{C}_0^{\ast,\ell}(I),\mathcal{C}_0^{\ast,G}(I),0)\},\,\,\,\ast=\mathrm{f},\mathrm{b}\\
\widetilde\lambda_I^j(\sigma) &=
\widetilde\Phi_0\{\sigma,(\mathcal{G}_0^{j,L}(I),\mathcal{G}_0^{j,\ell}(I),\mathcal{G}_0^{j,G}(I),0)\},\,\,\, j=3,4\\
\end{split}
\end{equation}
where $\widetilde\Phi_0$ is defined in \eqref{eq:Flow:Circ:Time} and $\mathcal{C}_0^\ast$
and $\mathcal{G}_0^j$ are given in Corollary \ref{coro:NHIMCircular:Poincare}.


Recall that 
\[
\Delta H_\mathrm{ell}^{1,\pm}(\ell,L,g,G,t)=\Delta
H_\mathrm{ell}^{1,\pm}(\ell,L,g,G)e^{it}+\Delta
H_\mathrm{ell}^{1,\pm}(\ell,L,g,G)e^{-it}, 
\]
as defined in Corollary \ref{coro:ExpansioHamiltonian}, and that
the functions $\omega_\pm^\ast$ are defined in
\eqref{def:Omega0PlusMinus}.

\begin{lemma}\label{lemma:Outer:Elliptic}
  Assume Ansatz~\ref{ans:NHIMCircular}. The outer maps
$\mathcal{F}_{e_0}^{\mathrm{out},\ast}$ have the following expansion with
  respect to $e_0$:
  \begin{equation}\label{def:OuterMap:Elliptic}
    \mathcal{F}_{e_0}^{\mathrm{out},\ast}:\left(\begin{array}{c} I\\
        t
      \end{array}\right)\mapsto \left(\begin{array}{c} I+ e_0\left(B^{\ast,+} (I)e^{it}+B^{\ast,-} (I)e^{-it}\right) +\mathcal{O}\left(e_0^2\right)\\
        t+\mu\omega^{\ast}(I)+\mathcal{O}(e_0)
      \end{array}\right),\,\,\,\ast=\mathrm{f},\mathrm{b}.
  \end{equation}
  The functions $B^{\ast,\pm}(I)$ are defined as
\begin{equation}\label{def:Omega:PlusMinus}
\begin{split}
B^{\mathrm{f},\pm}(I)&=B_\mathrm{out}^{\mathrm{f},\pm}(I)+B_\mathrm{in}^{\mathrm{f},\pm}(I)e^{\pm i\mu\omega_\mathrm{out}^\mathrm{f}(I)}\\
B^{\mathrm{b},\pm}(I)&=B_\mathrm{in}^{\mathrm{b},\pm}(I)+B_\mathrm{out}^{\mathrm{b},\pm}(I)e^{\pm i\mu\omega_\mathrm{in}^\mathrm{b}(I)},
\end{split}
\end{equation}
where $\omega_\mathrm{out}^\mathrm{f}(I)$ and $\omega_\mathrm{in}^\mathrm{b}(I)$ are the functions defined in \eqref{def:Omega0:OuterPart} and \eqref{def:Omega0:InnerPart} respectively and
  \begin{align}\label{def:Omega:PlusMinus:Out:for}
    B^{\mathrm{f},\pm}_\mathrm{out}(I)=&\pm i\mu\lim_{T\rightarrow+\infty}\int_0^T\left(\frac{\Delta H^{1,\pm}_\mathrm{ell}\circ\gamma_I^\mathrm{f}(\sigma)}{-1+\mu\partial_G\Delta H_\mathrm{circ}\circ\gamma_I^\mathrm{f}(\sigma)}e^{\pm i\widetilde\gamma_I^\mathrm{f}(\sigma)}\right.\nonumber\\
    &\qquad\qquad\qquad\left.-\frac{\Delta H_\mathrm{ell}^{1,\pm}\circ\lambda_I^3(\sigma)}{-1+\mu\partial_G\Delta H_\mathrm{circ}\circ\lambda_I^3(\sigma)}e^{\pm i\left(\widetilde\lambda_I^3(\sigma)+\mu\omega_+^\mathrm{f}(I)\right)}\right) d\sigma\\
