Completeness of derived interleaving distances and sheaf quantization of non-smooth objects

Abstract

We develop sheaf-theoretic methods to deal with non-smooth objects in symplectic geometry. We show the completeness of a derived category of sheaves with respect to the interleaving distance and construct a sheaf quantization of a Hamiltonian homeomorphism. We also develop Lusternik–Schnirelmann theory in the microlocal theory of sheaves. With these new sheaf-theoretic methods, we prove an Arnold-type theorem for the image of a compact exact Lagrangian submanifold under a Hamiltonian homeomorphism.

A Hamiltonian stability with support conditions

In this appendix, we prove an estimate by the oscillation norm $\Vert H\Vert _{\textrm{osc},A}$ of H restricted to a non-empty closed subset A, in the context of the sheaf-theoretic energy estimate.

1.1 A.1 Sheaf quantization of 2-parameter Hamiltonian isotopies

We will use the sheaf quantization of a 2-parameter Hamiltonian isotopy. For that purpose, we first state the main result of [21] in a general form. Let N be a connected non-empty manifold and W a contractible open subset of ${\mathbb {R}}^n$ with the coordinate system $(w_1,\dots ,w_n)$ containing 0. Let us consider $\psi =(\psi _w)_{w \in W} :\mathring{T}^*N \times W \rightarrow \mathring{T}^*N$ be a homogeneous Hamiltonian isotopy, that is, a $C^\infty$-map satisfying (1) $\psi _w$ is homogeneous symplectic isomorphism for each $w \in W$ and (2) $\psi _0={\text {id}}_{\mathring{T}^*N}$. We can define a vector-valued homogeneous function $h :\mathring{T}^*N \times W \rightarrow {\mathbb {R}}^n$ by $h=(h_1,\dots ,h_n)$ with

$$\begin{aligned} \frac{\partial \psi _w}{\partial w_i} \circ \psi _{w}^{-1} = X_{h_i(\bullet ,w)}, \end{aligned}$$

(A.1)

where $X_{h_i(\bullet ,w)}$ is the Hamiltonian vector field of the function $h_i(\bullet ,w) :\mathring{T}^*N \rightarrow {\mathbb {R}}$. By using the function h, we define a conic Lagrangian submanifold $\Lambda _\psi$ of $\mathring{T}^*(N^2 \times W)$ by

$$\begin{aligned} \Lambda _\psi \,{:}{=}\,\left\{ \left( \psi _w(y;\eta ), (y;-\eta ), (u;-h(\psi _w(y;\eta ),w)) \right) \mathrel {}|\mathrel {}(y;\eta ) \in \mathring{T}^*N, w \in W \right\} \nonumber \\ \end{aligned}$$

(A.2)

The main theorem of [21] is the following.

Theorem A.1

([21, Thm. 3.7 and Rem. 3.9]) Let $\psi :\mathring{T}^*N \times W \rightarrow \mathring{T}^*N$ be a homogeneous Hamiltonian isotopy and set $\Lambda _\psi$ as above. Then there exists a unique simple object $\widetilde{K} \in \textsf{D}({\textbf{k}}_{N^2 \times W})$ such that ${{\,\mathrm{{\mathring{{{\,\mathrm{{{\text {SS}}}}\,}}}}}\,}}(\widetilde{K})=\Lambda _{\psi }$ and $\widetilde{K}|_{N^2 \times \{0\}} \simeq {\textbf{k}}_{\Delta _{N}}$.

For a non-homogeneous compactly supported Hamiltonian isotopy, we can associate a sheaf by homogenizing the isotopy. In the 1-parameter case, it is done as in Definition 3.10. Below we will explain how to homogenize a 2-parameter Hamiltonian isotopy.

