x1D716}_{1}\}\times M$ and whose $\mathbb{R}$ -components have minima that converge to negative infinity. This suggests that the curves break at $\unicode[STIX]{x1D70F}=-\infty$ along closed Reeb orbits of $\unicode[STIX]{x1D706}_{0}$ with period less than the limiting energy bound. Indeed, this is the case. This is a consequence of [Reference Albers, Fuchs and MerryAFM15, Theorem 5.3], which utilizes the compactness argument from [Reference Cieliebak and MohnkeCM05] and yields the following precise statement in the present setting.

Proposition 2.14 [Reference Albers, Fuchs and MerryAFM15, Theorem 5.3].

There are a subsequence $k_{n}$ and cylinders $C_{n}\subset U_{k_{n}}$ that are biholomorphically equivalent to the standard cylinders $[-L_{n},L_{n}]\times \mathbb{R}/\mathbb{Z}$ such that the lengths $L_{n}\rightarrow \infty$ and the curves $v_{k_{n}}|_{C_{n}}$ converge in $C_{\text{loc}}^{\infty }(\mathbb{R}\times \mathbb{R}/\mathbb{Z},\mathbb{R}\times M)$ to a trivial cylinder over a closed Reeb orbit of $\unicode[STIX]{x1D706}_{0}$ of period at most ${\mathcal{A}}_{H}(X)-{\mathcal{A}}_{H}(Y)$ .

The existence of this closed Reeb orbit of $\unicode[STIX]{x1D706}_{0}$ implies that $T_{\min }(\unicode[STIX]{x1D706}_{0})\leqslant {\mathcal{A}}_{H}(X)-{\mathcal{A}}_{H}(Y)$ and we have arrived at the desired contradiction.◻

Starting from the uniform bounds (30) and (31) and arguing as above, one also obtains $C^{0}$ -bounds for moduli spaces of the form $\widehat{{\mathcal{M}}}_{s}(X^{0},X^{1};F^{s})$ and $\widehat{{\mathcal{M}}}_{r,s}(X^{0},X^{1};F^{r,s})$ whenever ${\mathcal{A}}_{H^{0}}(X^{0})-{\mathcal{A}}_{H^{1}}(X^{1})+\text{cost}(H^{s})<T_{\min }(\unicode[STIX]{x1D706}_{0})$ and ${\mathcal{A}}_{H^{0}}(X^{0})-{\mathcal{A}}_{H^{1}}(X^{1})+\text{cost}(H^{r,s})<T_{\min }(\unicode[STIX]{x1D706}_{0})$ , respectively.

2.6 Floer trajectories: quotients and compactifications

Consider a regular Floer data set $F=(H,J)$ in $\mathbf{F}_{\text{reg}}$ and two distinct families $X$ and $Y$ in ${\mathcal{P}}_{\mathbb{R}/\mathbb{Z}}^{-}(H)$ at least one of which is simple. Since both $H$ and $J$ do not depend on $t$ , it follows that the $\mathbb{R}\times \mathbb{R}/\mathbb{Z}$ -action on $\widehat{{\mathcal{M}}}(X,Y;F)$ given by

$$\begin{eqnarray}(s,t)\ast u(\cdot ,\cdot )=u(\cdot +s,\cdot +t)\end{eqnarray}$$

is free. The quotient

$$\begin{eqnarray}{\mathcal{M}}(X,Y;F)=\widehat{{\mathcal{M}}}(X,Y;F)/(\mathbb{R}\times \mathbb{R}/\mathbb{Z})\end{eqnarray}$$

is then a smooth manifold. Let ${\mathcal{M}}^{k}(X,Y;F)$ be the submanifold of ${\mathcal{M}}(X,Y;F)$ which consists of all its components which have dimension $k$ . Given Proposition 2.13, the follow compactness statements are then standard.

Proposition 2.15. Suppose that ${\mathcal{A}}_{H}(X)-{\mathcal{A}}_{H}(Y)<T_{\min }(\unicode[STIX]{x1D706}_{0})$ . Then ${\mathcal{M}}^{0}(X,Y;F)$ is a compact manifold of dimension zero. If, in addition, both $X$ and $Y$ are simple, then ${\mathcal{M}}^{1}(X,Y;F)$ admits a compactification $\overline{{\mathcal{M}}}^{1}(X,Y;F)$ which is a one-dimensional manifold with boundary equal to

$$\begin{eqnarray}\mathop{\bigcup }_{Z\in {\mathcal{P}}_{\mathbb{R}/\mathbb{Z}}^{-}(H)}{\mathcal{M}}^{0}(X,Z;F)\times {\mathcal{M}}^{0}(Z,Y;F).\end{eqnarray}$$

