text:
\begin{document}

\maketitle

\begin{abstract}Consider the dynamics of two point masses on a surface of constant curvature subject to an attractive force analogue of Newton's inverse square law. When the distance between the bodies is sufficiently small, the reduced equations of motion may be seen as a perturbation of an integrable system. We define suitable action-angle coordinates to average these perturbing terms and observe dynamical effects of the curvature on the motion of the two-bodies.
\end{abstract}

\section{Introduction}

When the geometry of constant curvature spaces was brought to attention in the 19th century, mechanical problems in such spaces presented an opportunity for an interesting variation on familiar themes. In particular, studying the motion of point masses subject to extensions of Newton's usual (flat) inverse square force law to analogous `curved inverse square laws'  was soon undertaken, see e.g. the historical reviews in \cite{BMK2, DCS}.  Consequently, we speak of \textit{the} curved Kepler problem or \textit{the} curved $n$-body problem to refer to the mechanics of point masses in a space of constant curvature under such  nowadays customary extensions of Newton's inverse square force.

While the curved Kepler problem is quite similar to the flat Kepler problem --both being characterized by their conformance to analogues of Kepler's laws, or super-integrability-- there are striking differences between the curved 2-body problem and the usual 2-body problem. Namely, it has long been noticed that the absence of Galilean boosts prohibits straightforward reduction of a curved 2-body problem to a curved Kepler problem, and more recently it has been shown --as an application of Morales-Ramis theory in \cite{Shch}, and by numerically exhibiting complicated dynamics in \cite{BMK}-- that the curved 2-body problem, in contrast to the usual 2-body problem, is non-integrable. In this article we will apply classical averaging techniques to compare some features of certain orbits in these curved and flat 2-body dynamics.

When the two bodies are close, their motion is a small perturbation of the flat 2-body problem. Indeed, after reduction, we find their relative motion governed by a Kepler problem and additional perturbing terms which vanish as the distance between the bodies goes to zero. Consequently their motion at each instant is approximately along an \textit{osculating conic}: the conic section the two bodies would follow if these perturbing terms were ignored. The averaged, or \textit{secular}, dynamics take into account how the perturbing terms slowly deform these osculating conics.

Our main results, theorems \ref{thm:qper} and \ref{thm:per} below, describe quasi-periodic and periodic motions in the reduced curved 2-body problems (see fig.~\ref{fig:orbs}).

\begin{figure}[!htb]
\centering
\includegraphics[scale = .28]{Orbs1.png}
\caption{\footnotesize{(a) For two sufficiently close bodies on a sphere, there are motions (blue) along which the two bodies revolve nearly along conic sections which precess in the direction \textit{opposite} to the bodies (fast) motion around the conic. The angular momentum here is vertical, and the whole system rotates about the angular momentum axis. As the two bodies get closer to the equator, their rate of precession decreases and the eccentricity of their osculating conic increases until at the equator two types of (regularized) periodic collision orbits survive: one for which the two bodies bounce perpendicularly to the equator (red) and another for which they bounce along the equator. Similar results hold for two bodies in a negatively curved space, but now the precession is in the \textit{same} direction as the bodies motion around the conic (b) with vertical angular momentum and (c) for angular momentum 'to the right'. We expect some chaotic dynamics to occur near the collision orbits (remark \ref{rmrk:split}).}}\label{fig:orbs}
\end{figure}

We will begin in section \ref{sec:Kep} by recalling properties of the curved Kepler problem, with details in appendix \ref{app:DelPoi}.  As an aside, in seeking parametrizations of the curved Kepler orbits useful for averaging, we derive a 'curved Kepler equation' eq. (\ref{eq:KepEq}) that, as far as we can tell, is new.

The curved 2-body problem admits symmetries by the isometry group of the space of constant curvature. In section \ref{sec:2bdy}, we carry out a Meyer-Marsden-Weinstein (symplectic) reduction to formulate the equations of motion for the reduced curved 2-body problem (prop.~\ref{prop:red}), and define a scaling of the coordinates (prop.~\ref{prop:scal})  allowing us to focus our attention on the dynamics when the two bodies are close relative to the curvature of the space.

In section \ref{sec:SecDyn} we prove our two main results, establishing that the behaviour depicted in fig.~\ref{fig:orbs} holds for 'most' orbits when the two bodies in a given curved space are sufficiently close.

\begin{rmrk}
Our results are related to a remark (in section 4.1 of \cite{BMK}), in that determining the secular dynamics is a preliminary step towards a proof of non-integrability by 'Poincar\'e's methods'.   Upon defining curved analogues of the Delaunay and Poincar\'e coordinates, our methods are quite similar to those applied to the usual 3-body problem in \cite{FejQP}, although the setting here is simpler in the sense that we only need to average over one fast angle.
\end{rmrk}

\section{Curved Kepler problems}
\label{sec:Kep}
Newton's '$1/r$' potential may be characterized by either of the following properties:
\begin{itemize}
\item it is a (constant multiple) of the fundamental solution of  the 3-dimensional laplacian, $\Delta_{\R^3}$

\item the corresponding central force problem has orbits following Kepler's laws.
\end{itemize}
Generalizing these properties to spaces of constant positive (resp.~negative) curvature yields '$\cot\fe$' (resp.~'$\coth\fe$') potentials. We refer to \cite{Al} for a discussion of these long established potentials and their properties using central projection (fig. \ref{fig:CentProj}), presenting some details here in appendix \ref{app:DelPoi}. In particular, the curved Kepler problems are super integrable and on the open subset of initial conditions leading to bounded non-circular motions we have action-angle coordinates, $(L, \ell, G, g)$ where the energy depends only on $L$ (see prop. \ref{prop:DP}).

\begin{figure}[!htb]
\centering
\includegraphics[scale = .5]{gnomic.png}
\caption{\footnotesize{The curved Kepler dynamics --with \textit{fixed} center of attraction or 'sun' at $q_s$-- may be characterized by requiring that its \textit{unparametrized} trajectories centrally project to those of the flat Kepler problem in the tangent plane at $q_s$. Setting $\rho = |cq_s|$, the sphere has curvature $\kappa = 1/\rho^2$ and the hyperboloid (with the restricted Minkowski metric) has curvature $\kappa = - 1/\rho^2$.  We call $\fe := \angle (q_scq)$ (resp.~$\tilde\fe:= \angle(q_sc\tilde q)$ measured with the Minkowski inner product) the \textit{angular distance} from $q$ to $q_s$ (resp.~$\tilde q$ to $q_s$). The distance from $q$ on the sphere to $q_s$ is given by $\rho\fe$, while the distance from $\tilde q$ to $q_s$ on the hyperboloid is given by $\rho \tilde\fe$. We set $r:=|q_oq_s| = \rho \tan\fe = \rho\tanh\tilde\fe$.}}\label{fig:CentProj}
\end{figure}

In what follows, we will consider motions satisfying $\fe = O(\e)$ for all time, where $\e$ is a small parameter. Evidently, for these Kepler problems, such orbits are central projections of elliptic orbits in the plane. Also, to avoid repetitive arguments and computations, we will give details for the positive curvature case and remark on the analogous results for negative curvature.

\section{Curved two body problems}\label{sec:2bdy}

Let $S^2$ be the sphere of curvature $\kappa$, with  $\|\cdot\|_\kappa$ the norm given by its constant curvature metric. Consider two point masses, $$(q_1,q_2) =:q\in (S^2\times S^2)\backslash \{|\cot \fe| = \infty\} =: Q,$$ on this sphere of masses $m_1, m_2>0$ with $\fe$ the angular distance between $q_1$ and $q_2$. Normalize the masses so that $m_1 + m_2 = 1$. We consider the Hamiltonian flow on $T^*Q$ of: $$F := \frac{\|p_1\|_\kappa^2}{2m_1} + \frac{\|p_2\|_\kappa^2}{2m_2} - \frac{m_1m_2}{\rho}\cot\fe.$$

The tangential components of the forces are of equal magnitude and directed towards eachother along the arc $\stackrel{\frown}{q_1q_2}$. By letting $\vec q_j$ be the position vector of $q_j$ from the center of the sphere, the components of the \textit{angular momentum vector}: $$\vec C := m_1\vec q_1\times \dot{\vec q}_1 + m_2 \vec q_1\times \dot{\vec q}_2$$ are first integrals. They correspond to the symmetry by the diagonal action, $(q_1,q_2)\mapsto (gq_1, gq_2)$, of $\SO_3$ on $Q$. Since $\fe\ne 0, \pi$ in $Q$, this $\SO_3$ action on each level set $\{\fe = cst.\}\subset Q$ is free and transitive. By choosing a representative configuration (see fig.~\ref{fig:com}) in each level set of $\fe$ we obtain a slice of this group action and have: \[
    Q\cong I\times \SO_3,~~I = (0,\pi)\ni \fe,\]  such that the $\SO_3$ action is by left multiplication on the second factor.

\begin{figure}[!htb]
\centering
\includegraphics[scale = .5]{com.png}
\caption{\footnotesize{Our choice of representative configuration. Embed the sphere of curvature $\kappa$ in $\R^3$, centered at the origin and take an orthonormal basis $\hat i, \hat j, \hat k$ with $\vec C = C\hat k$. We take $q^o(\fe):=( -\rho\sin m_2\fe \hat j + \rho\cos m_2\fe \hat k, \rho\sin m_1\fe \hat j + \rho\cos m_1\fe \hat k)$. Given $q\in Q$ with angular distance $\fe$ between the two bodies, there is a unique element $g\in \SO_3$ with $q = gq^o(\fe)$.}}\label{fig:com}
\end{figure}

\subsection{Reduced equations of motion} An application of symplectic reduction (see e.g. \cite{ACM} App. 5) gives:

\begin{prop}\label{prop:red} Fix the angular momentum $\vec C\ne 0$. The reduced 2-body dynamics on a sphere of curvature $\kappa = \frac{1}{\rho^2}$ takes place in $T^*I\times\mathcal{O}_\mu$, where $\mathcal{O}_\mu$ is a co-adjoint orbit in $\mathfrak{so}_3^*$. It is the Hamiltonian flow of:
$$F_{red} = Kep_\kappa + \kappa(\frac{\|\vec C\|^2}{2} - p_\theta^2) + O(\fe),~~~dp_\fe\wedge d\fe + dp_\theta\wedge d\theta,$$ for $|p_\theta|<\|\vec C\|$, and with $Kep_\kappa = \frac{p_\fe^2 + p_\theta^2/\sin^2\fe}{2\rho^2m_1m_2} - \frac{m_1m_2}{\rho}\cot\fe$, the Hamiltonian for a curved Kepler problem.
\end{prop}

\begin{proof} We first find the mass weighted metric, $\langle (\vec v_1, \vec v_2), (\vec u_1, \vec u_2)\rangle = m_1 \vec v_1\cdot \vec u_1 + m_2 \vec v_2\cdot \vec u_2$, in terms of our identification $Q\cong I\times \SO_3$ (fig.~\ref{fig:com}). Since the metric is invariant under left translations, it suffices to determine it at the identity (our representative configurations, $q^o$, in fig.~\ref{fig:com}).

Let $X_1, X_2, X_3$ be the infinitesimal symmetry vector fields generated by rotations about the $\hat i, \hat j, \hat k$ axes. Together with $\del_\fe$, they frame $Q$. We compute:
\begin{align*}
\langle \del_\fe, \del_\fe\rangle_{q^o} &= \rho^2m_1m_2 & \langle X_1, X_1\rangle_{q^o} &= m_1\|\hat i \times \vec q_1^o\|^2 + m_2\|\hat i \times \vec q_2^o\|^2 = \rho^2\\
\langle X_2, X_2\rangle_{q^o} &= \rho^2(m_1\cos^2m_2\fe + m_2\cos^2m_2\fe) & \langle X_3, X_3\rangle_{q^o} &= \rho^2(m_1\sin^2m_2\fe + m_2\sin^2m_2\fe)
\end{align*}
 \[   \langle X_2, X_3\rangle_{q^o} = \rho^2(m_1\cos m_2\fe\sin m_2\fe - m_2\cos m_1\fe\sin m_1\fe), \]
 all other inner products being zero. In terms of $Q$'s co-frame, $\{d\fe, X^j\}$, dual to the above frame, the metric is given by inverting its $\{\del_\fe, X_j\}$ matrix representation:
 \begin{align*}
\langle d\fe, d\fe\rangle_{q^o} &= \frac{1}{\rho^2m_1m_2} & \langle X^1, X^1\rangle_{q^o} &= \frac{1}{\rho^2}\\
\langle X^2, X^2\rangle_{q^o} &= \frac{m_1\sin^2m_2\fe + m_2\sin^2m_2\fe}{m_1m_2\rho^2\sin^2\fe} & \langle X^3, X^3\rangle_{q^o} &= \frac{m_1\cos^2m_2\fe + m_2\cos^2m_2\fe}{m_1m_2\rho^2\sin^2\fe}
\end{align*}
 \[   \langle X^2, X^3\rangle_{q^o} = \frac{m_2\cos m_1\fe\sin m_1\fe - m_1\cos m_2\fe\sin m_2\fe }{m_1m_2\rho^2\sin^2\fe}. \]

We identify $T^*Q\cong T^*I\times (\SO_3\times \mathfrak{so}_3^*)$ by \textit{right} translation: $\alpha_g\in T_g^*SO_3\mapsto (g, R_g^*\alpha_g)$. In these coordinates, the symplectic lift of $\SO_3$'s left action on $Q$ is given by: $g\cdot (\fe, p_\fe, h, \mu) = (\fe, p_\fe, gh, Ad_{g^{-1}}^*\mu)$ with associated moment map: $J(\fe, p_\fe, g, \mu) =  \mu$.  The reduced dynamics takes place on the quotient $P_\mu := J^{-1}(\mu)/G_\mu\cong T^*I\times\mathcal{O}_\mu$, realized by:  $(\fe, p_\fe, g, \mu)\mapsto (\fe, p_\fe, Ad_g^*\mu)$. The reduced symplectic form on $P_\mu$ is $dp_\fe\wedge d\fe + \Omega$, where $\Omega_\nu(ad_\xi^*\nu, ad_\eta^*\nu) = \nu([\xi, \eta])$ is the Kirillov-Kostant form on $\mathcal{O}_\mu$.