Let $(G_{s',s})_{(s',s) \in I^2}$ be a 2-parameter family of compactly supported smooth functions on $T^*M$. A 2-parameter family of diffeomorphisms $(\phi _{s',s})_{(s',s)\in I^2}$ is determined by $\phi _{s',0}={\text {id}}_{T^*M}$ and $\frac{\partial \phi _{s',s} }{\partial s}\circ \phi _{s',s}^{-1}=X_{G_{s',s}}$, where $X_{G_{s',s}}$ is the Hamiltonian vector field corresponding to the function $G_{s',s}$. We set $\widehat{G}_{s',s}(x,t;\xi ,\tau ) \,{:}{=}\,\tau G_{s',s}(x;\xi /\tau )$ and define a 2-parameter homogeneous Hamiltonian isotopy $\widehat{\phi }=(\widehat{\phi }_{s',s})_{s',s}$ by

$$\begin{aligned} {\left\{ \begin{array}{ll} \widehat{\phi }_{s',0}={\text {id}}_{\mathring{T}^*(M \times {\mathbb {R}}_t)}, \\ \frac{\partial \widehat{\phi }_{s',s}}{\partial s} \circ \widehat{\phi }_{s',s}^{-1} = X_{\widehat{G}_{s',s}}. \end{array}\right. } \end{aligned}$$

(A.3)

Then, we have

$$\begin{aligned} \Lambda _{\widehat{\phi }}= & {} \left\{ \left( \widehat{\phi }_{s',s}(y;\eta ), (y;-\eta ), (s'; -\widehat{F}_{s',s}(\widehat{\phi }_{s',s}(y;\eta ))), (s;- \widehat{G}_{s',s}(\widehat{\phi }_{s',s}(y;\eta ))) \right) \bigg | \right. \nonumber \\{} & {} \quad \left. (y;\eta ) \in \mathring{T}^*(M \times {\mathbb {R}}), s',s \in I \right\} , \end{aligned}$$

(A.4)

where the 2-parameter family of homogeneous functions $(\widehat{F}_{s',s})_{s',s}$ is determined by $\frac{\partial \widehat{\phi }_{s',s}}{\partial s'} \circ \widehat{\phi }_{s',s}^{-1}=X_{\widehat{F}_{s',s}}$. By the construction of $\widehat{\phi }$, there exists a 2-parameter family of timewise compactly supported functions $(F_{s',s})_{(s',s) \in I^2}$ satisfying $\widehat{F}_{s',s}(x,t;\xi ,\tau )=\tau F_{s',s}(x;\xi /\tau )$ and $\frac{\partial \phi _{s',s} }{\partial s'}\circ \phi _{s',s}^{-1}=X_{F_{s',s}}$. A calculation in [39] or [35] (see also [3]) shows that

$$\begin{aligned} \frac{\partial F_{s',s}}{\partial s}=\frac{\partial G_{s',s}}{\partial s'}-\{F_{s',s},G_{s',s}\}, \end{aligned}$$

(A.5)

where $\{-,-\}$ is the Poisson bracket. In this case, we can apply Theorem A.1 to the homogeneous Hamiltonian isotopy $\widehat{\phi }$ and obtain a simple object $\widetilde{K}$ satisfying ${{\,\mathrm{{\mathring{{{\,\mathrm{{{\text {SS}}}}\,}}}}}\,}}(\widetilde{K})=\Lambda _{\widehat{\phi }}$. By using the map $q :(M \times {\mathbb {R}})^2 \times I^2 \rightarrow M^2 \times {\mathbb {R}}_t \times I^2, (x_1,t_1,x_2,t_2,s',s) \mapsto (x_1,x_2,t_1-t_2,s',s)$, we also obtain an equivalence similar to (3.26). Hence, we can define $K \in \textsf{D}({\textbf{k}}_{M^2 \times {\mathbb {R}}\times I^2})$ by the condition ${{\,\mathrm{{\mathring{{{\,\mathrm{{{\text {SS}}}}\,}}}}}\,}}(K)=q_d q_\pi ^{-1} (\Lambda _{\widehat{\phi }})$ and $\mathcal {K}\,{:}{=}\,P_l(K) \in \mathcal {D}(M^2 \times I^2)$, which we call the sheaf quantization of $(\phi _{s',s})_{(s',s)\in I^2}$.