Now consider two admissible and nondegenerate Hamiltonians $H^{0}$ and $H^{1}$ , regular Floer data $F^{0}=(H^{0},J^{0})$ and $F^{1}=(H^{1},J^{1})$ and regular Floer continuation data $F^{s}\in \mathbf{F}^{s}(F^{0},F^{1})$ in $\mathbf{F}_{\text{reg}}^{s}$ . For families $X^{0}\in {\mathcal{P}}_{\mathbb{R}/\mathbb{Z}}^{-}(H^{0})$ and $X^{1}\in {\mathcal{P}}_{\mathbb{R}/\mathbb{Z}}^{-}(H^{1})$ , at least one of which is simple, the manifold $\widehat{{\mathcal{M}}}_{s}(X^{0},X^{1};F^{s})$ admits a free $\mathbb{R}/\mathbb{Z}$ -action,

$$\begin{eqnarray}t\ast u(\cdot ,\cdot )=u(\cdot ,\cdot +t).\end{eqnarray}$$

The quotient

$$\begin{eqnarray}{\mathcal{M}}_{s}(X^{0},X^{1};F^{s})=\widehat{{\mathcal{M}}}_{s}(X^{0},X^{1};F^{s})/(\mathbb{R}/\mathbb{Z})\end{eqnarray}$$

is then a smooth manifold and we let ${\mathcal{M}}_{s}^{k}(X^{0},X^{1};F^{s})$ be the collection of its $k$ -dimensional components. In this case we get the following compactness result.

Proposition 2.16. Suppose that ${\mathcal{A}}_{H^{0}}(X^{0})-{\mathcal{A}}_{H^{1}}(X^{1})+\text{cost}(H^{s})<T_{\min }(\unicode[STIX]{x1D706}_{0})$ . Then ${\mathcal{M}}_{s}^{0}(X^{0},X^{1};F^{s})$ is compact. If, in addition, both $X^{0}$ and $X^{1}$ are simple and $\text{cost}(H^{s})<-{\mathcal{A}}_{H}(X^{0})$ , then ${\mathcal{M}}_{s}^{1}(X^{0},X^{1};F^{s})$ admits a compactification $\overline{{\mathcal{M}}}_{s}^{1}(X^{0},X^{1};F^{s})$ which is a one-dimensional manifold whose boundary is

$$\begin{eqnarray}\displaystyle & & \displaystyle \mathop{\bigcup }_{Y^{0}\in {\mathcal{P}}_{\mathbb{R}/\mathbb{Z}}^{-}(H^{0})}{\mathcal{M}}^{0}(X^{0},Y^{0};F^{0})\times {\mathcal{M}}_{s}^{0}(Y^{0},X^{1};F^{s})\nonumber\\ \displaystyle & & \displaystyle \quad \cup \mathop{\bigcup }_{Y^{1}\in {\mathcal{P}}_{\mathbb{R}/\mathbb{Z}}^{-}(H^{1})}{\mathcal{M}}_{s}^{0}(X^{0},Y^{1};F^{s})\times {\mathcal{M}}^{0}(Y^{1},X^{1};F^{1}).\nonumber\end{eqnarray}$$

Note that the condition $\text{cost}(H^{s})<-{\mathcal{A}}_{H}(X^{0})$ is needed to ensure that the orbits $Y_{0}$ that appear in the expression above have negative action.

Finally, for regular Floer homotopy data $F^{r,s}=(H^{r,s},J^{r,s})\in \mathbf{F}^{r,s}(F^{0},F^{1})$ , we set

$$\begin{eqnarray}{\mathcal{M}}_{r,s}(X^{0},X^{1};F^{r,s})=\widehat{{\mathcal{M}}}_{r,s}(X^{0},X^{1};F^{s})/(\mathbb{R}/\mathbb{Z})\end{eqnarray}$$

and define ${\mathcal{M}}_{r,s}^{k}(X^{0},X^{1};F^{r,s})$ as above. For $k=0$ , we get the following.

For $k=1$ , there are two important versions of the relevant compactness statement to state.

Version 1: A closed homotopy of homotopies. We say that the homotopy data $F^{r,s}$ is closed if for some $\mathbf{R}>0$ ,

$$\begin{eqnarray}F^{r,s}=\left\{\begin{array}{@{}ll@{}}F^{0,s}\in \mathbf{F}_{\text{reg}}^{s}(F^{0},F^{1})\quad & \text{for all }r\leqslant -\mathbf{R},\\ F^{1,s}\in \mathbf{F}_{\text{reg}}^{s}(F^{0},F^{1})\quad & \text{for all }r\geqslant \mathbf{R}.\end{array}\right.\end{eqnarray}$$

Proposition 2.18. Suppose that $F^{r,s}=(H^{r,s},J^{r,s})$ is regular and closed and that ${\mathcal{A}}_{H^{0}}(X^{0})-{\mathcal{A}}_{H^{1}}(X^{1})+\text{cost}(H^{r,s})<T_{\min }(\unicode[STIX]{x1D706}_{0})$ . If $X^{0}$ and $X^{1}$ are both simple and $\text{cost}(H^{r,s})<|{\mathcal{A}}_{H}(X^{0})|$ , then ${\mathcal{M}}_{r,s}^{1}(X^{0},X^{1};F^{r,s})$ admits a compactification $\overline{{\mathcal{M}}}_{r,s}^{1}(X^{0},X^{1};F^{r,s})$ which is a one-dimensional manifold with boundary equal to