By left invariance of the two body Hamiltonian, $F$, we have:  $F(\fe, p_\fe, g, \mu) = F(\fe, p_\fe, e, Ad_g^*\mu)$, or letting $\nu = Ad_g^*\mu = \nu_j X^j(q^o)$, and $m = m_1m_2$, we have:  \begin{equation}\label{eq:Fred} F_{red} = \frac{p_\fe^2}{2m\rho^2} - \frac{m}{\rho}\cot\fe + \frac{\nu_1^2}{2\rho^2} + \nu_2^2\frac{m_1\sin^2m_2\fe + m_2\sin^2m_1\fe}{2m\rho^2\sin^2\fe}\end{equation} \[ + \nu_3^2\frac{m_1\cos^2m_2\fe + m_2\cos^2m_1\fe}{2m\rho^2\sin^2\fe} + \nu_2\nu_3\frac{m_2\cos m_1\fe\sin m_1\fe - m_1\cos m_2\fe\sin m_2\fe}{m\rho^2\sin^2\fe}.\]

Now, using that $\frac{\sin^2ax}{\sin^2x}$ is an analytic function around $x=0$, with expansion $a^2 + O(x^2)$, and that the $\nu_2\nu_3$ coefficient is analytic around $\fe = 0$ of $O(\fe)$, and $\nu_1^2 + \nu_2^2 + \nu_3^2 = \|\vec C\|^2$, we have:

\[ F_{red} = \frac{p_\fe^2}{2m\rho^2} + \frac{\nu_3^2}{2m\rho^2\sin^2\fe} - \frac{m}{\rho}\cot\fe + \kappa(\frac{\|\vec C\|^2}{2} - \nu_3^2) + O(\fe).\]

Finally, recall that the Kirillov-Kostant form is given by $\Omega = dp_\theta \wedge d\theta$ when we parametrize $\mathcal{O}_\mu$ -- away from the poles $|p_\theta| = \|\vec C\|$ -- as $(\sqrt{\|\vec C\|^2 - p_\theta^2}\sin\theta, \sqrt{\|\vec C\|^2 - p_\theta^2}\cos\theta, p_\theta) = (\nu_1, \nu_2, \nu_3)$.
\end{proof}

\begin{rmrk}\label{rmrk:eqMass} For equal masses, the expression for $F_{red}$ may be simplified by taking $\psi = \fe/2, p_\psi = 2p_\fe$. One has: $F_{red} = Kep_\kappa + \frac{\tan\psi}{8\rho} + \kappa\frac{\|\vec C\|^2 - p_\theta^2}{2}(1 + (\tan\psi\cos\theta)^2)$, where $Kep_\kappa = \frac{1}{2\rho^2}(p_\psi^2 + \frac{p_\theta^2}{\sin^2\psi}) - \frac{\cot\psi}{8\rho}$.
\end{rmrk}

\begin{rmrk} In \cite{Shch}, symplectic reduction was also applied to reduce the curved 2-body problems. However, for purposes of comparison, it was not clear to us what representative configuration or slice of the group action was used, which led us to present a reduction here 'from scratch'.  Our choice of representative (fig.~\ref{fig:com}) was motivated by the mass metric being more diagonal, namely requiring: $\langle \del_\fe, X_j\rangle = 0$.
\end{rmrk}

Next, we introduce a small parameter, $\e$, to study the dynamics when $\fe = O(\e)$:

\begin{prop}\label{prop:scal}
The dynamics on the open set $\fe = O(\e)$ may be reparametrized as that of:
$$\hat F_{red} = Kep_{\e^2} +  Per,~~\hat\omega = d\hat L\wedge d\hat\ell + d\hat G\wedge d\hat g$$ where $Kep_{\e^2} = - \frac{m^3}{2\hat L^2} + \frac{\e^2 \hat L^2}{2m}$, $m = m_1m_2$, and $Per = \e^2\left(\frac{\hat C^2}{2} - \hat G^2 + (m_2-m_1)O(\e) + O(\e^2)\right)$.
\end{prop}

\begin{proof} Take Delaunay coordinates (app. \ref{prop:DP}), $(L,G,\ell, g)$,  for the $Kep_\kappa$ term of $F_{red}$.  Let $L^2 =: \rho\e\hat L^2, G^2 =: \rho\e\hat G^2, \|\vec C\|^2 =: \rho\e \hat C^2$, so that $\fe = O(\e)$. The dynamics of $\hat F_{red} := \rho\e F_{red}$ with symplectic form $\hat\omega := d\hat L\wedge d\ell + d\hat G\wedge dg = \omega/\sqrt{\rho\e}$, corresponds to the time reparametrization: $(\rho\e)^{3/2}\hat t = t$, where $\hat t$ is the new time. Note that the mean anomaly, $\hat\ell$, for $Kep_{\e^2}$ is the same as the mean anomaly, $\ell$, for the $Kep_{\kappa}$ term, as can be easily checked by comparing $d\ell = \del_LKep_\kappa~dt$ and $d\hat\ell = \del_{\hat L} Kep_{\e^2}~d\hat t$.
\end{proof}

\begin{rmrk} The scaling above includes the possibility of taking $\e = 1/\rho$, and in place of imagining the bodies as close, view their distance remaining bounded as the curvature of the space goes to zero.
\end{rmrk}

\begin{rmrk}
When the curvature is negative, eq. (\ref{eq:Fred}) has the trigonometric functions replaced by their hyperbolic counterparts, and with   $\mathfrak{so}_{2,1}^*$'s coadjoint orbits:  $\nu_1^2 + \nu_2^2 - \nu_3^2 = \|\vec C\|_{2,1}^2$ replacing the coadjoint orbits of $\mathfrak{so}_3^*$.  One arrives at $\hat F_{red} = Kep_{-\e^2} + \e^2(\frac{\|\hat C\|_{2,1}^2}{2} + \hat G^2 + (m_2-m_1)O(\e) + O(\e^2))$.
\end{rmrk}

\section{Secular dynamics}
\label{sec:SecDyn}

The \textit{secular} or averaged dynamics refers to the dynamics of the Hamiltonian: \[\langle F_{red}\rangle := \frac{1}{2\pi}\int_0^{2\pi} F_{red}(L,\ell, G, g)~d\ell. \] The secular dynamics is integrable, $L$ being a first integral. We will use the brackets to denote a functions averaged over $\ell$, namely $\langle f\rangle = \frac{1}{2\pi}\int_0^{2\pi} f~d\ell$. The dynamical relevance of the secular Hamiltonian is due to its appearance in a series of 'integrable normal forms' for the reduced curved 2-body dynamics:

\begin{prop} Consider the scaled Hamiltonian, $(\hat F_{red}, \hat\omega)$, of proposition \ref{prop:scal}. For each $k\in\N$, there is a symplectic change of coordinates, $(\psi^k)^*\hat\omega = \hat\omega$, with:
\[\hat F_{red}\circ\psi^k = Kep_{\e^2} +  \langle Per\rangle + \langle F^1\rangle + ... + \langle F^{k-1}\rangle + F^{k}\]
and each $F^j = O(\e^{2j+3})$. When the masses are equal, $F^j = O(\e^{2j + 4})$.
\end{prop}
\begin{proof} The iterative process to determine the sequence of $\psi^k$ and $F^k$'s is not new (see \cite{FejQP}). For the first step, let $\psi^1$ be the time 1-flow of a 'to be determined' Hamiltonian $\chi$. By Taylor's theorem:
\[ \hat F_{red}\circ \psi^1 = Kep_{\e^2} + Per + \{Kep_{\e^2}, \chi\} + \{Per, \chi\} + \int_0^1 (1-t)\frac{d^2}{dt^2}(\hat F_{red}\circ\psi^t)~dt.\]
Let $\tilde Per := Per - \langle Per\rangle$. Note that, while $Per = O(\e^2)$, we have $\tilde Per = O(\e^3)$. Taking $\hat n \chi := \int_0^\ell \tilde Per(\hat L, l, \hat G, g)~dl,$ where $\hat n = \del_{\hat L}Kep_{\e^2}$, we have: \[ Per + \{Kep_{\e^2}, \chi\} = \langle Per\rangle,~~\chi = O(\e^3),\] so that for $\e$ sufficiently small $\psi^1$ is defined. Finally, integrating by parts, one has \[ F^1 := \{Per, \chi\} + \int_0^1 (1-t)\frac{d^2}{dt^2}(\hat F_{red}\circ\psi^t)~dt = \int_0^1 \{\langle Per\rangle + t\tilde Per, \chi\}\circ \psi^t~dt = O(\e^5).\]
\end{proof}

\begin{rmrk}\label{rmrk:kSec} The truncated, or \textit{$k$'th order secular system}: $Sec^k:=Kep_{\e^2} +  \langle Per\rangle + \langle F^1\rangle + ... + \langle F^{k-1}\rangle$ is integrable, since it does not depend on $\ell$. If the sequence $\psi^k$ converged, the curved 2-body problem would be integrable, although by \cite{Shch} we know there is no such convergence. For each fixed $k$ however, the $k$'th order secular system approximates the true motion over long time scales in a region of configurations with the 2-bodies sufficiently close.
\end{rmrk}

In appendix \ref{app:Per}, we compute an expansion of $\langle Per\rangle$ in powers of $\e$, eq. (\ref{eq:PerExp}). Via this expansion, one finds two unstable periodic orbits (red in fig.~\ref{fig:orbs}), whose stable and unstable manifolds coincide for the integrable secular dynamics. The complicated dynamics which would result from the splitting of these manifolds is a likely mechanism responsible for the non-integrability of the curved 2-body problem.

By ignoring the $O(\fe)$ terms in $F_{red}$, the Keplerian orbits experience precession (see fig.~\ref{fig:orbs}): \[ \dot G = 0,~~\dot g = - 2\kappa G.\] When the curvature is positive, the conic precesses in a direction opposite to the particles (fast) motion around the conic, while when the curvature is negative the precession occurs in the same direction as the particles motion. We will check that for $\fe$ sufficiently small, many such orbits survive the perturbation of including the $O(\fe)$ terms: they may be continued to orbits of the reduced curved 2-body problem and then lifted to orbits of the curved 2-body problem.

\subsection{Continued orbits}

Applying a KAM theorem from \cite{Fej}, establishes certain quasi-periodic motions of the reduced problem. Introducing some notation, for $\gamma>0, \tau>1$, let \[ D_{\gamma,\tau}:= \{ v\in\R^2~:~|\mathbf{k}\cdot v|\ge \frac{\gamma}{(|k_1|+|k_2|)^\tau}, \forall \mathbf{k} = (k_1,k_2)\in\Z^2\backslash \mathbf{0}\} \] be the $(\gamma,\tau)$-\textit{Diophantine frequencies}. Take \[\mathbf{\hat B}:= \{ (\hat L, \hat G) ~:~ 0 < \hat G \le \hat L < 1\} \] and let \[(I,\theta) = (I_1,I_2,\theta_1, \theta_2) = (\hat L, \hat G, \ell, g) + O(\e^3)\] be action-angle variables for some $k$th order secular system, with $\mathbf{B} = \{ (I_1, I_2) :  (\hat L, \hat G)\in \mathbf{\hat B}\}\subset \R^2$ equipped with the Lebesgue measure, $Leb$, restricted to $\mathbf{B}$.

The reduced curved 2-body problem has, for $\fe$ sufficiently small,  a positive measure of invariant tori along each of which the motions are quasi-periodic. More precisely:

\begin{theorem}\label{thm:qper}
Fix $\tau >1, \hat\gamma >0, m>2$ and set $\gamma := \e^m\hat\gamma$. There exists $\e_0>0$ such that for each $0<\e<\e_0$, we have:

(i) A positive measure set, $d_{\gamma, \tau}\subset \mathbf{B}$,

(ii) for each $I^o\in d_{\gamma,\tau}$, there is a local symplectic change of coordinates, $\psi^o$, for which $\psi^o(I^o,  \theta)$ is an invariant torus of $\hat F_{red}$ with frequency in $D_{\gamma,\tau}$.

(iii) As $\e\to 0$, $Leb(d_{\gamma,\tau})\to Leb(\mathbf{B})$ and $\psi^o\to id$ (in the Whitney $C^\infty$ topology).
\end{theorem}
\begin{proof} We verify the hypotheses of \cite{Fej}.  Let $(I,\theta)$ be action-angle variables for a $k$'th order secular system (remark \ref{rmrk:kSec}) with $2k>2m-3$. Setting $s := I^o\in \mathbf{B}$, and $r := I - I^o$, we have by Taylor expansion:
\[ F_s^o := Sec^k = c_s^o + \alpha_s^o\cdot r + O(r^2) \] where $c_s^o := Sec_k(I^o), \alpha_s^o := dSec^k(I^o) = O(1, \e^2)$. Likewise, we take $F_s:=\hat F_{red}(r,\theta)$ around $s = I^o$. Since we have chosen $2k + 1>2m$, we have \[|F_s - F_s^o| = o(\gamma^2)\] for $\e$ sufficiently small. So  (\cite{Fej} Thm.~15 and remark 21) there exists a map $\mathbf{B}\ni s\mapsto \alpha_s\in \R^2$, with $|\alpha_s - \alpha_s^o|<<1$ in the $C^\infty$-Whitney topology such that \textit{provided} $\alpha_s\in D_{\gamma,\tau}$, we have $\hat F_{red}\circ\psi^o = c_s + \alpha_s\cdot r + O(r^2;\theta)$ for some local symplectic map $\psi^o$. It remains to check that $d_{\gamma,\tau}:=\{ s~:~\alpha_s\in D_{\gamma,\tau}\}$ is non-empty. Since $m>2$ and $\alpha_s = O(1, \e^2)$ we have (\cite{Fej} Cor.~29) \[ Leb(\mathbf{B}\backslash d_{\gamma,\tau}) = O(\e^{\frac{m-2}{\mu}})\] for some constant $\mu>0$. So indeed, for $\e$ sufficiently small, $d_{\gamma,\tau}$ has positive measure and is non-empty.
\end{proof}

\begin{rmrk} As the reduced dynamics has two degrees of freedom, there is KAM-stability in the $\fe<<1$ regime: the invariant tori form barriers in energy level sets.
\end{rmrk}

One may also apply the implicit function theorem to establish periodic orbits of long periods for the reduced dynamics:

\begin{theorem}\label{thm:per}
Let $m,n\in\N$. Then for $m$ sufficiently large, there exist periodic orbits of the reduced curved 2-body problem for which the bodies make $m$ revolutions about their osculating conics, while the osculating conic precesses around the origin $n$ times.
\end{theorem}
\begin{proof}
We will consider non-equal masses, a similar argument applies for equal masses. After iso-energetic reduction, the equations of motion for $\hat G, g$ are eq. (\ref{eq:PerExp}):
\[ \frac{d\hat G}{d\ell} = \e^3\mathfrak{m} \sin g + O(\e^4),~~~\frac{dg}{d\ell} = -2\e^2\hat G + O(\e^3),\] where $\mathfrak{m}(\hat L, \hat G)\ne 0$ for non-circular motions. Consider the 'long time return map':
\[ \overline P_\e (G^o, g^o) :=  \int_0^{2\pi/\e^2} (\frac{1}{\e}\frac{d\hat G}{d\ell}, ~\frac{dg}{d\ell})~d\ell,\] the integral taken over an orbit with initial condition $G^o, g^o$.  When $\overline P_\e(G^o, g^o) = (0, -2\pi n)$ and $1/\e^2 = m\in\N$, we have a periodic orbit as described in the theorem. The map $\overline P_\e$ is analytic in $\e$, with
 \[\overline P_0(G^o, g^o) = (\mathfrak{m}\sin g^o, ~ - 2G^o).\]
By the implicit function theorem, $g^o\equiv 0~\mod \pi, G^o = \pi n$, continues to solutions of $\overline P_\e(G^\e, g^\e) = (0, -2\pi n)$ for $0<\e<\e_o$, yielding periodic orbits of long periods for those $\e$ with  $ 1/\e^2 = m\in\N$.
\end{proof}

\begin{rmrk}
When the curvature is positive the precession is, as with the quasi-periodic motions, in the opposite direction to the particles motion around the conic, while for negative curvature in the same direction.
\end{rmrk}

\subsection{Lifted orbits}

The continued orbits are the image under a near identity symplectic transformation of certain orbits of $F_{red}$ with the $O(\fe)$ terms neglected: \[ H_{red} :=  Kep_\kappa + \kappa(\frac{\|\vec C\|^2}{2} - G^2). \] We describe how the precessing Keplerian orbits of $H_{red}$ lift to $T^*Q$.

In the reduced space, $T^*I\times\mathcal{O}_\mu$, the orbits of $H_{red}$ are given by:
\[ (p_\fe(t), \fe(t), Ad_{g(t)}^*\mu)\]
where $g(t) = R_{\hat i}(\lambda)R_{\hat k}(\theta(t))$ is a rotation by $\theta(t)$ about the $\hat k$-axis followed by a rotation by $\lambda$ about the $\hat i$-axis. Moreover, $(\fe(t), \theta(t))$ describe a precessing orbit of the curved Kepler problem having $G = \|\vec C\|\cos \lambda$ for its angular momentum.