1.2 Statement and proof

For a closed subset A of $T^*M$, we define a full subcategory $\mathcal {D}_A(M)$ of $\mathcal {D}(M)$ by

$$\begin{aligned} \mathcal {D}_A(M) \,{:}{=}\,\{ F \in \mathcal {D}(M) \mid {{\,\mathrm{{{\text {SS}}}}\,}}(F) \cap \{ \tau >0\} \subset \rho _t^{-1}(A) \}, \end{aligned}$$

(A.6)

where $\rho _t :T^*M \times T^*_{\tau >0}{\mathbb {R}}_t \rightarrow T^*M, (x,t;\xi ,\tau ) \mapsto (x;\xi /\tau )$.

Let $\mathcal {K}^H \in \mathcal {D}(M^2 \times I)$ be the sheaf quantization associated with a timewise compactly supported function $H :T^*M \times I \rightarrow {\mathbb {R}}$ and $F \in \mathcal {D}_A(M)$ with A being a closed subset of $T^*M$. Then we get $\mathcal {K}^H \bullet F \in \mathcal {D}(M \times I)$ and find that

$$\begin{aligned} \mathcal {K}^H_s \bullet F \simeq (\mathcal {K}^H \bullet F)|_{M \times \{s\} \times {\mathbb {R}}_t} \in \mathcal {D}_{\phi ^H_s(A)}(M) \quad \text {for any}\, s \in I. \end{aligned}$$

(A.7)

We shall estimate the distance between $F \in \mathcal {D}_A(M)$ and $\mathcal {K}^H_1 \bullet F \in \mathcal {D}_{\phi ^H_1(A)}(M)$ up to translation. See also Remark A.4 for a more straightforward but weaker case.

Theorem A.2

Let A be a non-empty closed subset of $T^*M$ and $F \in \mathcal {D}_A(M)$. Moreover, let $H :T^*M \times I \rightarrow {\mathbb {R}}$ be a timewise compactly supported function. Then for a continuous function $f:I\rightarrow {\mathbb {R}}$, one has

$$\begin{aligned} \begin{aligned}&d_{\mathcal {D}(M)}(F,T_{-c} \mathcal {K}^H_1 \bullet F) \\&\quad \le {} \int _0^1 \left( \max \left\{ \max _{p \in \phi ^H_s(A)}H_s(p), f(s) \right\} -\min \left\{ \min _{p \in \phi ^H_s(A)}H_s(p), f(s)\right\} \right) ds \end{aligned}\qquad \end{aligned}$$

(A.8)

where $c=\int _0^1 f(s) ds$.

Remark A.3

If we take $f\equiv 0$, we obtain

$$\begin{aligned} \begin{aligned}&d_{\mathcal {D}(M)}(F, \mathcal {K}^H_1 \bullet F) \\&\quad \le {} \int _0^1 \left( \max \left\{ \max _{p \in \phi ^H_s(A)}H_s(p), 0 \right\} -\min \left\{ \min _{p \in \phi ^H_s(A)}H_s(p), 0\right\} \right) ds. \end{aligned} \end{aligned}$$

(A.9)

Let $c \in {\mathbb {R}}$ be a real number satisfying

$$\begin{aligned} \int _0^1 \min _{p \in \phi ^H_s(A)}H_s(p) ds\le c \le \int _0^1 \max _{p \in \phi ^H_s(A)}H_s(p) ds. \end{aligned}$$

(A.10)

Then we can take f such that $c=\int _0^1 f(s) ds$ and

$$\begin{aligned} \min _{p \in \phi ^H_s(A)}H_s(p) \le f(s)\le \max _{p \in \phi ^H_s(A)}H_s(p) \end{aligned}$$