$$\begin{eqnarray}\displaystyle & & \displaystyle {\mathcal{M}}_{s}^{0}(X^{0},X^{1};F^{0,s})\cup {\mathcal{M}}_{s}^{0}(X^{0},X^{1};F^{1,s})\nonumber\\ \displaystyle & & \displaystyle \quad \cup \mathop{\bigcup }_{Y^{0}\in {\mathcal{P}}_{\mathbb{R}/\mathbb{Z}}^{-}(H^{0})}{\mathcal{M}}^{0}(X^{0},Y^{0};F^{0})\times {\mathcal{M}}_{r,s}^{0}(Y^{0},X^{1};F^{r,s})\nonumber\\ \displaystyle & & \displaystyle \quad \cup \mathop{\bigcup }_{Y^{1}\in {\mathcal{P}}_{\mathbb{R}/\mathbb{Z}}^{-}(H^{1})}{\mathcal{M}}_{r,s}^{0}(X^{0},Y^{1};F^{r,s})\times {\mathcal{M}}^{0}(Y^{1},X^{1};F^{1}).\nonumber\end{eqnarray}$$

Version 2: A half-open homotopy of homotopies. We now consider a more explicit homotopy of homotopies with an open end. Let $H^{0}$ , $H^{1}$ and $G$ be nondegenerate and admissible Hamiltonians. Consider two admissible homotopies, $H_{0}^{s}$ from $H^{0}$ to $G$ , and $H_{1}^{s}$ from $G$ to $H^{1}$ . Now consider a homotopy of homotopies of the form

$$\begin{eqnarray}H_{0\#1}^{r,s}=\left\{\begin{array}{@{}ll@{}}H_{0}^{s+\unicode[STIX]{x1D709}(r)}\quad & \text{for }s\leqslant 0,\\ H_{1}^{s-\unicode[STIX]{x1D709}(r)}\quad & \text{for }s>0,\end{array}\right.\end{eqnarray}$$

where $\unicode[STIX]{x1D709}(r)$ is a smooth, positive and nondecreasing function which equals $r$ for $r\gg \unicode[STIX]{x1D709}(0)$ and which equals $\unicode[STIX]{x1D709}(0)$ for $r\leqslant \unicode[STIX]{x1D709}(0)/2$ . It is easy to check that this is an admissible homotopy of homotopies from $H^{0}$ to $H^{1}$ if we choose $\unicode[STIX]{x1D709}(0)$ to be sufficiently large.

Fixing regular Floer data sets $F^{0}=(H^{0},J^{0})$ , $F^{1}=(H^{1},J^{1})$ and $F^{G}=(G,J^{G})$ , we extend these to Floer continuation data sets

$$\begin{eqnarray}F_{0}^{s}=(H_{0}^{s},J_{0}^{s})\in \mathbf{F}^{s}(F^{0},F^{G})\end{eqnarray}$$

and

$$\begin{eqnarray}F_{1}^{s}=(H_{1}^{s},J_{1}^{s})\in \mathbf{F}^{s}(F^{G},F^{1}),\end{eqnarray}$$

which we use to form the Floer homotopy data set

$$\begin{eqnarray}F_{0\#1}^{r,s}=\left\{\begin{array}{@{}ll@{}}F_{0}^{s+\unicode[STIX]{x1D709}(r)}\quad & \text{for }s\leqslant 0,\\ F_{1}^{s-\unicode[STIX]{x1D709}(r)}\quad & \text{for }s>0\end{array}\right.\end{eqnarray}$$

in $\mathbf{F}^{r,s}(F^{0},F^{1})$ . Perturbing these, if necessary, we may assume that the three previous data sets are all regular.

Proposition 2.19. Suppose that ${\mathcal{A}}_{H^{0}}(X^{0})-{\mathcal{A}}_{H^{1}}(X^{1})+\text{cost}(H^{r,s})<T_{\min }(\unicode[STIX]{x1D706}_{0})$ . If $X^{0}$ and $X^{1}$ are both simple and $\text{cost}(H_{0\#1}^{r,s})<-{\mathcal{A}}_{H}(X^{0}),$ then the manifold

$$\begin{eqnarray}{\mathcal{M}}_{r,s}^{1}(X^{0},X^{1};F_{0\#1}^{r,s})\end{eqnarray}$$

admits a compactification

$$\begin{eqnarray}\overline{{\mathcal{M}}}_{r,s}^{1}(X^{0},X^{1};F_{0\#1}^{r,s})\end{eqnarray}$$