We take $\vec C = C\hat k$ (fig.~\ref{fig:com}), so that the isotropy is by rotations about the $\hat k$-axis. The ambiguity in the reduced orbit is $Ad_{g(t)}^*Ad_{h(t)}^*\mu$ where $h(t) = R_{\hat k}(\omega(t))$. Hence the lifted orbit, on $T^*Q$, is of the form:
\[(*)~~~ (\fe(t), p_\fe(t),  h(t)g(t), \mu).\]

The Hamiltonian $H_{red}$ is the symplectic reduction of $H$ on $T^*Q$ given by $H(\fe, p_\fe, g, \mu) = H_{red}(\fe, p_\fe, Ad_g^*\mu)$. Requiring the lifted curve $(*) $ to satisfy the equations of motion of $H$, imposes the condition: \[\dot\omega = \kappa \|\vec C\|.\]

In summary, a precessing Keplerian orbit with angular momentum $G$ lifts to an orbit with angular momentum $\vec C = C\hat k$ on $Q$ where the two bodies follow precessing Keplerian orbits centered (we use the notion of 'center' from fig.~\ref{fig:com} here) on the colatitude, $\lambda$, satisfying $C\cos\lambda = G$. The whole system is rotated with angular speed $\kappa C$ about the $\hat k$ axis.

\begin{rmrk}
As $\lambda\to \pi/2$, the center of the system approaches the equator and the bodies move along more eccentric conic sections until at the equator we have collision orbits. The rate of precession decreases as one moves closer to the equator, where the collision orbits cease to precess (see remark \ref{rmrk:split}, for a finer description of the equatorial behaviour). When $C > L$, there is a maximal value of $\lambda$, and such orbits are constrained to bands around the equator.
\end{rmrk}

The true motions of the curved 2-body problem established in the previous section are qualitatively the same as such lifted orbits of $H_{red}$, performing small oscillations around them.

\appendix

\section{Action-angle coordinates for the curved Kepler problem}
\label{app:DelPoi}

\subsection{Curved conics}

The Kepler problem on a sphere of radius $\rho$ with a fixed 'sun', $q_s$, of mass $M$ and particle, $q\in S^2$, of mass $m$ is given by the Hamiltonian flow of:
\[    Kep_\kappa := \frac{\|p\|_\kappa^2}{2m} - \frac{mM}{\rho}\cot\fe\] on $T^*(S^2\backslash \{|\cot\fe| = \infty\})$. Here $\|\cdot \|_\kappa$ is the norm induced by the metric of constant curvature $\kappa = 1/\rho^2$ on $S^2$ and $\fe$ is the angular distance from $q_s$ to $q$ (see figure \ref{fig:CentProj}). For a negatively curved space, one replaces $\cot\fe$ with $\coth\fe$.

Because the force is central, letting $\vec q\in\R^3$ be the position of the particle from the center, $c$, of the sphere and $\hat k$ a unit vector from $c$ to $q_s$, the \textit{angular momentum}: \begin{equation}\label{eq:am}
    G:=m(~\vec q\times \dot{\vec q}~)\cdot \hat k
\end{equation} is a first integral. It corresponds to the rotational symmetry about the $\Vec{cq_s}$ axis.

\begin{rmrk}\label{rem:coords}
In spherical coordinates, $\vec q = \rho (\sin\fe\cos\theta, \sin\fe\sin\theta, \cos\fe)$, we have:
$\|p\|_\kappa^2 = \frac{1}{\rho^2}(p_\fe^2 + \frac{p_\theta^2}{\sin^2\fe})$, and $G = m\rho^2\sin^2\fe\dot\theta = p_\theta$.
In these coordinates, it is not hard to show analytically that the curved Kepler trajectories centrally project to flat Kepler orbits (fig.~\ref{fig:CentProj}). Indeed,  setting $R := 1/r = \frac{\cot\fe}{\rho}$, one finds $\frac{d^2R}{d\theta^2} + R = m^2M/G^2$, so that the $\theta$-parametrized orbits are: $r = \frac{G^2/m^2M}{1 + e\cos(\theta - g)}$, with $e$ and $g$ being constants of integration. Here $\nu := \theta - g$ is the true anomaly and $g$ is the \textit{argument of pericenter}.
\end{rmrk}

The curved Kepler trajectories are in fact  conic sections on the sphere having a focus at $q_s$ (see fig.~\ref{fig:curvedConic}). One may express energy and momentum in terms of geometric parameters of such spherical conics.

\begin{figure}[!htb]
\centering
\includegraphics[scale = .4]{SpherConic.png}
\caption{\footnotesize{An ellipse on the sphere, with foci at $q_s$ and $f$ is the set of points $q$ for which $|qq_s| + |qf| = 2\alpha$ is constant. We call $\alpha$ its \textit{semi-major axis}, the midpoint, $q_c$, between the foci its \textit{center}, and the length $|bq_c| =:\beta$ its \textit{semi-minor axis}. The number, $\epsilon$, for which $|q_sq_c| = \alpha\epsilon$ is called the \textit{eccentricity}. When $\epsilon\ne 0$, the closest point to $q_s$ is called the \textit{pericenter}, $pc$, and furthest, $ac$, the \textit{apocenter}. The angle $\nu := \angle(pc,q_s,q)$ is called the \textit{true anomaly} of $q$.}}\label{fig:curvedConic}
\end{figure}

\begin{prop}\label{prop:geomcoords}(see \cite{Al})\label{prop:enmom} Consider an orbit of the curved Kepler problem along a spherical conic with a focus at $q_s$ (fig.~\ref{fig:curvedConic}). Let $\alpha$ be the semi-major axis and $\beta$ the semi-minor axis of this conic. The orbit has energy and momentum:
\[Kep_\kappa = -\frac{mM}{\rho}\cot\frac{2\alpha}{\rho},\]
\[G^2 = m^2M\rho\tan^2\frac{\beta}{\rho}\cot\frac{\alpha}{\rho}.\]
\end{prop}

\begin{proof}
Consider the spherical triangle $\Delta(f,q_s,q)$ with sidelengths $2\alpha\epsilon, \rho\fe, 2\alpha - \rho\fe$ and interior angle $\pi - \nu$ opposite to side $\stackrel{\frown}{fq}$ (see fig.~\ref{fig:curvedConic}). The cosine rule of spherical trigonometry yields: $r = \rho\tan\fe = \frac{p^2}{1 + e\cos\nu},$ where $p^2 = \rho\frac{\cos\frac{2\alpha\epsilon}{\rho} - \cos\frac{2\alpha}{\rho}}{\sin\frac{2\alpha}{\rho}}$ and $e = \frac{\sin\frac{2\alpha\epsilon}{\rho}}{\sin\frac{2\alpha}{\rho}}$, so that the orbits are indeed curved conic sections. Comparing with the expression in remark \ref{rem:coords} yields $G^2 = m^2Mp^2$. The expression for $G$ in the proposition follows by using the relation: $\cos\frac{\beta}{\rho} = \frac{\cos\frac{\alpha}{\rho}}{\cos\frac{\alpha\epsilon}{\rho}}$ (consider the right spherical triangle $\Delta(q_s,q_c,b)$), to simplify.

To obtain the expression for $Kep_\kappa$, observe that $\fe_p := \frac{\alpha(1-\epsilon)}{\rho}$ and $\fe_a := \frac{\alpha(1+\epsilon)}{\rho}$ are maximal and minimal values of $\fe$ over the trajectory having energy $Kep_\kappa =: h$. Consequently $p_\fe = 0$ at $\fe_{a,p}$ so we have two solutions of the equation: $h\cos 2\fe = \frac{mM}{\rho}\sin 2\fe + h - \frac{G^2}{\rho^2}.$ Adding and subtracting the above equation evaluated at $\fe_{a,p}$ yields: $Kep_\kappa = h = \frac{mM}{\rho}\frac{\sin 2\fe_a - \sin 2\fe_p}{\cos 2\fe_a - \cos 2\fe_p} = - \frac{mM}{\rho}\cot(\fe_a + \fe_p).$
\end{proof}

\begin{rmrk}
The sign of $G$ represents the orbits orientation, with positive $G$ for counterclockwise motion around $q_s$. The same arguments apply to bounded motions when the curvature is negative, replacing trigonometric functions with their hyperbolic counterparts. The energy for bounded motions in a space of curvature $\kappa = - 1/\rho^2$ is always less than $-mM/\rho$.
\end{rmrk}

\begin{rmrk}
The orbits of the curved Kepler problem colliding with $q_s$ may be 'regularized' similarly to the usual Kepler regularization, via an elastic bounce  \cite{ChavReg}.
\end{rmrk}

\subsection{Delaunay and Poincar\'e coordinates}

We have the following analogues of the Delaunay and Poincar\'e symplectic coordinates for the curved Kepler problem:

\begin{prop}\label{prop:DP} Let $G$ be the angular momentum, $g$ the argument of pericenter,
\[L^2 := m^2M\rho\tan\frac{\alpha}{\rho},~~L>0,\]  and $\ell$ be proportional to the area swept out by the \textit{orthogonal} projection along the conic from pericenter, scaled so that $\ell\in \R/2\pi\Z$. Then $(L, \ell, G, g)$ are symplectic (Delaunay) coordinates for bounded non-circular motions.  The variables: \[\Lambda = L, ~~\lambda = \ell + g, ~~\xi = \sqrt{2(L-|G|)}\cos g, ~~\eta = \sqrt{2(L-|G|)}\sin g,\] are symplectic (Poincar\'e) coordinates in a neighborhood of the circular motions.

The energy is given by:
\begin{equation}\label{eq:KepL} Kep_\kappa = -\frac{m^3M^2}{2L^2} + \kappa\frac{L^2}{2m}.\end{equation}
\end{prop}

\begin{proof}
The construction of these coordinates is almost identical as for the planar Kepler problem (see e.g. \cite{FejAA}). We only found some difference in the computation determining $L$, owing to the fact that in the curved case, the period as a function of energy (Kepler's third law) does not have such a simple expression.

For non-circular motions, we have symplectic coordinates $(H = Kep_\kappa, t, G, g)$, where $t$ is the time. Since orbits of fixed energy, $Kep_\kappa = H$, all have a common period, $T(H)$, we set $\ell = \frac{2\pi}{T(H)}t$ and seek a conjugate coordinate $L(H)$ to $\ell$, i.e. we want to integrate:
\[dL = \frac{T(H)}{2\pi}dH.\]
To integrate this expression, we make some changes of variable. For $H$ an energy value admitting bounded motions, let $\fe_c(H)$ be the angular distance of the circular orbit having energy $H$. Then: \[ H = - \frac{mM}{\rho}\cot 2\fe_c,~~G_cT(H) = 2\pi m\rho^2\sin^2\fe_c\] where $G_c^2 := \rho m^2M\tan\fe_c$ is the angular momentum of the circular solution. Note that  \[(*)~~~~H = -\frac{m^3M^2}{2G_c^2} + \kappa \frac{G_c^2}{2m}.\] We compute that $\frac{T}{2\pi}dH = dG_c$. So we take $L = G_c$, and have $L^2 = m^2M\rho\tan\frac{\alpha}{\rho}$ from $(*)$ and prop. \ref{prop:enmom}.
\end{proof}

\begin{rmrk}
The mean anomaly, $\ell$, is related to time by $d\ell = n~ dt$ where $n(L) := \frac{m^3M^2}{L^3} + \kappa\frac{L}{m}$. When the curvature is negative, the same arguments lead to $L^2 = m^2M\rho\tanh\frac{\alpha}{\rho}$, and the same expression, eq. (\ref{eq:KepL}), for the energy.
\end{rmrk}

\subsection{Other anomalies}\label{sec:anoms}

In averaging functions over curved Keplerian orbits, e.g.: determining $\frac{1}{2\pi}\int_0^{2\pi} f(q)~d\ell$, it is often useful to perform a change of variables, as the position on the orbit, $q$, does not have closed form expressions in terms of $\ell$. We collect here some parametrizations of Keplerian orbits.  Although not all are necessary for our main results, they may serve useful in other perturbative studies of the curved Kepler problem.

The position on the conic is given explicitely in terms of the true anomaly (remark \ref{rem:coords}). By conservation of angular momentum (recall $d\ell = n~dt$ with $ n = \frac{m^3M^2}{L^3} + \kappa \frac{L}{m}$):
\[ G~d\ell =n~m\rho^2\sin^2\fe ~d\nu. \]

One may centrally project a curved Kepler conic to the tangent plane at $q_s$ and then parametrize this planar conic by its eccentric anomaly, which we denote here by $u_o$. Letting $a, e, b$ be the semi-major axis, eccentricity and minor axis of this planar conic, the position is given by:
\begin{align}\label{eq:flatEApos}
r &= \rho\tan\fe = a(1 - e\cos u_o) & x &= r\cos\nu = a(\cos u_o - e) & y&=r\sin\nu = b\sin u_o
\end{align}
 and one has: \begin{equation}\label{eq:flatEA}
    \sqrt{\frac{M}{a}}~d\ell =n~\rho\sin\fe\cos\fe~du_o.\end{equation}

Some more parametrizations arise naturally when one \textit{orthogonally} projects the curved Kepler ellipse onto the tangent plane at $q_s$. The time-parametrized motion along this  plane curve sweeps out area at a constant rate, however it is now a quartic curve: the locus of a 4th order polynomial in the plane. This quartic may be seen naturally as an elliptic curve and then parametrized by Jacobi elliptic functions. Taking $R = \rho\sin\fe, X = R\cos\nu, Y = R\sin\nu$, the quartic is the projection to the $XY$-plane of the intersection of the quadratic surfaces:
\[  R^2 = X^2 + Y^2, ~~~~ (R  + eX)^2 = p^2(1 - \kappa R^2), \]
where $p, e$ are as in the proof of prop. \ref{prop:geomcoords}.  Consequently we find the parametrization:
\begin{align*}
R &= \rho\sin\frac{\alpha}{\rho}k' \text{nd}_k w - \rho\cos\frac{\alpha}{\rho} k \text{cd}_k w & X &= \rho\sin\frac{\alpha}{\rho}k' \text{cd}_k w - \rho\cos\frac{\alpha}{\rho} k \text{nd}_k w & Y&= \rho\tan\frac{\beta}{\rho}\cos\frac{\alpha}{\rho}\text{sd}_k w.
\end{align*}
where $k = \sin\frac{\alpha\epsilon}{\rho}, k' = \cos\frac{\alpha\epsilon}{\rho}$. Since the area is swept at a constant rate, one computes:
\[ \rho\sin\frac{\alpha}{\rho}~d\ell = \rho\sin\fe~dw.\]
 which integrates to give a 'curved Kepler equation', i.e. the relation between position and time through: \begin{equation}\label{eq:KepEq}
    \ell = \arccos \text{cd}_k w - \frac{\cot\frac{\alpha}{\rho}}{2}\log \frac{1 + k\text{sn}_k w}{1 - k\text{sn}_k w}
\end{equation}
It turns out that a geometric definition of eccentric anomaly, $u$ (see fig.~\ref{fig:eccanom}), is the Jacobi amplitude of $w$: \[ du = \text{dn}_k w~dw,\] which can be established by using spherical trigonometry to give the position in terms of $u$, and some straightforward, although tedious, simplification.

\begin{figure}[!htb]
\centering
\includegraphics[scale = .4]{eccanom.png}
\caption{\footnotesize{A geometric eccentric anomaly, $u$, for a (non-circular) Keplerian conic on the sphere. One inscribes a circle around the conic and to a point $q$ on the conic assigns the angle $u :=\angle(pc, q_c, p)$, where $p$ is the intersection of the circle with the perpendicular dropped from $q$ to the major axis.}}\label{fig:eccanom}
\end{figure}

\begin{rmrk}
Projecting the spherical orbits from the south pole leads as well to quartic curves in the tangent plane at $q_s$, however along these quartics the time-parametrization is no longer by sweeping area at a constant rate --as it is for orthogonal projection-- which we found only led to complicated expressions. Another parametrization of the orbits is presented in \cite{Koz} eq. (28), and used to derive a different 'curved Kepler equation' from our eq. (\ref{eq:KepEq}).
\end{rmrk}

\section{Expansion of $\langle Per\rangle$}
\label{app:Per}

We will compute an expansion in powers of our small parameter $\e$ for $\langle Per\rangle$ (prop. \ref{prop:scal}) upto $O(\e^5)$, eq. (\ref{eq:PerExp}). This expansion may be found by using the 'flat eccentric anomaly', $u_o$, of eq. (\ref{eq:flatEA}) to average the terms.  With this approach, it is necessary to make use of the following formulas allowing one to translate between the major axis and eccentricity $(a,e)$ of the centrally projected planar conic and our Delaunay coordinates:
\[ a = \frac{L^2}{m^2}\left(\frac{1}{1 - \frac{\kappa}{m^4}L^2(L^2-G^2)}\right) = \rho\e \frac{\hat L^2}{m^2}(1 + O(\e^2)),~~~~~~e^2 = (1 - \frac{\hat G^2}{\hat L^2})(1 + \frac{\e^2}{m^4}\hat L^2\hat G^2)\]
(recall we set $m = m_1m_2$). Note that by eq. (\ref{eq:flatEApos}), $r = \rho\tan\fe = \rho O(\e)$, so indeed $\fe = \frac{r}{\rho} + O(\e^3)$ is $O(\e)$.