(A.11)

for any $s \in I$. Hence, by Theorem A.2, we get

$$\begin{aligned} d_{\mathcal {D}(M)}(F, T_{-c} \mathcal {K}^H_1 \bullet F) \le \int _0^1 \left( \max _{p \in \phi ^H_s(A)}H_s(p) - \min _{p \in \phi ^H_s(A)}H_s(p) \right) ds.\qquad \end{aligned}$$

(A.12)

For simplicity, we introduce a symbol for the right-hand side of (A.8). For a function $H :T^*M \times I \rightarrow {\mathbb {R}}$, a function $f :I \rightarrow {\mathbb {R}}$, and a non-empty closed subset A of $T^*M$, we set

$$\begin{aligned} B(H,f,A) \,{:}{=}\,\int _0^1 \left( \max \left\{ \max _{p \in \phi ^H_s(A)}H_s(p), f(s) \right\} -\min \left\{ \min _{p \in \phi ^H_s(A)}H_s(p), f(s) \right\} \right) ds.\nonumber \\ \end{aligned}$$

(A.13)

Proof of Theorem A.2

Let $\varepsilon >0$. We can take a smooth family $(\rho _{a,b}:{\mathbb {R}}\rightarrow {\mathbb {R}})_{a,b}$ of smooth functions parametrized by $a,b\in {\mathbb {R}}$ with $a\le b$ such that

• (1) $\rho _{a,b}(y)=y$ on a neighborhood of [a, b],

• (2) $a-\varepsilon \le \inf _y \rho _{a,b}(y) < \sup _y \rho _{a,b}(y) \le b+\varepsilon$.

Recall that I denotes an open interval containing the closed interval [0, 1]. We take smooth functions $M, m :I\rightarrow {\mathbb {R}}$ satisfying

$$\begin{aligned} \max _{p \in \phi ^H_s(A)}H_s(p) +\frac{\varepsilon }{2} \le M(s) \le \max _{p \in \phi ^H_s(A)}H_s(p) +\varepsilon \end{aligned}$$

(A.14)

and

$$\begin{aligned} \min _{p \in \phi ^H_s(A)}H_s(p) -\varepsilon \le m(s) \le \min _{p \in \phi ^H_s(A)}H_s(p)-\frac{\varepsilon }{2}. \end{aligned}$$

(A.15)

Fix $R>0$ sufficiently large so that $R> \max _{p,s}H_s(p) -\min _{p,s}H_s(p) +2\varepsilon$. Define $a(s', s)\,{:}{=}\,m(s)-Rs', b(s',s)\,{:}{=}\,M(s)+Rs'$ for $(s',s)\in I^2$. We may assume that $I\subset (-\frac{\varepsilon }{2R},+\infty )$ by taking I smaller if necessary, and hence that

$$\begin{aligned} a(s', s) \le \min _{p \in \phi ^H_s(A)}H_s(p) \le \max _{p \in \phi ^H_s(A)}H_s(p) \le b(s',s) \end{aligned}$$

(A.16)

for all $(s',s)\in I^2$. Take a smooth function $\tilde{f}:I\rightarrow {\mathbb {R}}$ such that $\Vert \tilde{f}-f\Vert _{C^0} \le \varepsilon$. By shrinking I, we may assume that $\bigcup _{s} {{\,\mathrm{{\text {supp}}}\,}}(H_s)$ is relatively compact. Then we can also take a compactly supported smooth cut-off function $\chi :T^*M\rightarrow [0,1]$ such that $\chi \equiv 1$ on a neighborhood of $\bigcup _{s} {{\,\mathrm{{\text {supp}}}\,}}(H_s)$. Using these functions, we define a function $G=(G_{s',s})_{s',s \in I^2} :T^*M \times I^2 \rightarrow {\mathbb {R}}$ by

$$\begin{aligned} G_{s',s}\,{:}{=}\,\left( \rho _{a(s',s),b(s',s)}\circ H_s - (1-s') \tilde{f}(s)\right) \chi . \end{aligned}$$

(A.17)