which is a one-dimensional manifold whose boundary is

$$\begin{eqnarray}\displaystyle & & \displaystyle {\mathcal{M}}_{s}^{0}(X^{0},X^{1};F_{0\#1}^{0,s})\nonumber\\ \displaystyle & & \displaystyle \quad \cup \mathop{\bigcup }_{Z\in {\mathcal{P}}_{\mathbb{R}/\mathbb{Z}}^{-}(G)}{\mathcal{M}}_{s}^{0}(X^{0},Z;F_{0}^{s})\times {\mathcal{M}}_{s}^{0}(Z,X^{1};F_{1}^{s})\nonumber\\ \displaystyle & & \displaystyle \quad \cup \mathop{\bigcup }_{Y^{0}\in {\mathcal{P}}_{\mathbb{R}/\mathbb{Z}}^{-}(H^{0})}{\mathcal{M}}^{0}(X^{0},Y^{0};F^{0})\times {\mathcal{M}}_{r,s}^{0}(Y^{0},X^{1};F_{0\#1}^{r,s})\nonumber\\ \displaystyle & & \displaystyle \quad \cup \mathop{\bigcup }_{Y^{1}\in {\mathcal{P}}_{\mathbb{R}/\mathbb{Z}}^{-}(H^{1})}{\mathcal{M}}_{r,s}^{0}(X^{0},Y^{1};F_{0\#1}^{r,s})\times {\mathcal{M}}^{0}(Y^{1},X^{1};F^{1}).\nonumber\end{eqnarray}$$

2.7 Floer theory for finely tuned Hamiltonians

Throughout this section we consider a fixed nondegenerate rigid constellation ${\mathcal{C}}_{\unicode[STIX]{x1D706}_{0},\unicode[STIX]{x1D6FC}}(T)$ .

2.7.1 Homology

To every Hamiltonian $H$ which is finely tuned to ${\mathcal{C}}_{\unicode[STIX]{x1D706}_{0},\unicode[STIX]{x1D6FC}}(T)$ and every regular Floer data set $F=(H,J)\in \mathbf{F}_{\text{reg}}$ , we associate a version of Floer homology. Let ${\mathcal{P}}_{\unicode[STIX]{x1D6FC},\mathbb{R}/\mathbb{Z}}^{-}(H)$ be the set of $\mathbb{R}/\mathbb{Z}$ -families of closed $1$ -periodic orbits of $H$ which have negative action and which represent the class $\unicode[STIX]{x1D6FC}$ . Since $H$ is tuned, each family $X$ in ${\mathcal{P}}_{\unicode[STIX]{x1D6FC},\mathbb{R}/\mathbb{Z}}^{-}(H)$ corresponds to a unique $\mathbb{R}/\mathbb{Z}$ -family of closed Reeb orbits $\unicode[STIX]{x1D6E4}_{X}$ in ${\mathcal{C}}_{\unicode[STIX]{x1D706}_{0},\unicode[STIX]{x1D6FC}}(T)$ and vice versa. Since each family $\unicode[STIX]{x1D6E4}_{X}$ in ${\mathcal{C}}_{\unicode[STIX]{x1D706}_{0},\unicode[STIX]{x1D6FC}}(T)$ is simple, so are the families $X$ in ${\mathcal{P}}_{\unicode[STIX]{x1D6FC},\mathbb{R}/\mathbb{Z}}^{-}(H)$ .

Define the chain group by

$$\begin{eqnarray}\operatorname{CF}(H;\unicode[STIX]{x1D6FC})=\text{Span}_{\mathbb{Z}/2\mathbb{Z}}\{X:X\in {\mathcal{P}}_{\unicode[STIX]{x1D6FC},\mathbb{R}/\mathbb{Z}}^{-}(H)\}\end{eqnarray}$$

and the corresponding boundary map $\unicode[STIX]{x2202}_{F}:\operatorname{CF}(H;\unicode[STIX]{x1D6FC})\rightarrow \operatorname{CF}(H;\unicode[STIX]{x1D6FC})$ to be the linear operator defined on generators by

$$\begin{eqnarray}\unicode[STIX]{x2202}_{F}(X)=\mathop{\sum }_{Y\in {\mathcal{P}}_{\unicode[STIX]{x1D6FC},\mathbb{R}/\mathbb{Z}}^{-}(H)}\#{\mathcal{M}}^{0}(X,Y;F)Y.\end{eqnarray}$$

Here, and in what follows, for any finite set ${\mathcal{M}}$ the notation $\#{\mathcal{M}}$ will denote the number of elements modulo 2. By the definition of finely tuned, we have $\unicode[STIX]{x1D6E5}(H)<T_{\min }(\unicode[STIX]{x1D706}_{0})$ (see Lemma 2.7). It then follows from Proposition 2.15 that $\unicode[STIX]{x2202}_{F}$ is well defined and satisfies $\unicode[STIX]{x2202}_{F}\circ \unicode[STIX]{x2202}_{F}=0$ . Standard arguments imply that the resulting homology is independent of the choice of regular $J\in {\mathcal{J}}(H)$ and so we denote this homology by $\operatorname{HF}(H;\unicode[STIX]{x1D6FC})$ .