By Taylor expansion of eq. (\ref{eq:Fred}) in $\fe$, and then exchanging $\fe$ to $\frac{r}{\rho}$, we have:
\begin{equation}\label{eq:Tayexp}
Per = \e^2\left( \frac{\hat C^2}{2} - \hat G^2 + \frac23(m_2-m_1)\hat G\sqrt{\hat C^2 - \hat G^2}  \frac{r\cos\theta}{\rho}  + ((\hat C^2 - \hat G^2)\cos^2\theta - \hat G^2)\sigma\frac{r^2}{\rho^2}\right) + O(\e^5),\end{equation}
where we set $\sigma = \frac{1 -(m_1^3 + m_2^3)}{6}$. So, to determine $\langle Per\rangle = \frac{1}{2\pi}\int_0^{2\pi} Per~d\ell$ upto $O(\e^5)$, it remains to find:
\[
\langle r\cos\theta \rangle,  ~~~~~~~ \langle r^2\cos^2\theta \rangle,  ~~~~~~~ \langle r^2 \rangle.
\]
By eq. (\ref{eq:flatEA}), $d\ell = n\sqrt{a}\frac{r}{1 + \kappa r^2} du_o = n\sqrt{a} r (1 - \kappa r^2 + O(\e^4))~du_o$, where
\[ n\sqrt{a} = \frac{m^2}{\rho\e\hat L^2} + O(\e).\]

Hence:
\[ \langle r\cos\theta \rangle = \frac{n\sqrt{a}}{2\pi} \int_0^{2\pi}r^2\cos\theta -  \kappa r^3\cos\theta ~du_o + O(\e^3).\]
Using $\theta = \nu + g$, and the expressions for $r, r\cos\nu, r\sin \nu$ from eq. (\ref{eq:flatEApos}), we obtain:
\[ \langle r\cos\theta \rangle = a^2 e n \sqrt{a}\cos g \left( -\frac32 + \kappa a ( 2 + \frac{e^2}{2}) \right). \]
Or, in terms of the (scaled) Delaunay coordinates:
\begin{equation}\label{eq:avg1}
\langle r\cos\theta \rangle = \rho\e \cos g~ \hat L\sqrt{\hat L^2 - \hat G^2}\left( - \frac32 + \frac{\e}{2\rho m^2}(5\hat L^2 - \hat G^2) \right) + O(\e^3).
\end{equation}
Likewise, one computes:
\begin{align}\label{eq:avg2}
\langle r^2\cos^2\theta \rangle &= \frac{\rho^2\e^2}{2m^4}\hat L^2\left( 2\hat L^2 + (3 + 5\cos 2g)(\hat L^2 - \hat G^2)\right) + O(\e^4),  & \langle r^2 \rangle&= \frac{\rho^2\e^2}{2m^4}\hat L^2(5\hat L^2 - 3\hat G^2) + O(\e^4).
\end{align}

Combining eqs. (\ref{eq:avg1}), (\ref{eq:avg2}) with the averaged eq. (\ref{eq:Tayexp}), yields:
\begin{equation}\label{eq:PerExp}
    \begin{split}
\langle Per\rangle = \e^2\left(\frac{\hat C^2}{2} - \hat G^2\right) + \e^3\left(m_\Delta\hat L\hat G\sqrt{(\hat C^2 - \hat G^2)(\hat L^2 - \hat G^2)}\cos g\right)  \\
+ \e^4\left[ \tilde m \hat L^2 \left( (\hat C^2 - \hat G^2)(2\hat L^2 + (\hat L^2 - \hat G^2)(3 + 5\cos 2g)) - \hat G^2(5\hat L^2 - 3\hat G^2)\right) \right. \\
\left.  - \frac{m_\Delta}{3\rho m^2}\hat L\hat G\sqrt{(\hat L^2 - \hat G^2)(\hat C^2 - \hat G^2)}(5\hat L^2 - \hat G^2)\cos g  \right] + O(\e^5),\\
    \end{split}
\end{equation}
where we set $m_\Delta = m_1-m_2, \tilde m = \frac{1 - m_1^3 - m_2^3}{12m^4}$, and $m = m_1m_2$.

\begin{rmrk}
When the masses are equal, one may avoid the need to take expansions as we have done here by using the simplified expression in remark \ref{rmrk:eqMass}, and making use of the different anomalies presented in section \ref{sec:anoms} to obtain explicit expressions.
\end{rmrk}

Taking into account higher order terms of $\langle Per\rangle$ leads to finer descriptions of the orbits. For example above we have for the most part worked at order $\e^2$, at which we see the precession properties described for our continued orbits. Considering the order 3-terms (or order 4-terms with equal masses), one arrives at a more precise description of the near collision orbits. Namely, upon fixing a value of $L$, the phase portrait of the secular dynamics on the $(G,g)$ cylinder near $G= 0$ is qualitatively as that of the pendulum: there are two unstable (saddle) fixed points with $G = 0, g = 0, \pi$, having a heteroclinic connection. These fixed points lift to the red collision orbits of fig.~\ref{fig:orbs}. One corresponds to a collision orbit with body 1 in the northern hemisphere and body 2 in the southern hemisphere while the other collision orbit has body 2 in the northern hemisphere.

\begin{rmrk}\label{rmrk:split}
For the true dynamics, one expects a splitting of these stable and unstable manifolds. If one could establish a transversal intersection between these manifolds, one would obtain random motions near these collision orbits along the equator of the following form.  For any sequence $s_1,s_2,...$ with $s_k\in \{1,2\}$, there would exist an orbit for which, during the time interval $[nT, (n+1)T]$  the two bodies are closely following the periodic collision orbit with body $s_n$ in the northern hemisphere, where $T>0$ is some sufficiently long time period.
\end{rmrk}


text:
\begin{document}

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\title[$n$-braid action on the free group of rank $n$]{Actions of the n-strand braid groups on the free group of rank n which are similar to the Artin representation}
\thanks{T.I. was partially supported by the Grant-in-Aid for Research Activity start-up, Grant Number 25887030.}
\author{Tetsuya Ito}
\address{Research Institute for Mathematical Sciences, Kyoto University
Kyoto University, Sakyo-ku, Kyoto, Japan}
\email{tetitoh@kurims.kyoto-u.ac.jp}
\subjclass[2010]{Primary~20F36, Secondary~57M25}
\urladdr{http://www.kurims.kyoto-u.ac.jp/~tetitoh/}
\keywords{Braid groups, Local $\Aut(F_{n})$ representation, group-valued invariants.}

\begin{abstract}
We classify an action of the $n$-strand braid group on the free group of rank $n$ which is similar to the Artin representation in the sense that the $i$-th generator $\sigma_{i}$ of $B_{n}$ acts so that it fixes all free generators $x_{j}$ except $j = i,i+1$. We determine all such representations and discuss knot invariants coming from such representations.
\end{abstract}
\maketitle

\section{Introduction}

Let $B_{n}$ be the braid group of $n$-strands, with standard generators $\sigma_{1},\ldots,\sigma_{n-1}$ and let $F_{n}$ be the free group of rank $n$ generated by $x_{1},\ldots,x_{n}$. Throughout the paper, we consider {\em right actions}.

The {\em Artin representation} $\rho_{\sf Artin}$ is a standard action of $B_{n}$ on $F_{n}$, given by
 \begin{gather*}
 (x_{j}) \rho_{\sf Artin}(\sigma_{i}) =
\begin{cases}
x_{i}x_{i+1}x_{i}^{-1} & (j=i),\\
x_{i} & (j=i+1), \\
x_{j} & \text{(Otherwise)}.
\end{cases}
\end{gather*}
A remarkable feature of $\rho_{\sf Artin}$ is that $\rho_{\sf Artin}(\sigma_i)$ preserves all free generators of $F_{n}$ except $x_{i}$ and $x_{i+1}$, and that $\rho_{\sf Artin}(\sigma_i)$ sends $x_{i}$ and $x_{i+1}$ to words on $\{x_{i},x_{i+1}\}$.
In this paper we study an $\Aut(F_{n})$-representation of the braid group $B_{n}$ having the same property.

For $i=1,\ldots,n-1$, we say an automorphism $\otau: F_{n} \rightarrow F_{n}$ is {\em $i$-local} if $\otau(x_{j}) = x_{j}$ for $j \neq i,i+1$ and $\otau ( \langle x_{i},x_{i+1} \rangle ) = \langle x_{i},x_{i+1} \rangle$ where $ \langle x_{i},x_{i+1} \rangle$ represents the subgroup of $F_{n}$ generated by $x_{i}$ and $x_{i+1}$. In other words, $\otau \in \Aut(F_{n})$ is $i$-local if and only if there exists $\tau \in \Aut(F_{2})$ such that
\[ \otau = \textsf{id} * \tau * \textsf{id} : F_{n}= F_{i-1}*F_{2}*F_{n-i-1} \rightarrow F_{i-1}*F_{2}*F_{n-i-1} = F_{n}. \]
We say $\tau$ is the {\em core} of $\otau$, and write $\otau = T^{i}(\tau)$.

\begin{definition}
An $\Aut(F_{n})$ representation of the braid group $\Theta: B_{n} \rightarrow \Aut(F_{n})$ is {\em local} if $\Theta(\sigma_{i})$ is $i$-local for all $i=1,2,\ldots,n-1$.
\end{definition}

For a local $\Aut(F_{n})$ representation $\Theta$, we denote the core of $\Theta(\sigma_{i})$ by $\tau^{\Theta}_{i}$ (or simply $\tau_{i}$), and we will say that $(\tau_{1},\ldots,\tau_{n-1}) \in \Aut(F_{n})^{n-1}$ {\em defines} $\Theta$.

A local $\Aut(F_{n})$ representation with $\tau_{1}= \tau_{2} = \cdots = \tau_{n-1}$ was introduced by Wada \cite{w} and called a {\em Wada-type representation}.
In \cite[Corollary 1.2]{i} we determined all Wada-type representations (actually we have classified more general objects, called a solution of certain variant of set-theoretical Yang-Baxter equation).
We proved that, up to certain natural symmetries, there are exactly seven types of Wada-type representations as Wada conjectured in \cite{w}.

In this paper we give a classification of local $\Aut(F_{n})$-representations:
In Theorem \ref{thm:main} we list up all local $\Aut(F_{3})$-representations, and give an oriented, labelled graph whose edge-path describes all local $\Aut(F_{n})$-representations.

One motivation of studying and introducing a local $\Aut(F_{n})$ representation comes from knot theory. By imitating the presentation of knot groups using the Artin representations \cite{b}, one obtains a group-valued invariant of knots and links from Wada-type representations \cite{w}. By generalizing the Artin representation in a different manner, Crisp and Paris constructed a group-valued invariant of knots and links \cite{cp}, and in \cite{i1} the author used a quandle variant of Crisp-Paris' construction to define a quandle-valued knot invariant. However, there are few applications of these invariants. Unfortunately, the classification of Wada-type representations shows that in most cases the group-valued invariants from Wada-type representations are determined by the fundamental group of the double branched covering of knots, so they are less interesting.

We can use a local $\Aut(F_{n})$-representation to construct a group-valued invariant in a similar manner (Proposition \ref{prop:ginvariant}). Unfortunately, our classification again shows that group-valued invariants from local $\Aut(F_{n})$-representations are nothing new, because they coincide with the group-valued invariants from a Wada-type representation of the same type (see Theorem \ref{theorem:ginvariant}).

Nevertheless, the classification and idea of local $\Aut(F_{n})$ representations are interesting in its own right. First, a local $\Aut(F_{n})$ representation can be seen as a generalization of Yang-Baxter equation.
 Second, an idea of local $\Aut(F_{n})$ representations to produce knot invariants can be investigated further, although the local $\Aut(F_{n})$ representation themselves cannot produce new interesting invariants. In the theory of quandles or state-sum invariants (see \cite{cjkls}), one treats all crossing in the same manner, in the sense that we assign the relations or weights of the same form at each crossings. It is an interesting problem to try to generalize such invariants by giving \emph{different} types of relations or weights at each crossing. Finally, investigating a homomorphism from $B_{n}$ to $\Aut(F_{n})$  would be interesting and will help us to understand relationships between these two groups.

\section{Classification of local $\Aut(F_{n})$ braid representations}
\label{sec:classification}

\subsection{Statement of the Classification theorem}

For $\tau,\kappa \in \Aut(F_{2})$ and $i,j \in \{1,\ldots,n-1\}$, $T^{i}(\tau)T^{j}(\kappa) = T^{j}(\kappa)T^{i}(\kappa)$ if $|i-j|>1$, so $(\tau_{1},\ldots, \tau_{n-1})\in \Aut(F_{2})^{n-1}$ defines a local $\Aut(F_{n})$ representation if and only if $(\tau_{i},\tau_{i+1})$ defines a local $\Aut(F_{3})$ representation for each $i$.
Hence to classify local $\Aut(F_{n})$ representations, it is sufficient to consider the case $n=3$. From now on, we study the problem when $(\tau,\kappa) \in \Aut(F_{2})^{2}$ defines a local $\Aut(F_{3})$ braid representation.

Let $F_{2}$ be the free group of rank two generated by $\{a,b\}$.
For a pair of automorphisms $(\tau,\kappa) \in \Aut(F_{2})^{2}$ that defines a local $\Aut(F_{3})$ representation $\Theta$, let $A,B,C,D$ be the reduced words representing $\tau(a),\tau(b),\kappa(a),\kappa(b)$, respectively. We will often say reduced words $(A,B,C,D)$ {\em define} a local $\Aut(F_{3})$ braid representation.

To give a precise statement of the classification, first we observe symmetries of local $\Aut(F_{3})$ representation derived from symmetries of the free groups and the braid groups, which we will call \emph{natural symmteries}.