A 2-parameter family $(\phi _{s',s})_{(s',s)\in I^2}$ of Hamiltonian diffeomorphisms is determined by $\phi _{s',0}={\text {id}}_{T^*M}$ and $\frac{\partial \phi _{s',s} }{\partial s}\circ \phi _{s',s}^{-1}=X_{G_{s',s}}$, where $X_{G_{s',s}}$ is the Hamiltonian vector field corresponding to the function $G_{s',s}$. Note that $G_{1,s}=H_s$ and $\phi _{s',s}$ is independent of $s'$ on a neighborhood U of A. Moreover, we have

$$\begin{aligned}{} & {} \int _0^1 \left( \max _{p \in T^*M} G_{0,s}(p) - \min _{p \in T^*M} G_{0,s}(p) \right) ds \nonumber \\{} & {} \quad \le {} \int _0^1 \left( \max _{p \in T^*M} \left( \rho _{m(s),M(s)} \circ H_s(p) - \tilde{f}(s) \right) \chi (p) - \min _{p \in T^*M} \left( \rho _{m(s),M(s)} \circ H_s(p) - \tilde{f}(s) \right) \chi (p) \right) ds\nonumber \\{} & {} \quad \le {} \int _0^1 \left( \max \left\{ \max _{p \in \phi ^H_s(A)} \left( H_s(p) - \tilde{f}(s) \right) , 0 \right\} -\min \left\{ \min _{p \in \phi ^H_s(A)} \left( H_s(p) - \tilde{f}(s) \right) , 0\right\} \right) ds +2\varepsilon \nonumber \\{} & {} \quad = {} \int _0^1 \left( \max \left\{ \max _{p \in \phi ^H_s(A)} H_s(p), \tilde{f}(s) \right\} -\min \left\{ \min _{p \in \phi ^H_s(A)} H_s(p), \tilde{f}(s) \right\} \right) ds +2\varepsilon \nonumber \\{} & {} \quad \le {} B(H,f,A) +4\varepsilon . \end{aligned}$$

(A.18)

For $s' \in I$, we set $G_{s'} \,{:}{=}\,G_{s',\bullet } :T^*M \times I \rightarrow {\mathbb {R}}$. Then, by Theorem 5.1 and the natural inequality for the distance with respect to functorial operations (see (3.11)), we obtain

$$\begin{aligned} d_{\mathcal {D}(M)}(F, \mathcal {K}^{G_0}_1 \bullet F) = d_{\mathcal {D}(M)}(\mathcal {K}^0_1 \bullet F, \mathcal {K}^{G_0}_1 \bullet F) \le B(H,f,A) +4\varepsilon .\qquad \end{aligned}$$

(A.19)

We set $\tilde{c}(s)\,{:}{=}\,\int _0^s \tilde{f}(t) dt$ and claim that $\mathcal {K}^{G_0}_1 \bullet F \simeq T_{-\tilde{c}(1)}\mathcal {K}^{G_1}_1 \bullet F$. By the result recalled in appendix A.1, we can construct the sheaf quantization $\mathcal {K}\in \mathcal {D}(M^2 \times I^2)$ of the 2-parameter family of diffeomorphisms $(\phi _{s',s})_{s',s}$. We shall use the same notation as in appendix A.1. Then, $F_{s',0}=0$ and $F_{\bullet ,s}|_{\phi _{1,s}(U)\times I}:\phi _{1,s}(U)\times I\rightarrow {\mathbb {R}}, (p,s')\mapsto F_{s',s}(p)$ is locally constant for each s. By (A.5), we find that $\frac{\partial F_{s',s}}{\partial s}=\frac{\partial G_{s',s}}{\partial s'}=\tilde{f}(s)$ on $\bigcup _s \phi _{1,s}(U)\times I\times \{s\}$ and that $F_{s',s} = \int _0^s \tilde{f}(t) dt=\tilde{c}(s)$ there. We define $\mathcal {H}\,{:}{=}\,\mathcal {K}\bullet F \in \mathcal {D}(M \times I^2)$. Then, by the microsupport estimate, we have