2.7.2 Continuation maps

Let $H^{0}$ and $H^{1}$ both be finely tuned to ${\mathcal{C}}_{\unicode[STIX]{x1D706}_{0},\unicode[STIX]{x1D6FC}}(T)$ . We now construct tools which allow us to compare $\operatorname{HF}(H^{0};\unicode[STIX]{x1D6FC})$ and $\operatorname{HF}(H^{1};\unicode[STIX]{x1D6FC}).$ Let

(34) $$\begin{eqnarray}\unicode[STIX]{x1D6E5}_{s}(H^{0},H^{1})=\min \biggl\{\frac{T_{\min }(\unicode[STIX]{x1D706}_{0},\unicode[STIX]{x1D6FC})}{2},T_{\min }(\unicode[STIX]{x1D706}_{0})-\unicode[STIX]{x1D6E5}(H^{0},H^{1})\biggr\}.\end{eqnarray}$$

By Lemma 2.7, $\unicode[STIX]{x1D6E5}_{s}(H^{0},H^{1})>0.$ Consider an admissible homotopy $H^{s}$ from $H^{0}$ to $H^{1}$ such that

(35) $$\begin{eqnarray}\text{cost}(H^{s})<\unicode[STIX]{x1D6E5}_{s}(H^{0},H^{1}).\end{eqnarray}$$

Perturbing $H^{s}$ if necessary, we choose regular Floer continuation data $F^{s}=(H^{s},J^{s})$ between regular Floer data $F^{0}=(H^{0},J^{0})$ and $F^{1}=(H^{1},J^{1})$ and define the linear map

$$\begin{eqnarray}\unicode[STIX]{x1D703}_{F^{s}}:\operatorname{CF}(H^{0};\unicode[STIX]{x1D6FC})\rightarrow \operatorname{CF}(H^{1};\unicode[STIX]{x1D6FC})\end{eqnarray}$$

on generators by

$$\begin{eqnarray}\unicode[STIX]{x1D703}_{F^{s}}(X^{i})=\mathop{\sum }_{Y^{j}\in {\mathcal{P}}_{\unicode[STIX]{x1D6FC},\mathbb{R}/\mathbb{Z}}^{-}(H^{1})}\#{\mathcal{M}}_{s}^{0}(X^{i},Y^{j};F^{s})Y^{j}.\end{eqnarray}$$

Proposition 2.16 implies that $\unicode[STIX]{x1D703}_{F^{s}}$ is a well-defined chain map. The usual arguments again imply that the resulting map in homology is independent of the choice of $J^{s}$ and so we denote this map by

$$\begin{eqnarray}\unicode[STIX]{x1D6E9}_{H^{s}}:\operatorname{HF}(H^{0};\unicode[STIX]{x1D6FC})\rightarrow \operatorname{HF}(H^{1};\unicode[STIX]{x1D6FC}).\end{eqnarray}$$

A convex combination of two homotopies that satisfy (35) also satisfies the same bound. Hence, the usual homotopy of homotopies argument in Floer theory can be used to show that the map $\unicode[STIX]{x1D6E9}_{H^{s}}$ does not depend on the choice of homotopy $H^{s}$ with cost less than $\unicode[STIX]{x1D6E5}_{s}(H^{0},H^{1})$ .

Lemma 2.20. If $H^{s}$ and $\widetilde{H}^{s}$ are two admissible homotopies from $H^{0}$ to $H^{1}$ with cost less than $\unicode[STIX]{x1D6E5}_{s}(H^{0},H^{1})$ , then the maps $\unicode[STIX]{x1D6E9}_{H^{s}}$ and $\unicode[STIX]{x1D6E9}_{\widetilde{H}^{s}}$ are equal.

In the present setting, the usual composition rule for continuation maps has the following form.

Lemma 2.21. Suppose that $H^{0}$ , $H^{1}$ and $H^{2}$ are Hamiltonians that are finely tuned to ${\mathcal{C}}_{\unicode[STIX]{x1D706}_{0},\unicode[STIX]{x1D6FC}}(T)$ and that $H_{10}^{s}$ is an admissible homotopy from $H^{0}$ to $H^{1}$ and $H_{21}^{s}$ is an admissible homotopy from $H^{1}$ to $H^{2}$ . If

(36) $$\begin{eqnarray}\displaystyle & \displaystyle \text{cost}(H_{10}^{s})<\unicode[STIX]{x1D6E5}_{s}(H^{0},H^{1}), & \displaystyle\end{eqnarray}$$

(37) $$\begin{eqnarray}\displaystyle & \displaystyle \text{cost}(H_{21}^{s})<\unicode[STIX]{x1D6E5}_{s}(H^{1},H^{2}) & \displaystyle\end{eqnarray}$$

and

(38) $$\begin{eqnarray}\text{cost}(H_{10}^{s})+\text{cost}(H_{21}^{s})<\unicode[STIX]{x1D6E5}_{s}(H^{0},H^{2}),\end{eqnarray}$$

then there is an admissible homotopy $H_{20}^{s}$ from $H^{0}$ to $H^{2}$ with cost at most $\text{cost}(H_{10}^{s})+\text{cost}(H_{21}^{s})$ such that

$$\begin{eqnarray}\unicode[STIX]{x1D6E9}_{H_{20}^{s}}=\unicode[STIX]{x1D6E9}_{H_{21}^{s}}\circ \unicode[STIX]{x1D6E9}_{H_{10}^{s}}.\end{eqnarray}$$

With these tools in place we can now begin the process of identifying the Floer homology groups associated to different finely tuned Hamiltonians. As above, the arguments we use are standard, but are complicated by the need to manage the cost of homotopies at each step.