In this paper we use the following notation. For words $V$ and $W$ we will write $V \equiv W$ if they are the same as words, and will write $V=W$ if they represents the same element in the free group. Thus, $abb^{-1} \not \equiv a$ but $abb^{-1} = a$, for example.\\

\begin{description}
\item[Inverse symmetry]
 {$ $}\\
Let $\tau'$ and $\kappa'$ be the core of $\Theta(\sigma_{1}^{-1})$ and $\Theta(\sigma_{2}^{-1})$ respectively.
Then $(\tau',\kappa')$ also defines a local $\Aut(F_{3})$ representation $\Theta^{-}$ which we call the {\em inverse representation} of $\Theta$.\\

\item[Swap symmetry]
 {$ $}\\
Let $A^{\sigma}$ be the word over $\{a^{\pm 1},b^{\pm 1}\}$ obtained by interchanging the letters $a$ and $b$. For example, $A^{\sigma} \equiv b^{-2}ab$ if $A \equiv a^{-2}ba$. $B^{\sigma}, C^{\sigma}$ and $D^{\sigma}$ are defined similarly.
Then $(D^{\sigma},C^{\sigma},B^{\sigma},A^{\sigma})$ also defines a local $\Aut(F_{3})$ representation $\Theta^{\sigma}$ which we call the {\em swap-dual representation} of $\Theta$.\\

\item[Backward symmetry]
 {$ $}\\
Let $\overleftarrow{A}$ be a reduced word obtained by reading $A$
backward. For example, $\overleftarrow{A} \equiv a^{2}bab^{-1}$ if $A \equiv b^{-1}aba^{2}$. $\overleftarrow{B}$, $\overleftarrow{C}$ and $\overleftarrow{D}$ are defined similarly. Then $(\overleftarrow{A}, \overleftarrow{B}, \overleftarrow{C}, \overleftarrow{D})$ defines a local $\Aut(F_{3})$ braid representation $\overleftarrow{\Theta}$ which we call the {\em backward-dual representation} of $\Theta$. \\

\end{description}

It is directly checked that these three symmetries commute each other. That is, $(\Theta^{-})^{\sigma}$, the swap-dual of the inverse $\Theta^{-}$ coincide with the $(\Theta^{\sigma})^{-}$, the inverse of the swap-dual $\Theta^{\sigma}$.
So we simply write $\Theta^{-\sigma}$ to represent the swap and inverse dual of $\Theta$, for example. Thus for each local $\Aut(F_{3})$ representation $\Theta$, there may be eight different local $\Aut(F_{3})$-representations by considering natural symmetries.

The following is the main result of this paper.

\begin{theorem}[Classification of local $\Aut(F_{3})$ braid representation.]
\label{thm:main}
Up to natural symmetries, reduced words $(A,B,C,D)$ on $\{a^{\pm1},b^{\pm1}\}$ defines a local $\Aut(F_{3})$ braid representation if and only if $(A,B,C,D)$ is one of the following.
\begin{enumerate}
\item[($T$)] $(A,B,C,D)=(a,b,a,b)$.
\item[($T'$)] $(A,B,C,D)=(a,b^{-1},a^{-1},b)$.
\item[($A_{r}\,$-- 1)]  $(A,B,C,D)=(a^{r}b a^{-r}, a, a^{r}ba^{-r}, a)$ \ $(r \in \Z_{\geq 0})$.
\item[($A_{r}\,$-- 2)]  $(A,B,C,D) = (a^{r}ba^{-r}, a, a^{r} b^{-1}a^{-r},a^{-1})$ \ $(r \in \Z_{\geq 0})$.
\item[($A_{r}\,$-- 3)]  $(A,B,C,D) =  (a^{r}b^{-1}a^{-r}, a^{-1}, a^{-r} b^{-1}a^{r}, a^{-1})$ \ $(r \in \Z_{\geq 0})$.

\item[($B\,$-- 1)] $(A,B,C,D) = (b^{-1},a , b^{-1}, a)$.
\item[($B\,$-- 2)] $(A,B,C,D) = (b^{-1},a , b, a^{-1})$.

\item[($C\,$-- 1)] $(A,B,C,D) = (ab^{-1}a,a,ab^{-1}a,a)$.
\item[($C\,$-- 2)] $(A,B,C,D) = (ab^{-1}a, a, aba, a^{-1})$.
\item[($C\,$-- 3)] $(A,B,C,D) = (aba, a^{-1}, aba, a^{-1})$.

\item[($D\,$-- 1)] $(A,B,C,D) = (a^{-1}b^{-1}a,b^{2}a,a^{-1}b^{-1}a,b^{2}a)$.
\item[($D\,$-- 2)] $(A,B,C,D) = (aba^{-1},b^{2}a^{-1},a^{-1}b^{-1}a,b^{2}a)$.
\item[($D\,$-- 3)] $(A,B,C,D) = (a^{-1}b^{-1}a,b^{2}a,a^{-1}ba,a^{-1}b^{2})$.
\item[($D\,$-- 4)] $(A,B,C,D) = (aba^{-1},b^{2}a^{-1},a^{-1}ba,a^{-1}b^{2})$.

\end{enumerate}

\end{theorem}

It is convenient to express the classification of local $\Aut(F_{n})$ representations by using an oriented graph $\Gamma$ in Figure \ref{fig:classification}.
For each vertex $v$ of $\Gamma$, we assign an element of $\Aut(F_{2})$, $\tau_{v}: F_{2} \rightarrow F_{2}$ by indicating a pair of reduced words $(\tau_v(a),\tau_v(b))$. Two vertices $v$ and $w$ are connected by an oriented edge $e$ oriented from $v$ to $w$ if and only if $(\tau_{v},\tau_{w})$ defines a local $\Aut(F_{3})$ representation. Up to natural symmetries, $(\tau_v,\tau_w)$ appears in the list in Theorem \ref{thm:main} so for each edge, for convenience we assign a labeling to indicate the correspondence to the list in Theorem \ref{thm:main}: for example, $\overleftarrow{(A_{r}-1)}$ means the backward-dual of the local $\Aut(F_{3})$-representation $(A_{r}-1)$ in Theorem \ref{thm:main}.

Up to natural symmetries, the set of local $\Aut(F_{n})$ representations is identified with the set of oriented edge-path of length $(n-2)$ in $\Gamma$: For an oriented edge-path $\gamma$ of $\Gamma$, assume that $\gamma$ passes the vertices $v_{1},v_{2},\ldots,v_{n-1}$ in this order. Then $(\tau_{v_{1}},\ldots,\tau_{v_{n-1}}) \in \Aut(F_{2})^{n-1}$ defines a local $\Aut(F_{n})$ representation $\Theta_{\gamma}$. Conversely, for a local $\Aut(F_{n})$ representation $\Theta$, take a vertex $v_{i}$ $(i=1,\ldots,n-1)$ of $\Gamma$ so that $\tau^{\Theta}_{i} = \tau_{v_{i}}$ $i=1,\ldots,n-1$. If we take $\gamma$ an oriented edge-path $\gamma$ of $\Gamma$ so that $\gamma$ passes the vertices $v_{1},\ldots,v_{n-1}$ in this order, then $\Theta = \Theta_{\gamma}$.

For each connected component of $\Gamma$, we assign the labeling $\{T,T',A_{r} (r\in \Z_{\geq 0}),C,D\}$ and we call a local $\Aut(F_{n})$ representation $\Theta$ is of type $\mathcal{L}$ if $\Theta$ corresponds to the oriented edge-path in the connected component labelled by $\mathcal{L} \in \{T,T',A_{r},B,C,D,\}$.

We also note:
\begin{theorem}
A local $\Aut(F_{n})$-representation which is not of type $(T)$, $(T')$, $(A_{0})$, $(B)$ is faithful.
\end{theorem}
\begin{proof}
The faithfulness of Wada-type representation is proven in \cite{sh}, by using the  theory of Dehornoy's ordering \cite{ddrw}: For every non-trivial braid $\beta$, either $\beta$ or $\beta^{-1}$ is represented by a word $w$ over $\{\sigma_{1}^{\pm 1},\ldots,\sigma_{n-1}^{\pm 1}\}$ which is $\sigma$-positive (there exists $i\in \{1,\ldots,n-1\}$ such that $w$ contains at least one $\sigma_{i}$ but does not contain any $\sigma_{1}^{\pm 1},\ldots,\sigma_{i-1}^{-1},\sigma_{i}^{-1}$.

Let $\Theta$ be a local $\Aut(F_{n})$ representation which is not of type $(T)$, $(T')$, $(A_{0})$, $(B)$ (note that a local $\Aut(F_{n})$ representation of type $(T)$, $(T')$, $(A_{0})$, $(B)$ are clearly non-faithful). A direct calculation (see \cite{sh}, for details) shows that for a braid $\beta$ represented by a $\sigma$-positive word, $\Theta(\beta) \neq 1$.
\end{proof}

\begin{figure}
\[
[T] \ \ \
\xymatrix{
(a,b) \ar@(ul,ur)^{(T)}
}
\ \ \ \ \ \ \ \ [T'] \ \
\xymatrix{
(a,b^{-1}) \ar[r]^{(T')} & (a^{-1},b)
}
\]

\vspace{0.3cm}

\[
\lower4ex\hbox{  $[A_r]$}
\ \ \ \
\xymatrix{
 (a^{r}ba^{-r},a) \ar@(ul,ur)^{(A_{r}-1)} \ar[rr]^{(A_{r}-2)}& & (a^{r}b^{-1}a^{-r},a^{-1}) \ar[d]^{(A_{r}-2)^{-\sigma}} \ar@/^2mm/[lld]^<<<<<<<{\;\;\;(A_{r}-3)}\\
 (a^{-r}b^{-1}a^{r},b) \ar[u]^{\overleftarrow{(A_{r}-2)}{}^{-\sigma}} \ar@/^3mm/[urr]^<<<<<{\;\;\;\overleftarrow{(A_{r}-3)}} & &
 (a^{-r}ba^{r},b) \ar[ll]^{\overleftarrow{(A_{r}-2)}}
\ar@(dl,dr)_{\overleftarrow{(A_{r}-1)}} \\
}\]

\[
[B] \ \ \
\xymatrix{
(b^{-1},a) \ar@(ul,ur)^{(B-1)}  \ar@/^2mm/[rr]^{(B-2)} &  & (b,a^{-1}) \ar@(ur,ul)_{(B-1)^{\sigma}} \ar@/^2mm/[ll]^{(B-2)^{-}}
}\]

\[\xymatrix{
[C] \ \ \
(ab^{-1}a,a) \ar@(ul,ur)^{(C-1)}  \ar@/^2mm/[rr]^{(C-2)} &  & (aba,a^{-1}) \ar@(ur,ul)_{(C-3)} \ar@/^2mm/[ll]^<<<<<{(C-2)^{-\sigma}}
}\]

\vspace{0.3cm}

\[
\lower9ex\hbox{$[D]$} \ \ \
\xymatrix{
  & (a^{-1}ba,b^{2}a) \ar@(ul,ur)^{(D-1)} \ar[rd]^{\;\;\overleftarrow{(D-2)}{}^{-\sigma}}&\\
(aba^{-1},b^{2}a^{-1}) \ar[ur]^{\overleftarrow{(D-3)}{}^{-\sigma}}
\ar@/^3mm/[rr]^{\overleftarrow{(D-4)}{}^{-\sigma}}
& & (a^{-1}ba,a^{-1}b^{2})\ar@/^3mm/[ll]^{(D-4)^{-\sigma}} \ar[dl]^{(D-3)^{-\sigma}} \\
 & (ab^{-1}a^{-1},ab^{2}) \ar@(dl,dr)_{(D-1)^{-\sigma}} \ar[ul]^{(D-2)^{-\sigma} \; \;}&
}\]
\caption{Graph $\Gamma$ describing the clasisfication of local $\Aut(F_{n})$-representations}
\label{fig:classification}
\end{figure}

\subsection{Classification: Outline}

Before proceeding to prove the classification theorem, we briefly explain an outline of the proof.

First of all, we review our notations.
A word $W$ over an alphabet $\mathcal{A}$ is \emph{reduced} if $W$ contains no subword of the form $aa^{-1}$ and $a^{-1}a$ $(a \in \mathcal{A})$. By removing subwords of the form $aa^{-1}$ or $a^{-1}a$, every word $W$ is changed to the unique reduced word which we denote by $\red(W)$. For two words $W$ and $V$, $W=V$ if and only if $\red(W) \equiv \red(V)$.

For a word $W$ over an alphabet $\{a^{\pm1},b^{\pm1}\}$, and words $A$ and $B$ over another alphabet $\mathcal{X}=\{x^{\pm 1}, y^{\pm 1}\ldots \}$, $W(A,B)$ represents the word over $\mathcal{X}$ obtained from $W$ by substituting each letter $a^{\pm 1}$ and $b^{\pm 1}$ with the word $A^{\pm 1}$ and $B^{\pm 1}$, respectively. For example, if $W \equiv aab$, $A \equiv xy$ and $B \equiv yx^{-1}$, then $W(A,B)\equiv xyxyyx^{-1}$.

For a word $W$ over $\{x^{\pm 1},y^{\pm 1},z^{\pm 1}, \ldots \}$, we will write
\[ W \equiv x^{k_1}y^{k_2}*\cdots \]
to say that $W$ is written as $W \equiv x^{k_1}y^{k_2}W'$, where $W' \not \equiv yW'', y^{-1}W''$. (Here $W'$ might be empty word.)
On the other hand, we will simply use dots, like
\[ W \equiv x^{k_1}y^{k_2} \cdots \]
to say that $W$ is written as $W \equiv x^{k_1}y^{k_2}W'$.
For example, $W \equiv x \cdots x$ just says that $W \equiv xW'x$ for some word $W'$, but if we write $W \equiv x * \cdots * x$, then it means that $W \equiv xW'x$ with $W' \not \equiv x^{\pm 1}W'',W''x^{\pm 1}$.

The proof of the classification theorem goes as follows. By direct computation, $(A,B,C,D)$ defines a local $\Aut(F_{3})$ representation if and only if the following four conditions are satisfied.

\begin{enumerate}
\item[{\bf[T]}:] $A ( A(x,y), C( B(x,y),z)) =  A (x, C(y,z) )$.
\item[{\bf[M]}:] $B ( A(x,y), C( B(x,y), z) ) = C ( B(x, C(y,z) ), D(y,z) )$.
\item[{\bf[B]}:] $D (B(x,y), z) = D ( B(x, C(y,z) ), D(y,z) )$.
\item[{\bf[Aut]}:] The homomorphism $\tau$ of $F_{2}=\langle a,b\rangle$ defined by $a \mapsto A$, $b \mapsto B$ is an automorphism. Similarly, the homomorphism $\kappa$ of $F_{2}$ defined by $a \mapsto C$, $b \mapsto D$ is an automorphism.
\end{enumerate}

In particular, the last condition {\bf[Aut]} implies that all elements $A,B,C,D$ are primitive, that is, they admit no non-trivial root in $F_{2}$.

Let $(A,B,C,D)$ be reduced words which defines a local $\Aut(F_{3})$ representation $\Theta$. The proof of classification theorem splits into the following two propositions which we will be proven in Section \ref{sec:proof}.

\begin{proposition}
\label{prop:caseI} If $B$ does not contain $a^{\pm 1}$, then $\Theta$ is of type  $(T)$ or $(T')$.
\end{proposition}

\begin{proposition}
\label{prop:caseIII}
If $B$ contains at least one $a^{\pm 1}$, then up to natural symmetries, $\Theta$ is of type $(A_{r})$, $(B)$, $(C)$ or $(D)$.
\end{proposition}

Finally we explain reasons why we are able to solve the equations {\bf[T]}, {\bf[M]}, and {\bf[B]}. An important feature of equations {\bf[T]} and {\bf[B]} is that the order of the length of words are different.