$$\begin{aligned} \begin{aligned} {{\,\mathrm{{{\text {SS}}}}\,}}(\mathcal {H})&\subset \left\{ \left( \widehat{\phi }_{s',s}(x,t;\xi ,\tau ), (s';-\tau \tilde{c}(s)), (s;-\tau G_{s',s}(\phi _{s',s}(x;\xi /\tau ))) \right) \; \bigg | \right. \\&\qquad \quad \left. (x,t;\xi ,\tau ) \in {{\,\mathrm{{\mathring{{{\,\mathrm{{{\text {SS}}}}\,}}}}}\,}}(F), s',s \in I \right\} \cup 0_{M \times {\mathbb {R}}\times I^2}. \end{aligned} \end{aligned}$$

(A.20)

Hence, $M \times {\mathbb {R}}\times I \times \{1\}$ is non-characteristic for $\mathcal {H}$ and we get

$$\begin{aligned} \begin{aligned} {{\,\mathrm{{{\text {SS}}}}\,}}(\mathcal {H}|_{M \times {\mathbb {R}}\times I \times \{1\}}) \subset&\left\{ \left( \widehat{\phi }_{s',1}(x,t;\xi ,\tau ), (s';- \tilde{c}(1) \tau ) \right) \; \bigg | \right. \\&\quad \left. (x,t;\xi ,\tau ) \in {{\,\mathrm{{\mathring{{{\,\mathrm{{{\text {SS}}}}\,}}}}}\,}}(F), s' \in I \right\} \cup 0_{M \times {\mathbb {R}}\times I}. \end{aligned} \end{aligned}$$

(A.21)

Define a diffeomorphism $\varphi :M \times {\mathbb {R}}\times I \xrightarrow {\sim }M \times {\mathbb {R}}\times I, (x,t,s') \mapsto (x,t-\tilde{c}(1)s', s')$. Then we have ${{\,\mathrm{{{\text {SS}}}}\,}}(\varphi _* \mathcal {H}|_{M \times {\mathbb {R}}\times I \times \{1\}}) \subset T^*(M \times {\mathbb {R}}) \times 0_{I}$, which shows $\varphi _* \mathcal {H}|_{M \times {\mathbb {R}}\times I \times \{1\}}$ is the pull-back of a sheaf on $M \times {\mathbb {R}}$ by [23, Prop. 5.4.5]. In particular,

$$\begin{aligned} \begin{aligned} \mathcal {K}^{G_0}_1 \bullet F&= \mathcal {H}|_{M \times {\mathbb {R}}\times \{0\} \times \{1\}} \\&\simeq (\varphi _* \mathcal {H}|_{M \times {\mathbb {R}}\times I \times \{1\}})|_{\{ s'=0\}} \\&\simeq (\varphi _* \mathcal {H}|_{M \times {\mathbb {R}}\times I \times \{1\}})|_{\{ s'=1\}} \\&\simeq T_{-\tilde{c}(1)}\mathcal {H}|_{M \times {\mathbb {R}}\times \{1\} \times \{1\}} = T_{-\tilde{c}(1)} \mathcal {K}^{G_1}_1 \bullet F. \end{aligned} \end{aligned}$$

(A.22)

Since $|c-\tilde{c}(1)| \le \varepsilon$, we have $d_{\mathcal {D}(M)}(T_{-c}\mathcal {K}^H_1 \bullet F, T_{-\tilde{c}(1)}\mathcal {K}^H_1 \bullet F) \le \varepsilon$. Combining the result above and noticing $G_1=H$, we obtain

$$\begin{aligned} \begin{aligned} d_{\mathcal {D}(M)}(F, T_{-c} \mathcal {K}^H_1 \bullet F)&\le d_{\mathcal {D}(M)}(F, T_{-\tilde{c}(1)} \mathcal {K}^H_1 \bullet F) + \varepsilon \\&= d_{\mathcal {D}(M)}(F, \mathcal {K}^{G_0}_1 \bullet F) + \varepsilon \\&\le B(H,f,A) +5\varepsilon . \end{aligned} \end{aligned}$$