Given a function $G:\mathbb{R}\times M\rightarrow \mathbb{R}$ , set

(39) $$\begin{eqnarray}\Vert G\Vert =\max _{\mathbb{R}\times M}G-\min _{\mathbb{R}\times M}G.\end{eqnarray}$$

Corollary 2.22. Suppose that the Hamiltonians $H^{0}$ and $H^{1}$ are finely tuned to ${\mathcal{C}}_{\unicode[STIX]{x1D706}_{0},\unicode[STIX]{x1D6FC}}(T)$ . If

(40) $$\begin{eqnarray}\displaystyle & \displaystyle \max _{\mathbb{R}\times M}(H^{1}-H^{0})<\unicode[STIX]{x1D6E5}_{s}(H^{0},H^{1}), & \displaystyle\end{eqnarray}$$

(41) $$\begin{eqnarray}\displaystyle & \displaystyle \max _{\mathbb{R}\times M}(H^{0}-H^{1})<\unicode[STIX]{x1D6E5}_{s}(H^{1},H^{0}) & \displaystyle\end{eqnarray}$$

and

(42) $$\begin{eqnarray}\Vert H^{1}-H^{0}\Vert <\unicode[STIX]{x1D6E5}_{s}(H^{0},H^{0}),\end{eqnarray}$$

then $\operatorname{HF}(H^{0};\unicode[STIX]{x1D6FC})$ and $\operatorname{HF}(H^{1};\unicode[STIX]{x1D6FC})$ are isomorphic.

Proof. Fix a smooth nondecreasing step function $\mathbf{step}:\mathbb{R}\rightarrow \mathbb{R}$ such that

$$\begin{eqnarray}\mathbf{step}(\unicode[STIX]{x1D70F})=\left\{\begin{array}{@{}ll@{}}0\quad & \text{for }\unicode[STIX]{x1D70F}\leqslant -1,\\ 1\quad & \text{for }\unicode[STIX]{x1D70F}\geqslant 0.\end{array}\right.\end{eqnarray}$$

Let

$$\begin{eqnarray}H^{s}=(1-\mathbf{step}(s))H^{0}+\mathbf{step}(s)H^{1}.\end{eqnarray}$$

We then have

$$\begin{eqnarray}\text{cost}(H^{s})=\max _{\mathbb{ R}\times M}(H^{1}-H^{0})\end{eqnarray}$$

and

$$\begin{eqnarray}\text{cost}(H^{-s})=\max _{\mathbb{ R}\times M}(H^{0}-H^{1}).\end{eqnarray}$$

By Lemma 2.21, there is an admissible homotopy $\widetilde{H}^{s}$ from $H^{0}$ to $H^{0}$ with

(43) $$\begin{eqnarray}\text{cost}(\widetilde{H}^{s})\leqslant \max _{\mathbb{ R}\times M}(H^{1}-H^{0})-\min _{\mathbb{ R}\times M}(H^{1}-H^{0})=\Vert H^{1}-H^{0}\Vert <\unicode[STIX]{x1D6E5}_{s}(H^{0},H^{0})\end{eqnarray}$$

such that

$$\begin{eqnarray}\unicode[STIX]{x1D6E9}_{\widetilde{H}^{s}}=\unicode[STIX]{x1D6E9}_{H^{-s}}\circ \unicode[STIX]{x1D6E9}_{H^{s}}:\operatorname{HF}(H^{0};\unicode[STIX]{x1D6FC})\rightarrow \operatorname{HF}(H^{0};\unicode[STIX]{x1D6FC}).\end{eqnarray}$$

By Lemma 2.20, the map $\unicode[STIX]{x1D6E9}_{\widetilde{H}^{s}}$ is the same as that corresponding to the constant homotopy from $H^{0}$ to itself and so $\unicode[STIX]{x1D6E9}_{\widetilde{H}^{s}}$ is an isomorphism. Thus, $\unicode[STIX]{x1D6E9}_{H^{s}}$ is injective. Applying the same argument to the composition $\unicode[STIX]{x1D6E9}_{H^{s}}\circ \unicode[STIX]{x1D6E9}_{H^{-}s}$ , we see that $\unicode[STIX]{x1D6E9}_{H^{s}}$ is also surjective.◻

Corollary 2.23. If $H^{0}$ and $H^{1}$ are finely tuned to ${\mathcal{C}}_{\unicode[STIX]{x1D706}_{0},\unicode[STIX]{x1D6FC}}(T)$ and $H^{1}$ is sufficiently $C^{1}$ -close to $H^{0}$ , then $\operatorname{HF}(H^{0};\unicode[STIX]{x1D6FC})$ is isomorphic to $\operatorname{HF}(H^{1};\unicode[STIX]{x1D6FC})$ .