The left side of {\bf[T]} is a word of the form $W(W(W(\bullet,\bullet),\bullet),\bullet)$, where $W$ is one of $A,B,C,D$. This implies that the length of the left side of  {\bf[T]} is order of $l^{3}$, where $l$ denotes the length of $W$.
On the other hand, the right side of {\bf[T]} is a word of the form $W(W(\bullet,\bullet),\bullet)$ so its length is order of $l^{2}$. This observation suggests that either $l$ is small or the left side of the word {\bf[T]} must admit many cancellations -- as the length $l$ increases, the number of necessary cancellations must grow quite rapidly.
On the other hand, the number of possible cancellations are limited because all words $A,B,C,D$ are reduced. Thus {\bf[T]} and {\bf[B]} implicitly and vaguely says that the length of the words cannot be large, or these words have strong symmetries which leads to many cancellations.

\begin{remark}
In \cite{i}, the author classified, the solutions of certain variant of the set-theoretical Yang-Baxter equation, which is essentially equal to Wada-type representations without requiring invertibility, the condition {\bf[Aut]}.
The main idea and the method in the proof Theorem \ref{thm:main} are almost the same as the proof of \cite[Theorem 1.1]{i}. However, thanks to {\bf[Aut]} we are often able to simplify arguments. We also remark that the classification of local $\Aut(F_{n})$ representations does not use the classification of Wada-type representations.
\end{remark}

\section{Proof of Classification theorem}
\label{sec:proof}
In this section we prove Theorem \ref{thm:main}, by proving Proposition \ref{prop:caseI} and Proposition \ref{prop:caseIII}.
Let $(A,B,C,D)$ be reduced words that defines a local $\Aut(F_{3})$-representation.

\subsection{Key observation}

Let us regard two words $C(y,z)$ and $D(y,z)$ as alphabets $\mC$ and $\mD$.
Temporary we regard the word $C(B(x,C(y,z)),D(y,z))$, the right side of {\bf[M]}, as a word over $\{x^{\pm 1}, \mC^{\pm 1}, \mD^{\pm 1} \}$.

Then we write $C(B(x,C(y,z)),D(y,z)) = C(B(x,\mC),\mD)$ in the form
\[ C(B(x,\mC),\mD) =  W_{0}(\mC,\mD) x^{n_{1}} W_{1}(\mC,\mD) x^{n_{2}}*\cdots *x^{n_{k}}W_{k}(\mC,\mD),\]
where $W_{i}$ and $n_{i}$ satisfy the following conditions.
\begin{enumerate}
\item[(i)] $W_{0}(\mC ,\mD)$ and $W_{k}(\mC,\mD)$ are reduced words over $\{\mC^{\pm 1},\mD^{\pm 1}\}$.
$W_{0}(\mC,\mD)$ is an empty word if and only if $B \equiv a^{\pm 1} \cdots$ and $C\equiv a^{\pm 1}\cdots$.
\item[(ii)] For $i=1,\ldots,k-1$, $W_{i}(\mC,\mD)$ is a reduced, non-empty word over $\{\mC^{\pm 1},\mD^{\pm 1}\}$.
\item[(iii)] $n_{1},\ldots,n_{k}$ are non-zero integers.
\end{enumerate}

Now we back to see $C(B(x,\mC),\mD)$ as a word over $\{x^{\pm 1}, y^{\pm 1}, z^{\pm 1}\}$, by substituting $\mC$ and $\mD$ with $C(x,y)$ and $D(x,y)$.
Let $W_{i}\equiv W_{i}(C(y,z),D(y,z))$ be a word over $\{ y^{\pm 1}, z^{\pm 1}\}$.
By \textbf{[Aut]}, $C(y,z)$ and $D(y,z)$ generate $F_{2}=\langle y,z\rangle$. Since $W_{i}(\mC,\mD)$ is a non-empty reduced word, as an element of $F_{2}=\langle y,z\rangle$, $W_{i} \neq 1$. This shows the following.

\begin{lemma}
\label{lemma:key}
\[ \red( C(B(x,C(y,z)),D(y,z)) ) \equiv \red(W_{0}) x^{n_{1}} \red(W_{1}) x^{n_{2}}*\cdots * x^{n_{k}} \red(W_{k}) \]
\end{lemma}

A similar calculation applies for $D(B(x,C(y,z)),D(y,z)))$.
Similarly, by regarding $A(x,y)$ and $B(x,y)$ as alphabets $\mA$ and $\mB$, we can apply a similar calculation for $A(A(x,y),C(B(x,y),z))$ and $B(A(A(x,y),C(B(x,y),z)))$.
We will often use (sometimes without referring) Lemma \ref{lemma:key} to compute reduced word expression of left or right sides of equations {\bf [T]}, {\bf [M]} and {\bf [B]}.

\subsection{The case $B$ contains no $a^{\pm 1}$}

\begin{proof}[Proof of Proposition \ref{prop:caseI}]
By {\bf[Aut]} and the hypothesis that $B$ does not contain $a^{\pm 1}$, $B \equiv b^{\beta}$ $(\beta=\pm 1)$.
By {\bf[Aut]}, this implies that $A$ must be of the form
\[ A \equiv b^{p}a^{\alpha} b^{p'} \;\;\; (p,p' \in \Z, \alpha \in \{\pm 1\}) . \]

By {\bf[T]}, we get
\[ C(y^{\beta},z)^{p} (y^{p}x^{\alpha}y^{p'})^{\alpha}C(y^{\beta},z)^{p'} = C(y,z)^{p} x^{\alpha} C(y,z)^{p'}. \]

By comparing the exponent of $x$, we get $\alpha^{2}= \alpha$ so $\alpha = 1$. Hence the equation {\bf[T]} is actually written as
\[ C(y^{\beta},z)^{p} y^{p} x y^{p'}C(y^{\beta},z)^{p'} = C(y,z)^{p} x C(y,z)^{p'}. \]
This equality holds if and only if two equalities
\begin{equation}
\label{eqn:A}  \left\{
\begin{array}{l}
C(y^{\beta},z)^{p} y^{p} = C(y,z)^{p} \\
y^{p'}C(y^{\beta},z)^{p'} =C(y,z)^{p'}
\end{array}
\right.
\end{equation}
hold.
Let $m$ be the exponent sum of the $y$ in $C(y,z)$. By comparing the exponent of $y$ in (\ref{eqn:A}), we get
\begin{gather*}
\begin{cases}
\beta mp+p=mp \\
\beta mp'+p'=mp'.
\end{cases}
\end{gather*}
If $p\neq 0$, then $\beta m+1=m$. Since $\beta =\pm 1$ and $m$ is an integer, this is impossible so $p=0$. Similalry, $p'=0$ hence we conclude $A \equiv a$.

To determine $C$ and $D$ we consider the following two cases according to the value of $\beta$. Let $\kappa:F_{2}\rightarrow F_{2}$ be an automorphism defined by $\kappa(a)=C$ and $\kappa(b)=D$.\\

{\bf Case 1: $\beta=+1$.}\\

By {\bf[M]} and {\bf[B]},
\[ \left\{
\begin{array}{l}
C(y,z) = C(C(y,z),D(y,z))\\
D(y,z) = D(C(y,z),D(y,z))
\end{array}\right.
\]
These equalities show that $\kappa^{2}=\kappa$, so $\kappa = id$.
Thus we get
\begin{itemize}
\item[$(T)$]  $(A,B,C,D) = (a,b,a,b)$.\\
\end{itemize}

{\bf Case 2: $\beta =-1$.}\\

By {\bf[M]} and {\bf[B]},
\[ \left\{
\begin{array}{l}
C(y^{-1},z)^{-1} = C(C(y,z)^{-1},D(y,z))\\
D(y^{-1},z) = D (C(y,z)^{-1},D(y,z)).
\end{array}\right.
\]
Let $\iota:F_{2} \rightarrow F_{2}$ be the involution defined by $\iota(a)=a^{-1}$, $\iota(b)=b$. Then the above equalities say that $\iota\kappa\iota = \kappa \iota\kappa$. Moreover, $\kappa$ has order at most two since  $\kappa^{2}=\kappa^{2}\iota\kappa \kappa^{-1} \iota = \iota \kappa \kappa^{-1} \iota = id$.

Let $G$ be the subgroup of $\Aut(F_{2})$ generated by $\iota$ and $\kappa$.
The relations $\kappa^{2}=\iota^{2}=id$ and $\iota\kappa\iota = \kappa \iota\kappa$ show that $G$ is a quotient of the symmetric group $S_{3}$ so $G$ is either $\Z_{2}$ or $S_{3}$.

Let $\pi: \Aut(F_{2}) \rightarrow \textrm{GL}(2;\Z)$ be the projection.
By direct calculation, $\pi(\kappa)=\left(\begin{array}{cc} \pm 1  &0 \\ 0 & \pm 1  \end{array}\right)$, so $\pi(G) \cong \Z_{2}$.
On the other hand, an order $3$ element of $\Aut(F_{2})$ is conjugate to $\rho:F_{2} \rightarrow F_{2}$ given by $\rho(a)=a^{-1}b, \rho(b)=a^{-1}$ \cite{mes}. This shows that $G$ cannot contain an element of order three, so $G$ is isomorphic to $\Z_{2}$. Hence $\kappa = \iota$ and we get
\begin{itemize}
\item[$(T')$]  $(A,B,C,D) = (a,b^{-1},a^{-1},b)$.
\end{itemize}
\end{proof}

\subsection{The case $B$ contains $a^{\pm 1}$}

In this section we prove Proposition \ref{prop:caseIII}, which treats the case $B$ contains at least one $a^{\pm 1}$.
If $C$ does not contain $b^{\pm 1}$, then the swap-dual of $\Theta$ satisfies the assumption of Proposition \ref{prop:caseI}, so we may also assume that $C$ contains at least one $b^{\pm 1}$.

\begin{proposition}
 \label{prop:caseII}
Under the assumption of Proposition \ref{prop:caseIII}, $A$ contains at least one $b^{\pm 1}$ and $D$ contains at least one $a^{\pm 1}$
\end{proposition}

\begin{proof}

We show $A$ contains at least one $b^{\pm 1}$. The assertion for $D$ is obtained by considering the swap-dual.

Assume contrary, $A$ does not contain $b^{\pm 1}$, so $A \equiv a^{\alpha}$ $(\alpha =\pm 1)$.
By the same argument as Proposition \ref{prop:caseI}, by \textbf{[T]} and \textbf{[Aut]} $\alpha= +1$ and $B \equiv a^{q}b^{\beta}a^{q'}$ $(\beta \in \{\pm 1\},\; q,q' \in \Z )$. Since $B$ contains at least one $a^{\pm 1}$, $qq' \neq 0$. By taking the backward-dual if necessary, we may assume that $q\neq 0$. Then the equation \textbf{[M]} is written as
\begin{equation}
\label{eqn:caseIIM}
x^{q}C(x^{q}y^{\beta}x^{q'},z)^{\beta}x^{q'} = C(x^{q}C(y,z)^{\beta}x^{q'},D(y,z))
\end{equation}

We show that the equation (\ref{eqn:caseIIM}) has no solutions.

Let $m$ be the exponent sum of $a$ in the word $C$.
By comparing the exponent of $x$ in (\ref{eqn:caseIIM}),
we have $(q+q')(m\beta +1) = m(q+q')$.
If $q+q' \neq 0$, then we get $m\beta+1 = m$. Since $m$ is an integer and $\beta \in \{ \pm 1\}$, this cannot happen so $q'=-q$.

Let us put
\[ C(x,y) \equiv x^{c}y^{r}*\cdots \equiv *\cdots y^{r'}x^{c'} \ \ (r,r' \neq 0). \]
Then the left side of (\ref{eqn:caseIIM}) is
\begin{gather*}
x^{q}C(x^{q}y^{\beta}x^{-q},z)^{\beta}x^{-q} =
\begin{cases}
x^{2q}y^{c}x^{-q}z^{r}*\cdots = \cdots *z^{r'}x^{q}y^{c'}x^{-2q} & (\beta = +1)\\
x^{2q}y^{-c'}x^{-q}z^{r'}*\cdots = \cdots *z^{-r}x^{q}y^{-c}x^{-2q}& (\beta = -1)\\
\end{cases}
\end{gather*}
whereas the right side of (\ref{eqn:caseIIM}) is
\[ C(x^{q}C(y,z)^{\beta}x^{-q},D(y,z)) =
x^{q}C(y,z)^{\beta c}x^{-q}D(y,z)^{r}\cdots = \cdots D(y,z)^{r'}x^{q}C(y,z)^{\beta c'}x^{-q}.
\]
By comparing the reduced word expression of both sides, we conclude (\ref{eqn:caseIIM}) has no solutions. For example, assume that $\beta = -1$, $c=0$ and $c' \neq 0$. Then the reduced word expression of the left side of (\ref{eqn:caseIIM}) is $x^{2q}*\cdots$, whereas the right side of (\ref{eqn:caseIIM}) begins with a letter $y^{\pm 1}$ or $z^{\pm 1}$, contradiction. The other cases are proven similarly.

\end{proof}

Summarizing, we have the following constraints for word $A,B,C$ and $D$:
\begin{gather}
\label{eqn:hypo}
\begin{cases}
A \text{ contains at least one } b^{\pm1}.\\
B \text{ contains at least one } a^{\pm1}.\\
C \text{ contains at least one } b^{\pm1}.\\
D \text{ contains at least one } a^{\pm1}.
\end{cases}
\end{gather}

To proceed further, we observe the following much stronger constraints for the words $B$ and $C$.
\begin{lemma}
\label{lem:restriction}
Under the assumption of Proposition \ref{prop:caseIII}, the reduced words $B$ and $C$ satisfy the following.
\begin{enumerate}
\item  $B \equiv b^{q}a^{\beta} b^{q'}, \;\;\; C \equiv a^{r}b^{\gamma} a^{r'}$  $(q,q',r,r' \in \Z,\  \beta,\gamma \in \{\pm 1\})$.
\item  $qq'rr'=0$.
\end{enumerate}
\end{lemma}
\begin{proof}
By (\ref{eqn:hypo}), we may put
\begin{gather*}
\begin{cases}
A \equiv a^{p}b^{p'}*\cdots & (p' \neq 0) \\
C \equiv a^{r}b^{\gamma}a^{r'}b^{\gamma'} *\cdots & (\gamma \neq 0).
\end{cases}
\end{gather*}

Assume that $r', \gamma' \neq 0$. Then $\red (C(x,y)^{p'}) \equiv x^{r_1} y^{\gamma_1} x^{r_2}y^{\gamma_2}*\cdots $ $(r_1,r_2,\gamma_1,\gamma_2 \neq 0)$, so
the right side of {\bf [T]} is
\[
\red (A(x,C(y,z)) \equiv x^{p}y^{r_1} z^{\gamma_1} y^{r_2}z^{\gamma_2}*\cdots.
\]
On the other hand, by Lemma \ref{lemma:key}, the left side of \textbf{[T]} is
\[
 \red (A(A(x,y),C(B(x,y),z))) = \red( A(x,y)^{p}B(x,y)^{r_{1}}) z^{\gamma_1} \red(B(x,y)^{r_{2}}) x^{\gamma_2}* \cdots,
\]
so $B(x,y)^{r_{2}} = y^{r_{2}}$. Since $r_{2} \neq 0$, this implies $B(x,y) \equiv y$, which contradicts (\ref{eqn:hypo}). Therefore $\gamma'$ should be zero, so $C \equiv a^{r}b^{\gamma}a^{r'}$. By considering the swap-dual we conclude $B \equiv b^{q}a^{\beta}b^{q'}$.