(A.23)

Since $\varepsilon >0$ is arbitrary, this completes the proof. $\square$

Remark A.4

Under the same assumption as in Theorem A.2, we can prove the weaker result

$$\begin{aligned} d_{\text {w-isom}}(F,T_{-c} \mathcal {K}^H_1 \bullet F) \le B(H,f,A) \end{aligned}$$

(A.24)

more straightforwardly, without the 2-parameter family, as follows. Here $d_{\text {w{-}isom}}$ denotes the pseudo-distance on $\mathcal {D}(M)$ defined by

$$\begin{aligned} d_{\text {w{-}isom}}(F,G) \,{:}{=}\,\inf \left\{ a+b \mathrel {}|\mathrel {}(F,G)~\text {is weakly}\, (a,b)\text {-isomorphic} \right\} . \end{aligned}$$

(A.25)

We set $\mathcal {H}\,{:}{=}\,\mathcal {K}^H \bullet F \in \mathcal {D}(M \times I)$. Then we have $\mathcal {H}|_{M \times {\mathbb {R}}_t \times \{0\}} \simeq F$ and $\mathcal {H}|_{M \times {\mathbb {R}}_t \times \{1\}} \simeq \mathcal {K}^H_1 \bullet F$. Moreover, by the microsupport estimate, we find that

$$\begin{aligned} {{\,\mathrm{{{\text {SS}}}}\,}}(\mathcal {H}) \subset T^*M \times \left\{ (t,s;\tau ,\sigma ) \;\bigg |\; -\max _{p \in \phi ^H_s(A)} H_s(p) \cdot \tau \le \sigma \le -\min _{p \in \phi ^H_s(A)} H_s(p) \cdot \tau \right\} .\nonumber \\ \end{aligned}$$

(A.26)

Let $\varepsilon >0$ and take a smooth function $\tilde{f}:I\rightarrow {\mathbb {R}}$ such that $\Vert \tilde{f}-f\Vert _{C^0} \le \varepsilon$. We define a function $\tilde{c} :I \rightarrow {\mathbb {R}}$ by $\tilde{c}(s) \,{:}{=}\,\int _0^s \tilde{f}(s')ds'$ and a function $\varphi :M \times {\mathbb {R}}_t \times I \rightarrow M \times {\mathbb {R}}_t \times I$ by $\varphi (x,t,s) \,{:}{=}\,(x,t-\tilde{c}(s),s)$. Then we have $\varphi _* \mathcal {H}|_{M \times {\mathbb {R}}_t \times \{0\}} \simeq F$, $\varphi _* \mathcal {H}|_{M \times {\mathbb {R}}_t \times \{1\}} \simeq T_{-\tilde{c}(1)}\mathcal {K}^H_1 \bullet F$, and

$$\begin{aligned} \begin{aligned}&{{\,\mathrm{{{\text {SS}}}}\,}}(\varphi _* \mathcal {H}) \\&\quad \subset {} T^*M \times \left\{ (t,s;\tau ,\sigma ) \;\bigg |\; - \left( \max _{p \in \phi ^H_s(A)} H_s(p) -\tilde{f}(s) \right) \cdot \tau \le \sigma \right. \\&\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \left. \le - \left( \min _{p \in \phi ^H_s(A)} H_s(p) - \tilde{f}(s) \right) \cdot \tau \right\} . \end{aligned} \end{aligned}$$

(A.27)