Proof. This follows easily from Corollary 2.22 since the hypotheses of the theorem are met for all $H^{1}$ sufficiently $C^{1}$ -close to $H^{0}$ . For example, if $H^{k}$ is a sequence of finely tuned Hamiltonians converging to $H^{0}$ in the $C^{1}$ -topology, then $\max _{\mathbb{R}\times M}(H^{k}-H^{0})\rightarrow 0$ whereas $\unicode[STIX]{x1D6E5}_{s}(H^{0},H^{k})\rightarrow \unicode[STIX]{x1D6E5}_{s}(H^{0},H^{0})>0$ .◻

Corollary 2.24. Suppose that the Hamiltonians $H^{0}$ and $H^{1}$ are finely tuned to ${\mathcal{C}}_{\unicode[STIX]{x1D706}_{0},\unicode[STIX]{x1D6FC}}(T)$ and that $H^{s}$ is an admissible homotopy from $H^{0}$ to $H^{1}$ such that for each $s$ the Hamiltonian $H^{s}$ is also finely tuned to ${\mathcal{C}}_{\unicode[STIX]{x1D706}_{0},\unicode[STIX]{x1D6FC}}(T)$ . Then the following statements hold.

1. (i) The groups $\operatorname{HF}(H^{s};\unicode[STIX]{x1D6FC})$ are isomorphic to one another for all $s$ .

2. (ii) If, in addition, $\text{cost}(H^{s})=0$ , then the map

   $$\begin{eqnarray}\unicode[STIX]{x1D6E9}_{H^{s}}:\operatorname{HF}(H^{0};\unicode[STIX]{x1D6FC})\rightarrow \operatorname{HF}(H^{1};\unicode[STIX]{x1D6FC})\end{eqnarray}$$
   is an isomorphism.

Proof. Reparameterizing if necessary, we may assume that $H^{s}=H^{0}$ for all $s\leqslant 0$ and $H^{s}=H^{1}$ for all ${\geqslant}1$ . It follows from Corollary 2.22, and continuity, that for each $s^{\prime }\in [0,1]$ there is a $\unicode[STIX]{x1D6FF}_{s^{\prime }}>0$ such that $\operatorname{HF}(H^{\unicode[STIX]{x1D70D}};\unicode[STIX]{x1D6FC})$ is isomorphic to $\operatorname{HF}(H^{s^{\prime }};\unicode[STIX]{x1D6FC})$ for all $\unicode[STIX]{x1D70D}\in (s^{\prime }-\unicode[STIX]{x1D6FF}_{s^{\prime }},s^{\prime }+\unicode[STIX]{x1D6FF}_{s^{\prime }})$ . Covering $[0,1]$ by finitely many such intervals, it follows that for all $s\in [0,1]$ the groups $\operatorname{HF}(H^{s};\unicode[STIX]{x1D6FC})$ are isomorphic to one another.

To prove the second assertion of the corollary, it suffices (by Lemma 2.20) to find an admissible homotopy $\widetilde{H}^{s}$ from $H^{0}$ to $H^{1}$ with $\text{cost}(\widetilde{H}^{s})<\unicode[STIX]{x1D6E5}_{s}(H^{0},H^{1})$ such that $\unicode[STIX]{x1D6E9}_{\widetilde{H}^{s}}:\operatorname{HF}(H^{0};\unicode[STIX]{x1D6FC})\rightarrow \operatorname{HF}(H^{1};\unicode[STIX]{x1D6FC})$ is an isomorphism. We will use $H^{s}$ and Lemma 2.21 to construct this $\widetilde{H}^{s}$ .

Arguing as above, and invoking the proof of Corollary 2.22, we can find numbers $s_{0}=1<s_{1}<\cdots s_{N}=1$ such that for $k=0,\ldots ,N-1$ , each linear homotopy

$$\begin{eqnarray}G_{k}^{s}=H^{s_{k}}(1-\mathbf{step}(s))+H^{s_{k+1}}\mathbf{step}(s)\end{eqnarray}$$

induces an isomorphism $\unicode[STIX]{x1D6E9}_{G_{k}^{s}}:\operatorname{HF}(H^{s_{k}};\unicode[STIX]{x1D6FC})\rightarrow \operatorname{HF}(H^{s_{k+1}};\unicode[STIX]{x1D6FC})$ . From $H^{s}$ , we can also construct a homotopy

$$\begin{eqnarray}H_{k}^{s}=H^{(s_{k}+(s_{k+1}-s_{k})\mathbf{step}(s))}\end{eqnarray}$$

from $H^{s_{k}}$ to $H^{s_{k+1}}$ which is admissible and cost free. It follows from Lemma 2.20 that $\unicode[STIX]{x1D6E9}_{H_{k}^{s}}=\unicode[STIX]{x1D6E9}_{G_{k}^{s}}$ , so that each $\unicode[STIX]{x1D6E9}_{H_{k}^{s}}$ is an isomorphism. On the other hand, since each $H_{k}^{s}$ is cost free and each $\unicode[STIX]{x1D6E5}_{s}(H^{s_{k}},H^{s_{k+1}})$ is positive, we can invoke Lemma 2.21  $N+1$ times to obtain a cost-free admissible homotopy $\widetilde{H}^{s}$ from $H^{0}$ to $H^{1}$ such that

$$\begin{eqnarray}\unicode[STIX]{x1D6E9}_{\widetilde{H}^{s}}=\unicode[STIX]{x1D6E9}_{H_{N-1}^{s}}\circ \cdots \circ \unicode[STIX]{x1D6E9}_{H_{0}^{s}}.\end{eqnarray}$$

This completes the proof. ◻

Now we get to our main invariance result.

