In this section we are going to assign to each line $l\in{\cal L}$ a symplectomorphism

$$\varphi_{l}\in\Gamma\left((0,+\infty),f_{l}^{*}\left(Symp\right)\right)$$

giving for each $t>0$ a transformation $\varphi_{l}(t):{\cal O}^{can}_{Y,f_{l}(t)}\to{\cal O}^{can}_{Y,f_{l}(t)}$. This symplectomorphism in local coordinates will belong to the subgroup $G_{\lambda}$ where $\lambda$ is the slope of $\alpha_{l}(t)$. More precisely, we demand that $\varphi_{l}(t)$ is of the form

$$\varphi_{l}(t)=\exp\{F_{l,t}(\xi^{-a}\eta^{-b}),\cdot\}\,\,,$$

where $\alpha_{l}(t)=adx+bdy$, operation $\{\cdot,\cdot\}$ is the Poisson bracket on ${\cal O}^{can}_{Y,f_{l}(t)}$ and $F_{l,t}(z)\in zK[[z]]$ is an analytic function of one variable satisfying the following condition. Let us consider the pullback (by the exponential map) of the function $F_{l,t}(\xi^{-a}\eta^{-b})$ to a section of the sheaf ${\cal O}^{can}$ on vector space $T_{f_{l}(t)}Y\simeq{\bf R}^{2}$ considered as a manifold with ${\bf Z}$-affine structure. Then this pullback should admit an analytic continuation from $0\in T_{f_{l}(t)}$ to the half-plane $P_{l,t}$, and obey there the bound

$$|F_{l,t}(\xi^{-a}\eta^{-b})|\leq\exp(-ord_{l}(t))\,\,.$$

Let us explain the construction of $\varphi_{l}(t)$, leaving the justification for the next sections.

Symplectomorphisms $\varphi_{l}$ are constructed by an inductive procedure. Let $l=l_{+}\in{\cal L}_{in}$ be (in standard affine coordinates) a line in the half-plane $y>0$ emerging from $(0,0)$ (there is another such line $l_{-}$ in the half-plane $y<0$). Assume that $t$ is sufficiently small. Then we define $\varphi_{l}(t)\in Symp_{f_{l}(t)}$ on topological generators $\xi,\eta$ by the formula (as in Section 8)

$$\varphi_{l}(t)(\xi,\eta)=(\xi(1+1/\eta),\eta)\,\,.$$

Notice that $\varphi_{l}(t)=\exp\{F(\eta^{-1}),\cdot\}$, where $F(z)=\sum_{n>0}(-1)^{n}z^{n}/n^{2}$ is convergent for $|z|<1$.

In order to extend $\varphi_{l}(t)$ to the interval $(0,t_{0})$, where $t_{0}$ is not small, we cover the corresponding segment of $l$ by open charts. Notice that change of affine coordinates transforms $\eta$ into a monomial multiplied by a constant from $K^{\times}$. Therefore $\eta$ extends analytically in a unique way to a global section over $(0,+\infty)$ of the sheaf $f_{l}^{\ast}(({\cal O}^{can})^{\times})$. Moreover, the norm $|\eta|$ strictly decreases as $t$ increases, and remains strictly smaller than $1$. Hence $F(\eta)$ can be canonically extended for all $t>0$.

Each symplectomorphism $\varphi_{l}(t)$ is defined by a series which converges in the half-plane $P_{l,t}$. Using the exponential map associated with the affine structure as well as estimates of $ord_{l}(t)$, we can extend analytically $\varphi_{l}(t)$ into a neighborhood of $f_{l}(t)$.

Let us now assume that $l_{1}$ and $l_{2}$ collide at $p=f_{l_{1}}(t_{1})=f_{l_{2}}(t_{2})$, generating the line $l\in{\cal L}_{com}$. Then $\varphi_{l}(0)$ is defined with the help of factorization theorem in the group $G$. More precisely, we set $\alpha_{i}:=\alpha_{l_{i}}(t_{i}),\,\,i=1,2$ and the angle $V$ to be the intersection of half-planes $P_{l_{1},t_{1}}\cap P_{l_{2},t_{2}}$. By construction elements $g_{0}:=\varphi_{l_{1}}(t_{1})$ and $g_{+\infty}:=\varphi_{l_{2}}(t_{2})$ belong respectively to $G_{0}$ and $G_{+\infty}$. Then we can use the factorization Theorem 6 and write down the formula

$$g_{+\infty}g_{0}={\textstyle\prod_{\to}}\left((g_{\lambda})_{\lambda\in[0,+\infty]_{\bf Q}}\right)=g_{0}\dots g_{1/2}\dots g_{1}\dots g_{+\infty}\,\,,$$

where $g_{\lambda}\in G_{\lambda}$ and the product on the right is in the increasing order. There is no clash of notations because it is easy to see that the boundary factors in the decomposition from above are indeed equal to $g_{0}$ and $g_{+\infty}$. Each term $g_{\lambda}$ with $0<\lambda=n_{1}/n_{2}<+\infty$ corresponds to the newborn line $l$ with the direction covector $n_{1}\alpha_{l_{1}}(t_{1})+n_{2}\alpha_{l_{2}}(t_{2})$. Then we set $\varphi_{l}(0):=g_{\lambda}$. This transformation is defined by a series which is convergent in a neighborhood of $p$, and using the analytic continuation as above, we obtain $\varphi_{l}(t)$ for $t>0$. The decomposition identity can be rewritten as

$$g_{0}\dots g_{1/2}\dots g_{1}\dots g_{+\infty}g_{0}^{-1}g_{+\infty}^{-1}=id$$

where each factor corresponds to half-lines at the collision point (see Figure 5), and the meaning of the identity is that the infinite composition of symplectomorphisms in the natural cyclic order on half-lines, is trivial.