Note that we may have $\max _{p \in \phi ^H_s(A)} H_s(p) -\tilde{f}(s) < 0$ and $\min _{p \in \phi ^H_s(A)} H_s(p) - \tilde{f}(s) >0$ in general. By applying Proposition 3.9, we obtain

$$\begin{aligned} \begin{aligned}&d_{\text {w{-}isom}}(F,T_{-\tilde{c}(1)}\mathcal {K}^H_1 \bullet F) \\&\quad = {} d_{\text {w{-}isom}}(\varphi _* \mathcal {H}|_{M \times {\mathbb {R}}_t \times \{0\}}, \varphi _* \mathcal {H}|_{M \times {\mathbb {R}}_t \times \{1\}}) \\&\quad \le {} \int _0^1 \left( \max \left\{ \max _{p \in \phi ^H_s(A)} \left( H_s(p) - \tilde{f}(s) \right) , 0 \right\} -\min \left\{ \min _{p \in \phi ^H_s(A)} \left( H_s(p) - \tilde{f}(s) \right) , 0\right\} \right) ds\\&\quad = {} \int _0^1 \left( \max \left\{ \max _{p \in \phi ^H_s(A)} H_s(p), \tilde{f}(s) \right\} -\min \left\{ \min _{p \in \phi ^H_s(A)} H_s(p), \tilde{f}(s) \right\} \right) ds \\&\quad \le {} \int _0^1 \left( \max \left\{ \max _{p \in \phi ^H_s(A)}H_s(p), f(s) \right\} -\min \left\{ \min _{p \in \phi ^H_s(A)}H_s(p), f(s)\right\} \right) ds + 2\varepsilon . \end{aligned} \end{aligned}$$

(A.28)

Hence, we have

$$\begin{aligned} \begin{aligned}&d_{\text {w{-}isom}}(F, T_{-c}\mathcal {K}^H_1 \bullet F) \le d_{\text {w{-}isom}}(F, T_{-\tilde{c}(1)} \mathcal {K}^H_1 \bullet F) + \varepsilon \\&\quad \le {} \int _0^1 \left( \max \left\{ \max _{p \in \phi ^H_s(A)}H_s(p), f(s) \right\} -\min \left\{ \min _{p \in \phi ^H_s(A)}H_s(p), f(s)\right\} \right) ds + 3\varepsilon , \end{aligned}\nonumber \\ \end{aligned}$$

(A.29)

which completes the proof.

For a timewise compactly supported function $H :T^*M \times I \rightarrow {\mathbb {R}}$ and a non-empty closed subset A of $T^*M$, we set

$$\begin{aligned} \Vert H\Vert _{\textrm{osc},A} \,{:}{=}\,\int _0^1 \left( \max _{p \in A} H_s(p) - \min _{p \in A} H_s(p) \right) ds. \end{aligned}$$

(A.30)

Proposition A.5

Let A be a non-empty closed subset of $T^*M$ and $F \in \mathcal {D}_A(M)$. Moreover, let $H :T^*M \times I \rightarrow {\mathbb {R}}$ be a timewise compactly supported function. Then there exists $c \in {\mathbb {R}}$ such that

$$\begin{aligned} d_{\mathcal {D}(M)}(F,T_{-c}\mathcal {K}^H_1 \bullet F) \le \Vert H\Vert _{\textrm{osc},A}. \end{aligned}$$

(A.31)

Proof

Using the technique in the proof of [45, Theorem 1.3], one can construct a function $H'$ such that $\phi ^H_1=\phi ^{H'}_1$ and

$$\begin{aligned} \begin{aligned}&\int _0^1 \left( \max _{p \in A}H_s(p) - \min _{p \in A}H_s(p) \right) ds \\&\quad ={} \int _0^1 \left( \max _{p \in \phi ^{H'}_s(A)}H'_s(p) - \min _{p \in \phi ^{H'}_s(A)}H'_s(p) \right) ds. \end{aligned} \end{aligned}$$

(A.32)

By Proposition 5.9, $\phi ^H_1=\phi ^{H'}_1$ implies $\mathcal {K}^H_1 \simeq \mathcal {K}^{H'}_1$. Hence, the result follows from Theorem A.2 (see also Remark A.3). $\square$