Proposition 2.25. The rank of $\operatorname{HF}(H;\unicode[STIX]{x1D6FC})$ is the same for every Hamiltonian $H$ that is finely tuned to the rigid constellation ${\mathcal{C}}_{\unicode[STIX]{x1D706}_{0},\unicode[STIX]{x1D6FC}}(T)$ .

Proof. We first observe that for all sufficiently small $\unicode[STIX]{x1D716}>0$ , every Hamiltonian in the space

$$\begin{eqnarray}{\mathcal{H}}_{\unicode[STIX]{x1D716}}(T)={\mathcal{H}}_{\unicode[STIX]{x1D716},T+\unicode[STIX]{x1D716},3(T+\unicode[STIX]{x1D716})/\unicode[STIX]{x1D716},0}\end{eqnarray}$$

is finely tuned to ${\mathcal{C}}_{\unicode[STIX]{x1D706}_{0},\unicode[STIX]{x1D6FC}}(T)$ . Moreover, there is an $\unicode[STIX]{x1D716}_{0}>0$ such that for any $\unicode[STIX]{x1D716},\unicode[STIX]{x1D716}^{\prime }<\unicode[STIX]{x1D716}_{0}$ and any Hamiltonians $H_{\unicode[STIX]{x1D716}}\in {\mathcal{H}}_{\unicode[STIX]{x1D716}}(T)$ and $H_{\unicode[STIX]{x1D716}^{\prime }}\in {\mathcal{H}}_{\unicode[STIX]{x1D716}^{\prime }}(T)$ , we have

$$\begin{eqnarray}\unicode[STIX]{x1D6E5}_{s}(H_{\unicode[STIX]{x1D716}},H_{\unicode[STIX]{x1D716}^{\prime }})>(T_{\min }(\unicode[STIX]{x1D706}_{0})+T_{\min }(\unicode[STIX]{x1D706}_{0},\unicode[STIX]{x1D6FC})-T)/2>0.\end{eqnarray}$$

It then follows from Corollary 2.22 that for all $\unicode[STIX]{x1D716}<\unicode[STIX]{x1D716}_{0}$ and every Hamiltonian $H_{\unicode[STIX]{x1D716}}$ in ${\mathcal{H}}_{\unicode[STIX]{x1D716}}(T)$ , the rank of $\operatorname{HF}(H_{\unicode[STIX]{x1D716}};\unicode[STIX]{x1D6FC})$ is the same.

By Corollary 2.24, it now suffices to show that given any finely tuned Hamiltonian $H$ there is an admissible homotopy $H^{s}$ which consists of finely tuned Hamiltonians and connects $H$ to some $H_{\unicode[STIX]{x1D716}}\in {\mathcal{H}}_{\unicode[STIX]{x1D716}}(T)$ with $\unicode[STIX]{x1D716}<\unicode[STIX]{x1D716}_{0}$ . The starting point of $H^{s}$ , $H$ , belongs to ${\mathcal{H}}_{a,b,c,\unicode[STIX]{x1D705}}$ for some $h$ in $\mathfrak{h}_{a,b,c}$ . For the end point $H_{\unicode[STIX]{x1D716}}$ , we choose $\unicode[STIX]{x1D716}<\unicode[STIX]{x1D716}_{0}$ small enough so that the following inequalities hold:

$$\begin{eqnarray}\unicode[STIX]{x1D716}<a,\quad T+\unicode[STIX]{x1D716}<b\quad \text{and}\quad 3(T+\unicode[STIX]{x1D716})/\unicode[STIX]{x1D716}>c.\end{eqnarray}$$

We now view $h$ as belonging to a smooth family of functions $h(A,B,C)$ such that $h(A,B,C)$ belongs to $\mathfrak{h}_{A,B,C}$ . The segments of the path $H^{s}$ will then be defined by varying the parameters $A$ , $B$ , $C$ and $\unicode[STIX]{x1D705}$ one at a time.

We begin with $\unicode[STIX]{x1D705}$ . The distinction between a tuned and a finely tuned Hamiltonian involves only the relationship between $a$ and $\unicode[STIX]{x1D705}$ and, if $a$ works for $\unicode[STIX]{x1D705}$ , then it also works for all smaller values of $\unicode[STIX]{x1D705}$ . So, the first segment of the path $H^{s}$ will be

$$\begin{eqnarray}s\in [0,1]\mapsto h(a,b,c)(e^{\unicode[STIX]{x1D70F}-(1-s)\unicode[STIX]{x1D705}}).\end{eqnarray}$$

The next segment increases $C$ from $c$ to $3(T+\unicode[STIX]{x1D716})/\unicode[STIX]{x1D716}$ . Again it follows easily from the definitions that the intermediate Hamiltonians remain finely tuned. Continuing in this way, we decrease $B$ from $b$ to $T+\unicode[STIX]{x1D716}$ and finally decrease $A$ from $a$ to $\unicode[STIX]{x1D716}$ . By joining these four segments in order and reparameterizing to smooth the transitions between them, we obtain the desired homotopy  $H^{s}$ .◻