To show (2), assume contrary that $qq'rr' \neq 0$. By Lemma \ref{lemma:key}, the right side of \textbf{[M]} is given by
\[
\red ( B(A(x,y),C(B(x,y),z) )
\equiv \left\{
\begin{array}{ll}
\red( B(x,y)^{r}) z^{\gamma} *\cdots & (q>0) \\
\red( B(x,y)^{-r'}) z^{-\gamma} *\cdots & (q<0).\\
\end{array}
\right.
\]
For a non-zero integer $M$, $\red( B(x,y)^{M}) \equiv y^{Q}x^{\pm l}*\cdots$, where $Q= q$ or $-q'$, so the right side of \textbf{[M]} is written as
\[
\red ( B(A(x,y),C(B(x,y),z) ) \equiv y^{Q}x^{\pm 1}*\cdots \;\;\;\; (Q = q,-q')
\]

On the other hand, by Lemma \ref{lemma:key}, the left side of \textbf{[M]} is of the form
\[
\red C ( B(x, C(y,z) ), D(y,z) )
\equiv \left\{
\begin{array}{ll}
\red( C(y,z)^{q}) x^{\pm1} *\cdots & (r>0) \\
\red( C(y,z)^{-q'}) x^{\pm1} *\cdots & (r<0)\\
\end{array}
\right.
\]
hence
\[ y^{Q} = C(y,z)^{Q'} \ \ \ (Q,Q' \in \{q,-q'\}). \]
Since $q,q'\neq 0$, this shows that $C(y,z)$ contains no $z^{\pm 1}$, which contradicts (\ref{eqn:hypo}).
\end{proof}

Now we are ready to complete the proof of Proposition \ref{prop:caseIII}.

\begin{proof}[Proof of Proposition \ref{prop:caseIII}]
By Lemma \ref{lem:restriction}, we already know $B \equiv b^{q}a^{\beta} b^{q'}$ and $C \equiv a^{r}b^{\gamma} a^{r'}$ $(q,q',r,r' \in \Z, \ \beta,\gamma \in \{\pm 1\})$. Since $qq'rr'=0$, by taking the swap and backward duals of $\Theta$ if necessary, we may assume that $q'=0$, namely, $B\equiv b^{q}a^{\beta}$.\\

{\bf Case 1: $q=0$.}\\

In this case, $B\equiv a^{\beta}$ $(\beta = \pm 1)$.
By \textbf{[B]},
\begin{equation}
\label{eqn:III-D}
 D(x^{\beta},z)= D (x^{\beta},D(y,z)).
\end{equation}

Assume that $D$ contains $b^{\pm 1}$.
Then the right side of (\ref{eqn:III-D}) contains at least one $y^{\pm 1}$, but the left side does not, contradiction. Thus $D$ cannot contain $b^{\pm 1}$, so by \textbf{[Aut]}, $D \equiv a^{\delta}$ $(\delta= \pm 1)$.

Now the equation \textbf{[M]} is written as
\begin{equation}
\label{eqn:III-A}
 A(x,y)^{\beta} = x^{\beta r}y^{\gamma\delta}x^{ \beta r'}.
\end{equation}

Now we study the following four cases according to the value $\beta$ and $\gamma\delta$.\\

{\it Subcase 1-1: $\beta=+1$ and $\gamma\delta=+1$.}\\

By (\ref{eqn:III-A}), $A(x,y) \equiv x^{r}yx^{r'}$.
The equation \textbf{[T]} is written as
\[ (x^{r}yx^{r'})^{r}x^{r}z^{\gamma}x^{r'}  (x^{r}yx^{r'})^{r'} = x^{r}y^{r}z^{\gamma}y^{r'}x^{r'}. \]
This is satisfied if and only if the following two equalities hold.
\begin{gather*}
\begin{cases}
(x^{r}yx^{r'})^{r}x^{r} = x^{r}y^{r}\\
x^{r'} (x^{r}yx^{r'})^{r'} = y^{r'}x^{r'}
\end{cases}
\end{gather*}

These equations are written as
\begin{gather*}
\begin{cases}
(x^{r}yx^{r'})^{r} = (x^{r}yx^{-r})^{r}\\
(x^{r}yx^{r'})^{r'} = (x^{-r'}yx^{r'})^{r'},
\end{cases}
\end{gather*}
so \textbf{[T]} is satisfied if and only if $r'=-r$, and we conclude
\begin{enumerate}
\item[($A_{r}-1$)]  $\;\;\;(A,B,C,D) = (a^{r}ba^{-r}, a , a^{r}ba^{-r},a) \;\;\; (r \in \Z_{\geq 0}) $.
\item[($A_{r}-2$)]  $\;\;\;(A,B,C,D) = (a^{r}ba^{-r}, a , a^{r}b^{-1}a^{-r},a^{-1}) \;\;\; (r \in \Z_{\geq 0})$.
\end{enumerate}
Here we remark that $(A_{-r}-1)=\overleftarrow{(A_{r}-1)}$ and $(A_{-r}-2)=\overleftarrow{(A_{r}-2)}$ so up to natural symmetry we can take $r \geq 0$.\\

{\it Subcase 1-2: $\beta=+1$ and $\gamma\delta=-1$.}\\

By (\ref{eqn:III-A}), $A(x,y) \equiv x^{r}y^{-1}x^{r'}$ so \textbf{[T]} is written as
\[ (x^{r}y^{-1}x^{r'})^{r}x^{-r'}z^{-\gamma}x^{-r} (x^{r}y^{-1}x^{r'})^{r'} = x^{r}y^{-r'}z^{-\gamma}y^{-r}x^{r'}. \]
Consequently we get two equations
\begin{gather*}
\begin{cases}
(x^{r}y^{-1}x^{r'})^{r}x^{-r'} = x^{r}y^{-r'}\\
x^{-r} (x^{r}y^{-1}x^{r'})^{r'} = y^{-r}x^{r'}
\end{cases}
\end{gather*}

The first equation is written as
\begin{equation}
\label{eqn:III-1-2}
(x^{r}y^{-1}x^{r'})^{r} = x^{r}y^{-r'}x^{r'}.
\end{equation}

If $r'=-r$, then (\ref{eqn:III-1-2}) is written as
\[ (x^{r}y^{-1}x^{-r})^{r} = (x^{r}y^{-1}x^{-r})^{-r}. \]
This shows $-r=r$, so $r=r'=0$.

If $r \neq -r'$, then the right side of (\ref{eqn:III-1-2}) is primitive, so $|r|\leq 1$. By direct calculations (\ref{eqn:III-1-2}) is satisfied only if $(r,r')=(1,1)$ or $(r,r')=(0,0)$.
In the case $(r,r')=(0,0)$ we get
\begin{enumerate}
\item[($B-1$)] $\;\;\; (A,B,C,D) = (b^{-1}, a , b ^{-1}, a)$.
\item[($B-2$)] $\;\;\;(A,B,C,D) = (b^{-1}, a , b, a^{-1})$.
\end{enumerate}
and in the case $(r,r')=(1,1)$ we get
\begin{enumerate}
\item[($C-1$)] $\;\;\; (A,B,C,D) = (ab^{-1}a,a , ab^{-1}a, a)$.
\item[($C-2$)] $\;\;\; (A,B,C,D) = (ab^{-1}a,a , aba, a^{-1})$. \\
\end{enumerate}

{\it Subcase 1-3: $\beta = -1$ and $\delta\gamma=+1$.}\\

By (\ref{eqn:III-A}), $A(x,y) \equiv x^{r'}y^{-1}x^{r}$ so \textbf{[T]} is written as
\[ (x^{r'}y^{-1}x^{r})^{r'}x^{r'}z^{-\gamma}x^{r} (x^{r'}y^{-1}x^{r})^{r} = x^{r'}y^{-r'}z^{-\gamma}y^{-r}x^{r}. \]
Consequently we get two equations
\begin{gather*}
\begin{cases}
(x^{r'}y^{-1}x^{r})^{r'}x^{r'} = x^{r'}y^{-r'}\\
x^{r} (x^{r'}y^{-1}x^{r})^{r} = y^{-r}x^{r}.
\end{cases}
\end{gather*}

As in the Subcase 1-1, these two equations are written as
\begin{gather*}
\begin{cases}
(x^{r'}y^{-1}x^{r})^{r'} = (x^{r'}y^{-1}x^{-r'})^{r'}\\
(x^{r'}y^{-1}x^{r})^{r} = (x^{-r}y^{-1}x^{r})^{r}.
\end{cases}
\end{gather*}
This shows $r=-r'$ and we get
\begin{enumerate}
\item [$(A_{r}-2)^{-\sigma}$] $\;\;\; (A,B,C,D) = (a^{-r}b^{-1}a^{r},a^{-1},a^{r}ba^{-r},a).$
\item [$(A_{r}-3)$] $\;\;\; (A,B,C,D) = (a^{-r}b^{-1}a^{r},a^{-1},a^{r}b^{-1}a^{-r},a^{-1}). $ \\
\end{enumerate}

{\it Subcase 1-4: $\beta=-1$, $\gamma\delta=-1$.}\\

By (\ref{eqn:III-A}), $A(x,y) \equiv x^{r'}yx^{r}$ so the equation \textbf{[T]} is written as
\[ (x^{r'}yx^{r})^{r'}x^{-r}z^{\gamma}x^{-r'} (x^{r'}yx^{r})^{r} = x^{r'}y^{r}z^{\gamma}y^{r'}x^{r}.\]
Consequently we get two equations
\begin{gather*}
\begin{cases}
(x^{r'}yx^{r})^{r'}x^{-r} = x^{r'}y^{r}\\
x^{-r'} (x^{r'}yx^{r})^{r} = y^{r'}x^{r}
\end{cases}
\end{gather*}

The first equation is written as
\begin{equation}
\label{eqn:III-1-4}
(x^{r'}yx^{r})^{r'} = x^{r'}y^{r}x^{r}
\end{equation}

If $r'=-r$, then (\ref{eqn:III-1-4}) is written as
\[ (x^{-r}yx^{r})^{-r} = (x^{-r}yx^{-r})^{r}. \]
Thus $-r=r$, so we conclude $r=r'=0$.
If $r \neq -r'$, then the right side of (\ref{eqn:III-1-4}) is primitive so $|r|\leq 1$. By direct calculation, (\ref{eqn:III-1-4}) is satisfied only if $(r,r')=(1,1)$ or $(r,r')=(0,0)$.

If $(r,r')=(0,0)$ then we get
\begin{itemize}
\item[$(B-2)^{-}$] $(A,B,C,D) = (b,a^{-1},b^{-1},a)$.
\item[$(B-1)^{\sigma}$] $(A,B,C,D) = (b,a^{-1},b,a^{-1})$.
\end{itemize}
and if $(r,r')=(1,1)$ and $(c,d)=(-1,1)$, then we get
\begin{enumerate}
\item[$(C-2)^{-\sigma}$] $(A,B,C,D) = (aba, a^{-1},ab^{-1}a, a)$.
\item[$(C-3)$] $ (A,B,C,D) = (aba, a^{-1},aba, a^{-1})$.\\
\end{enumerate}

{\bf Case 2: $q \neq 0$.}\\

Let us put $B \equiv y^{q}x^{\beta}$ and $D \equiv b^{s}a^{\delta}b^{s'}a^{\delta'}b^{s''}a^{\delta''}*\cdots$ $(\delta \neq 0)$.
\begin{claim}
$\delta,\delta',\delta'',\ldots \in \{\pm 1,0\}$.
\end{claim}
\begin{proof}[Proof of Claim]
Assume that $|\delta|>1$. Then the left side of {\bf [B]} is
\begin{equation}
\label{eqn:III2-Bl}
\red( D(B(x,y),z))
 \equiv \left\{ \begin{array}{ll}
z^{s}y^{q}x^{\delta}y^{q}x^{\delta}*\cdots & (\delta>1) \\
z^{s}x^{-\delta}y^{-q}x^{-\delta}*\cdots & (\delta<-1). \\
\end{array}
\right.
\end{equation}

On the other hand by Lemma \ref{lemma:key}, the right side of \textbf{[B]} is \begin{equation}
\label{eqn:III2-Br}
\red( D(B(x,C(y,z)),D(y,z)))
\equiv
\left\{ \begin{array}{ll}
\red(z^{s}C(y,z)^{q})x^{\delta}\red( C(y,z)^{q}) x^{\delta}*\cdots & (\delta>1) \\
z^{s} x^{-\delta} \red( C(y,z)^{-q})x^{-\delta}*\cdots & (\delta<-1). \\
\end{array}
\right.
\end{equation}

Hence by \textbf{[B]} $y^{q}=C(y,z)^{q}$.
 Since we have assumed $q\neq 0$, this implies $C(y,z) \equiv y$, which contradicts (\ref{eqn:hypo}). This proves $\delta= \pm 1$.
Similar arguments show $\delta',\delta'',\ldots \in \{\pm 1,0\}$.
\end{proof}

Then we consider the following two cases according to the value of $\delta$.\\

{\it Subcase 2-1: $\delta=+1$}\\

\begin{claim}
\label{claim:formD}
If $\delta=1$, then $D \equiv b^{s}a$ $(s\neq 0)$.
\end{claim}
\begin{proof}[Proof of Claim]

If $\delta=+1$, then by (\ref{eqn:III2-Bl}) and (\ref{eqn:III2-Br}) we get
\begin{equation}
\label{eqn:c-2-1} z^{s}y^{\delta} = D(y,z)^{s}C(y,z)^{\delta}.
\end{equation}

If $s=0$, then the equation (\ref{eqn:c-2-1}) gives $y^{q}=C(y,z)^{q}$ $(q\neq 0)$, contradictions (\ref{eqn:hypo}), hence $s\neq 0$.

Assume that $\delta'=-1$. This implies, in particular, $s'\neq 0$. Then the right side of \textbf{[B]} is written as
\[ \red( D(B(x,y),z)) \equiv z^{s}y^{q}x^{\beta}z^{s'}x^{-\beta}*\cdots \]
and the left side of equation \textbf{[B]} is
\[ \red( D(B(x,C(y,z)),D(y,z)))
\equiv
 \red (D(y,z)^{s}C(y,z)^{q}) x^{\beta}\red (D(y,z)^{s'}) x^{-\beta}* \cdots \]
so we get
\[ D(y,z)^{s'} = z^{s'}. \]
Since $s'\neq 0$, this implies $D(y,z)= z$, which contradicts (\ref{eqn:hypo}).

Similarly, if $\delta'=0$, then the equation \textbf{[B]} is written as
\[ z^{s}y^{q}x^{\beta} z^{s'} =  D(y,z)^{s}C(y,z)^{q}x^{\beta}D(y,z)^{s'} \]
so we get
\[ D(y,z)^{s'} = z^{s'}, \]
which leads to a contradiction unless $s'=0$, so we have $D \equiv b^{s}a$ as desired.

Finally assume that $\delta'=+1$. Then $s' \neq 0$, and equation \textbf{[B]} is written as
\[ z^{s}y^{q}x^{\beta} z^{s'}y^{q}x^{\beta} *\cdots =D(y,z)^{s}C(y,z)^{q}x^{\beta}D(y,z)^{s'}C(y,z)^{q}x^{\beta} *\cdots. \]
Hence we get
\[ z^{s'}y^{q} = D(y,z)^{s'}C(y,z)^{q}.\]

By applying this equality to equation (\ref{eqn:c-2-1}), we get \[ D(y,z)^{s-s'}=z^{s-s'}.\]
If $s-s' \neq 0$, then $D(y,z) \equiv z$, which contradicts (\ref{eqn:hypo}) hence $s=s'$.