    &\mp i\mu\lim_{T\rightarrow-\infty}\int_0^T\left(\frac{\Delta H_\mathrm{ell}^{1,\pm}\circ\gamma_I^\mathrm{f}(\sigma)}{-1+\mu\partial_G\Delta H_\mathrm{circ}\circ\gamma_I^\mathrm{f}(\sigma)}e^{\pm i\widetilde\gamma_I^\mathrm{f}(\sigma)}\right.\nonumber\\
    &\qquad\qquad\qquad\left.-\frac{\Delta H_\mathrm{ell}^{1,\pm}\circ\lambda_I^4(\sigma)}{-1+\mu\partial_G\Delta H_\mathrm{circ}\circ\lambda_I^4(\sigma)}e^{\pm i\left(\widetilde\lambda_I^4(\sigma)+\mu\omega_-^\mathrm{f}(I)\right)}\right)d\sigma,\nonumber
\end{align}
\begin{align}\label{def:Omega:PlusMinus:back}
  B_\mathrm{out}^{\mathrm{b},\pm}(I)=&\pm i\mu\lim_{T\rightarrow+\infty}\int_0^T\left(\frac{ \Delta H^{1,\pm}_\mathrm{ell}\circ\gamma_I^\mathrm{b}(\sigma)}{-1+\mu\partial_G\Delta H_\mathrm{circ}\circ\gamma_I^\mathrm{b}(\sigma)}e^{\pm i\widetilde\gamma_I^\mathrm{b}(\sigma)}\right.\nonumber\\
    &\qquad\qquad\qquad-\left.\frac{\Delta H_\mathrm{ell}^{1,\pm}\circ\lambda_I^4(\sigma)}{-1+\mu\partial_G\Delta H_\mathrm{circ}\circ\lambda_I^4(\sigma)}e^{\pm i\left(\widetilde\lambda_I^4(\sigma)+\mu\omega_+^\mathrm{b}(I)\right)}\right)d\sigma\\
    &\mp i\mu\lim_{T\rightarrow-\infty}\int_0^T\left(\frac{ \Delta H_\mathrm{ell}^{1,\pm}\circ\gamma_I^\mathrm{b}(\sigma)}{-1+\mu\partial_G\Delta H_\mathrm{circ}\circ\gamma_I^\mathrm{b}(\sigma)}e^{\pm i\widetilde\gamma_I^\mathrm{b}(\sigma)}\right.\nonumber\\
    &\qquad\qquad\qquad-\left.\frac{ \Delta H_\mathrm{ell}^{1,\pm}\circ\lambda_I^3(\sigma)}{-1+\mu\partial_G\Delta H_\mathrm{circ}\circ\lambda_I^3(\sigma)}e^{\pm i\left(\widetilde\lambda_I^3(\sigma)+\mu\omega_-^\mathrm{b}(I)\right)}\right)d\sigma,\nonumber
\end{align}
\begin{align}\label{def:Omega:PlusMinus:Inn}
  B_\mathrm{in}^{\mathrm{f},\pm}(I)=&\mp i\mu\int_0^{-12\pi}\frac{\Delta H^{1,\pm}_\mathrm{ell}\circ\lambda_I^4(\sigma)}{-1+\mu\partial_G\Delta H_\mathrm{circ}\circ\lambda_I^4(\sigma)}e^{\pm i\widetilde\lambda_I^4(\sigma)}d\sigma\\
B_\mathrm{in}^{\mathrm{b},\pm}(I)=&\mp\int_0^{-2\pi}\frac{\Delta
  H^{1,\pm}_\mathrm{ell}\circ\lambda_I^3(\sigma)}{-1+\mu\partial_G\Delta
  H_\mathrm{circ}\circ\lambda_I^3(\sigma)}e^{\pm i\widetilde\lambda_I^3(\sigma)}d\sigma. \nonumber
   \end{align}
 \end{lemma}
\begin{proof}
  Recall that the outer maps are the composition of
  two maps. Indeed, as explained in Section
  \ref{sec:Circular:Outer}, they are defined as
\[
 \begin{split}
  \mathcal{F}_{e_0}^{\mathrm{out},\mathrm{f}}=\mathcal{P}_{e_0}^6\circ\mathcal{S}_{e_0}^\mathrm{f}: \widetilde\Lambda_0^3\longrightarrow \widetilde\Lambda_0^3\\
  \mathcal{F}_{e_0}^{\mathrm{out},\mathrm{b}}=\mathcal{S}_{e_0}^\mathrm{b}\circ\mathcal{P}_{e_0}: \widetilde\Lambda_0^3\longrightarrow \widetilde\Lambda_0^3.
 \end{split}
\]