By iterating the above arguments for $\delta'',\delta''',\ldots$, we conclude that the possible form of the word $D(x,y)$ is $D(x,y)=y^{s}xy^{s}x \cdots y^{s}x=(y^{s}x)^{N}$.
By \textbf{[Aut]} $D(y,z)$ must be a primitive element, so $N=1$. This complete the proof of claim.
\end{proof}

By Claim \ref{claim:formD}, the equation (\ref{eqn:c-2-1}) is now in a simple form,
\begin{equation}
\label{eqn:c-2-1fin}
z^{s}y^{q} = (z^{s}y)^{s}(y^{r}z^{\gamma}y^{r'})^{q}.
\end{equation}
This equation can be solved directly.

\begin{claim}
\label{claim:solution}
The equation (\ref{eqn:c-2-1fin}) is satisfied only if  $(q,r,\gamma,r',s)=(2,-1,-1,1,2)$.
\end{claim}
\begin{proof}[Proof of Claim]
We have seen that $C \equiv a^{r}b^{\gamma}a^{r'}$ and $D \equiv b^{s}a$.

First we consider the case $r=-r'$. Then the equation (\ref{eqn:c-2-1fin}) is written as
\begin{equation}
\label{eqn:claim1} z^{s}y^{q} = (z^{s}y)^{s}y^{r}z^{\gamma q}y^{-r}
\end{equation}
If $r=0$, then $z^{s}y^{q} = (z^{s}y)^{s} z^{\gamma q}$. However, since $\gamma ,q \neq 0$ this is impossible so $r\neq 0$. Then we conclude $s>0$ and $r=-1$ because otherwise the right side of (\ref{eqn:claim1}) is already reduced so (\ref{eqn:claim1}) cannot be satisfied.
If $s=1$, then we get $z^{s}y^{q}=z^{s+ \gamma q}y^{-r}$. However, $\gamma,q \neq 0$ this is impossible.
If $s=2$, we get a solution $(q,r,\gamma,r',s)=(2,-1,-1,1,2)$.
Finally, if $s>2$ then $z^{s}y^{q}=(z^{s}y)^{s-2}z^{s}yz^{s+\gamma q}y$, it is impossible.

To complete the proof of claim, we show that there are no $(q,r,\gamma,r',s)$ satisfying (\ref{eqn:c-2-1fin}) if $r\neq -r'$. We prove this for the case $s>0,q>0$. The other cases are treated in a similar way.

In the case $s>0,q>0$ the equation (\ref{eqn:c-2-1fin}) is written as
\begin{equation}
\label{eqn:c-2-1claim} z^{s}y^{q} = (z^{s}y)^{s-1}z^{s}y^{r+1}z^{\gamma}y^{r'}(y^{r}z^{\gamma}y^{r'})^{q-1}.
\end{equation}
Since $r \neq -r'$, to satisfy (\ref{eqn:c-2-1claim}), we need $r =-1$ because otherwise the right side of (\ref{eqn:c-2-1claim}) is reduced. Hence we have
\[ z^{s}y^{q} = (z^{s}y)^{s-1}z^{s+\gamma}y^{r'}(y^{-1}z^{\gamma}y^{r'})^{q-1}\]

By \textbf{[Aut]}, $s$ and $\gamma$ should be coprime so $s + \gamma \neq 0$ if $|s| >1$. Hence if $s>1$, then the right side cannot be reduced further, so this equation cannot be satisfied.
Thus $s=1$, and we finally get
\[ zy^{q} = z^{1+\gamma}y^{r'}(y^{-1}z^{\gamma}y^{r'})^{q-1}. \]
If $q>1$, then the right side cannot be reduced further, and if $q=1$ we get $r'=+1$, which contradicts our assumption that $r \neq -r'$ because we have seen $r=-1$.
\end{proof}

By Claim \ref{claim:solution}, $B \equiv b^{2}a^{\beta}$, $C \equiv a^{-1}ba$ and $D\equiv b^{2}a$ ($\beta \in \{\pm 1 \}$). By \textbf{[M]}, $A^{\beta} = x^{\beta} y^{-1}x^{\beta}$, so we conclude
\begin{enumerate}
\item[$(D-1)$] $(A,B,C,D) = ( a^{-1}b^{-1}a, b^{2}a, a^{-1}b^{-1}a, b^{2}a)$.
\item[($D-2$)] $(A,B,C,D) = (aba^{-1},b^{2}a^{-1}, a^{-1}b^{-1}a, b^{2}a)$.\\
\end{enumerate}

{\it Subcase 2-2: $\delta=-1$.}\\

 In this case the equation \textbf{[B]} is given by
\[ z^{s}x^{-\beta}y^{-q} z^{s'}(y^{q}x^{\beta})^{\delta'} \cdots =D(y,z)^{s}x^{-\beta}C(y,z)^{-q}D(y,z)^{s'}(C(y,z)^{q}x^{\beta})^{\delta'} \cdots \]
 hence we get
\[ z^{s} = D(y,z)^{s}\]
If $s \neq 0$, this shows that $D(y,z) \equiv z$, which contradicts (\ref{eqn:hypo}). Hence $s=0$ and $D(x,y)$ is
\[ D(x,y) \equiv x^{-1}y^{s'}x^{\delta'}y^{s''} \cdots. \]
We consider the following three cases according to the value $\delta'$.\\

{\it Subcase 2-2-1: $\delta'=+1$.}\\

In this case $s' \neq 0$ and the equation \textbf{[B]} is
\[ x^{-\beta}y^{-q} z^{s'} y^{q}x^{\beta} z^{s''} (y^{q}x^{\beta})^{\delta''} \cdots =x^{-\beta}C(y,z)^{-q}D(y,z)^{s'}C(y,z)^{q}x^{\beta}D(y,z)^{s''} (C(y,z)^{q}x^{\beta})^{\delta''}\cdots \]
so we have
\begin{equation}
\label{eqn:c-2-2-1} y^{-q}z^{s'}y^{q} = C(y,z)^{-q}D(y,z)^{s'}C(y,z)^{q}.
\end{equation}
Moreover, if $s'' \neq 0$, then we further get
\begin{gather}
\label{eqn:c-2-2-11}
\begin{cases}
z^{s''} = D(y,z)^{s''} & (\delta'' = -1,0)\\
z^{s''}y^{q} = D(y,z)^{s''}C(y,z)^{q} & (\delta'' = +1)\\
\end{cases}
\end{gather}
However, (\ref{eqn:c-2-2-1}) and (\ref{eqn:c-2-2-11}) show $D(y,z)=z$, which contradicts (\ref{eqn:hypo}). Hence $s''=\delta''=\cdots = 0$ and $D\equiv a^{-1}b^{s'}$.
Then (\ref{eqn:c-2-2-1}) is written as
\[
y^{-q}z^{s'}y^{q} =
(y^{r}z^{\gamma}y^{r'})^{-q} (y^{-1}z^{s'})^{s'} (y^{r}z^{\gamma}y^{r'})^{q}
\]
By comparing the exponent of $y$, we conclude $s'=0$. However, since we have assumed $s' \neq 0$, this is a contradiction. \\

{\it Subcase 2-2-2: $\delta'=-1$. }\\

In this case $s' \neq 0$ and the equation \textbf{[B]} is
\[ x^{-\beta}y^{-q}z^{s'}x^{-\beta}y^{-q}z^{s''}(y^{q}x^{\beta})^{\delta''} \cdots =x^{-\beta}C(y,z)^{-q}D(y,z)^{s'}x^{-\beta}C(y,z)^{-q}D(y,z)^{s''} \cdots,\]
so we get
\begin{equation}
\label{eqn:c-2-2-2}
y^{-q}z^{s'} = C(y,z)^{-q}D(y,z)^{s'}.
\end{equation}
Then, by using a similar argument as in SubCase 2-1 (Claim \ref{claim:formD}), we conclude that $\delta''=0$ and $s'=s''$. Thus, $D \equiv a^{-1}b^{s'}a^{-1}b^{s'}$. However, by {\bf [Aut]} $D$ is primitive, contradiction.\\

{\it Subcase 2-2-3: $\delta'=0$.}\\

By \textbf{[B]}, we get equation
\begin{equation}
\label{eqn:solution'} y^{-q}z^{s'} = (y^{r}z^{\gamma}y^{r'})^{-q}(y^{-1}z^{s'})^{s'}.
\end{equation}

By a similar argument as Claim \ref{claim:solution}, (\ref{eqn:solution'}) is satisfied only if $(q,r,\gamma,r',s') = (2,-1,1,1,2)$ so $C\equiv a^{-1}ba$ and $D\equiv a^{-1}b^{2}$. By \textbf{[M]}, $A(x,y)^{\beta}=x^{-\beta}y^{-1}x^{\beta}$, hence we conclude
\begin{enumerate}
\item[$(D-3)$] $(A,B,C,D) = (a^{-1}ba, b^{2}a, a^{-1}ba, a^{-1}b^{2})$.
\item[$(D-4)$] $(A,B,C,D) = (ab^{-1}a^{-1}, b^{2}a^{-1}, a^{-1}ba, a^{-1}b^{2})$.
\end{enumerate}
\end{proof}

\section{Group-valued invariants of knots and links}
\label{sec:invariants}

We close the paper by giving a short discussion on group-valued invariants of knots and links from local $\Aut(F_{n})$ representations.

\begin{definition}
We say a local $\Aut(F_{n})$ braid representation $\Theta$ defined by $(\tau_{1},\ldots, \tau_{n-1}) \in \Aut(F_{2})^{n-1}$ has the {\em stabilization properties} if the following two conditions are satisfied:
\begin{description}
\item[S1] The group $G_{i} = \langle a,b \: | \: (b)\tau_{i} = b \rangle$ is an infinite cyclic group generated by $a=b$.
\item[S2] For any $m>n$, there exist a local $\Aut(F_{m})$ representation $\Theta^{*}$ which extends $\Theta$: That is, there is a local $\Aut(F_{m})$ representation $\Theta^{*}$ defined by $(\tau_{1},\ldots, \tau_{n-1}, \tau_{n},\ldots, \tau_{m-1})$.
\end{description}
\end{definition}

Let $L$ be an oriented link in $S^{3}$ represented as a closure of an $n$-braid $\beta$, and let $\Theta$ be a local $\Aut(F_{n})$ representation having the stabilization properties. Let $G_{\Theta}(\beta)$ be a group given by the presentation
\[ G_{\Theta}(\beta) = \left\langle x_{1},\ldots,x_{n} \: | \: (x_{i})[\Theta(\beta)] = x_{i} \;\;\; (i=1,\ldots, n) \right\rangle. \]

\begin{proposition}[Group-valued invariant of links]
\label{prop:ginvariant}
The group $G_{\Theta}(\beta)$ does not depend on a choice of closed braid representatives of $L$, hence $G_{\Theta}(\beta)$ defines a group-valued invariant of $L$.
\end{proposition}
\begin{proof}
By Markov theorem (see \cite{b}, for example) the closures of two braids $\alpha$ and $\beta$ represent the same oriented link if and only if $\alpha$ is converted to $\beta$ by applying following two operations.
\begin{description}
\item[Conjugation] $\alpha \rightarrow \gamma^{-1}\alpha\gamma$ where $\alpha,\gamma \in B_{n}$.
\item[(De)Stabilization] $\alpha \leftrightarrow \alpha\sigma_{n}^{\pm 1}$ where $\alpha \in B_{n}$.
\end{description}

Assume that $\Theta$ be a local $\Aut(F_{n})$ representation defined by $(\tau_{1},\ldots,\tau_{n-1}) \in \Aut(F_{2})^{n-1}$.
By definition, the group $G_{\Theta}(\beta)$ does not change under the conjugacy, so it is sufficient to show the invariance under the stabilization.

Take an arbitrary local $\Aut(F_{n+1})$ representation $\Theta^{*}$ which extends $\Theta$. That is, we assume that $\Theta^{*}$ is defined by $(\tau_{1},\ldots,\tau_{n-1},\tau_{n}) \in \Aut(F_{2})^{n}$ for some $\tau_{n} \in \Aut(F_{2})$. Such $\Theta^{*}$ exists from the assumption {\bf S2}. Then
\[ [\Theta^{*}(\beta \sigma_{n}^{\pm 1})] (x_{i}) =
\left\{
\begin{array}{ll}
(x_{i})[\Theta(\beta)] & i \neq n,n+1, \\
(x_{n})[\Theta(\beta) \tau_{n}^{\pm 1}] & i=n, \\
(x_{n+1})\tau_{n}^{\pm 1} & i=n+1.
\end{array}
\right.
\]
By {\bf S1}, the relation $(x_{n+1})\tau_{n}^{\pm 1} = x_{n+1}$ implies $(x_{n})\tau_{n}^{\pm 1} = x_{n} = x_{n+1}$, hence two groups $G_{\Theta}(\beta)$ and $G_{\Theta^{*}}(\beta\sigma_{n}^{\pm 1})$ are isomorphic.
\end{proof}

A priori, it is not clear whether there is a local $\Aut(F_{n})$-representation having stabilization properties. However, the classification theorem shows almost all local $\Aut(F_{n})$-representations satisfy the stabilization properties.
\begin{corollary}
If $\Theta$ is a local $\Aut(F_{n})$-representation which is not of type $(T)$ or $(T')$, then $\Theta$ satisfies the stabilization properties.
\end{corollary}

To take into account of a choice of local $\Aut(F_{n})$ representations and closed braid representative, it is convenient to formulate the group-valued invariants as follows.

\begin{definition}[Group-valued invariants]
Let $L$ be an oriented link in $S^{3}$ and $\wTheta$ be a non-trivial local $\Aut(F_{\infty})$-representation defined by a sequence $(\tau_{1},\tau_{2},\ldots,)$. The \emph{group-valued invariant} $G_{\wTheta}(L)$ is a group $G_{\wTheta}(L) = G_{\Theta_{n}}(\beta)$
where $\beta$ is an $n$-braid whose closure is $L$ and $\Theta_{n}$ is a local $\Aut(F_{n})$-representation defined by $(\tau_{1},\ldots,\tau_{n})$.
\end{definition}

As we have mentioned in introduction, at first glance we have obtained many new group-valued invariants. However, classification theorem also tells us that group-valued invariants for certain local $\Aut(F_{n})$-representations are nothing new.

\begin{theorem}
\label{theorem:ginvariant}
Two local $\Aut(F_{\infty})$ representations defines the same invariant if they are of the same type. In particular, a group-valued invariant from local $\Aut(F_{\infty})$ representations coincides with the group-valued invariant from the corresponding Wada-type representation.
\end{theorem}
\begin{proof}
Assume that $\wTheta$ and $\wTheta'$ are of the same type defined by $(\tau_{1},\ldots)$ and $(\tau'_1,\ldots)$, respectively.
Represent $L$ a closure of $n$-braid $\beta$, and let $\Theta_{n}, \Theta'_{n}$ be the  local $\Aut(F_{n})$-representations defined by $(\tau_{1},\ldots,\tau_{n-1})$ and  $(\tau'_{1},\ldots,\tau'_{n-1})$, respectively.
From the Classification theorem of local $\Aut(F_{n})$-representation, by taking sufficiently large $N$, there is a local $\Aut(F_{N})$-representations $\Phi$ defined by a sequence
\[ (\underbrace{\tau_1,\ldots,\tau_{n-1}}_{n-1},\ldots,\underbrace{\tau'_{1},\ldots,\tau'_{n-1}}_{n-1}) \in \Aut(F_{2})^{N}. \]
Let $\textsf{sh}:B_{n}\rightarrow B_{N}$ be the homomorphism defined by $\textsf{sh}(\sigma_{i}) = \sigma_{i+(N-n)}$.
The closures of $N$-braids $(\beta \sigma_{n}\sigma_{n+1}\cdots\sigma_{N-1})$ and $\sigma_{1}\cdots \sigma_{N-n}\textsf{sh}(\beta)$ are $L$, hence
\[
G_{\wTheta}(L) \cong G_{\Phi}(\beta \sigma_{n}\sigma_{n+1}\cdots\sigma_{N-1}) \cong G_{\Phi}(\sigma_{1}\cdots \sigma_{N-n}\textsf{sh}(\beta)) \cong G_{\wTheta'}(L).
\]
\end{proof}

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