Here it will be convenient to work in local coordinates $(x,y)=(\log|\xi|,\log|\eta|)$ on $Y$.

Let $(x_{0},y_{0})\in{\bf R}^{2}$ be a point, $\alpha_{1},\alpha_{2}\in({\bf Z}^{2})^{\ast}$ be $1$-covectors such that $\alpha_{1}\wedge\alpha_{2}>0$. Denote by $V=V_{(x_{0},y_{0}),\alpha_{1},\alpha_{2}}$ the closed angle

$$\{(x,y)\in{\bf R}^{2}|\langle\alpha_{i},(x,y)-(x_{0},y_{0})\rangle\geq 0,i=1,2\,\}\,\,.$$

Let ${\cal O}(V)$ be a $K$-algebra consisting of series $f=\sum_{n,m\in{\bf Z}}c_{n,m}\xi^{n}\eta^{m}$, such that $c_{n,m}\in K$ and for all $(x,y)\in V$ we have:

1. 1.

   if $c_{n,m}\neq 0$ then $\langle(n,m),(x,y)-(x_{0},y_{0})\rangle\leq 0$, where we identified $(n,m)\in{\bf Z}^{2}$ with a covector in $(T_{p}^{\ast}Y)^{{\bf Z}}$;

2. 2.

   $\log|c_{n,m}|+nx+my\to-\infty$ as long as $|n|+|m|\to+\infty$.

Algebra ${\cal O}(V)$ is a Poisson algebra with respect to the bracket $\{\xi,\eta\}=\xi\eta$.

For an integer covector $\mu=adx+bdy\in({\bf Z}^{2})^{*}$ we denote by $R_{\mu}$ the monomial $\xi^{a}\eta^{b}$.

Let us consider a pro-nilpotent Lie algebra ${\bf g}:={\bf g}_{\alpha_{1},\alpha_{2},V}\subset{\cal O}(V)$ consisting of series

$$f=\sum_{n_{1},n_{2}\geq 0,n_{1}+n_{2}>0}c_{n_{1},n_{2}}R_{\alpha_{1}}^{-n_{1}}R_{\alpha_{2}}^{-n_{2}}$$

satisfying the condition

$$\log|c_{n,m}|-n_{1}\langle\alpha_{1},(x,y)\rangle-n_{2}\langle\alpha_{2},(x,y)\rangle\leq 0\,\,\,\,\forall\,(x,y)\in V\,\,.$$

The latter condition is equivalent to $\log|c_{n,m}|-\langle n_{1}\alpha_{1}+n_{2}\alpha_{2},(x_{0},y_{0})\rangle\leq 0$.

Lie algebra ${\bf g}$ admits a filtration by Lie subalgebras ${\bf g}^{\geq k},k\in{\bf Z},k\geq 1$, ${\bf g}={\bf g}^{\geq 1}$, such that ${\bf g}^{\geq k}$ consists of the above series which satisfy the condition $n_{1}+n_{2}\geq k$. Clearly $[{\bf g}^{\geq k_{1}},{\bf g}^{\geq k_{2}}]\subset{\bf g}^{\geq k_{1}+k_{2}}$, and ${\bf g}=\varprojlim_{k\to+\infty}{\bf g}/{\bf g}^{\geq k}$.

Thus, ${\bf g}$ is a topological complete pro-nilpotent Lie algebra over $K$. We denote by $G$ the corresponding pro-nilpotent Lie group $\exp({\bf g})$. It inherits the filtration by normal subgroups $G^{\geq k}$ obtained from the corresponding Lie algebras.

For each $\lambda\in[0,+\infty]_{{\bf Q}}:={\bf Q}_{\geq 0}\cup\infty$ we define a Lie subalgebra

$${\bf g}_{\lambda}=\left\{\sum_{n_{1},n_{2}}c_{m,n}R_{\alpha_{1}}^{-n_{1}}R_{\alpha_{2}}^{-n_{2}}\in{\bf g}\,|\,\,c_{n_{1},n_{2}}\in K,\,\,{n_{2}\over{n_{1}}}=\lambda\,\right\}\,\,.$$

Each ${\bf g}_{\lambda}$ is an abelian Lie algebra. It carries the induced filtration by Lie algebras ${\bf g}_{\lambda}^{\geq k}={\bf g}_{\lambda}\cap{\bf g}^{\geq k}$. Denote by $G_{\lambda}=\exp({\bf g}_{\lambda})$ the corresponding pro-nilpotent group.

Proof. Indeed, for the monomial $R_{\alpha_{1}}^{-n_{1}}R_{\alpha_{2}}^{-n_{2}}\in{\bf g}_{\lambda}$ which maps non-trivially to the quotient ${\bf g}_{\lambda}/{\bf g}^{\geq k}_{\lambda}$ we have: $n_{1}+n_{2}\leq k,n_{1}/n_{2}=\lambda$, where $n_{1},n_{2}$ are non-negative integers. There are finitely many such non-negative integers $n_{1}$ and $n_{2}$. $\blacksquare$

It follows from the Lemma that we have a natural isomorphism of vector spaces $\prod_{\lambda\in[0,+\infty]_{{\bf Q}}}{\bf g}_{\lambda}/{\bf g}_{\lambda}^{\geq k}\to{\bf g}/{\bf g}^{\geq k}$, hence the map

$$(f_{\lambda})_{\lambda\in[0,+\infty]_{{\bf Q}}}\mapsto\sum_{\lambda}f_{\lambda}=\sum_{i=1}^{N_{i}}f_{\lambda_{i}},\,\,\,\,\mbox{ where }f_{\lambda}\in{\bf g}_{\lambda}/{\bf g}_{\lambda}^{\geq k}\,\,\,\forall\lambda\in[0,+\infty]_{{\bf Q}}$$

is well-defined and gives rise (after taking the projective limit as $k\to+\infty$) to the isomorphism ${\bf g}\simeq\prod_{\lambda\in[0,+\infty]_{{\bf Q}}}{\bf g}_{\lambda}$.

In a similar way we define the map $\prod_{\to}:\prod_{\lambda\in[0,+\infty]_{{\bf Q}}}G_{\lambda}\to G$, the product is taken with respect to the natural order on ${\bf Q}$. Namely, for any $k\geq 1$ we define

$${\textstyle\prod_{\to}^{(k)}}:\prod_{i=1}^{N_{k}}G_{\lambda_{i}}/G_{\lambda_{i}}^{\geq k}\to G/G^{\geq k}\,,\,(g_{1},\dots,g_{N_{k}})\mapsto g_{1}\dots g_{N_{k}},\,\mbox{ for }g_{i}\in G_{\lambda_{i}}/G_{\lambda_{i}}^{\geq k}$$

and then set $\prod_{\to}:=\varprojlim_{k}\prod_{\to}^{(k)}\,$.

Proof. Let $k\geq 1$ be an integer. We claim that $\prod_{\to}^{(k)}$ is a bijection of sets (this implies the proposition by taking the projective limit as $k\to+\infty$). We will prove the bijection by induction in $k$ . Case $k=1$ is obvious because all the groups under considerations are trivial.

We would like to prove that $\prod_{\to}^{(k+1)}$ is a bijection assuming that $\prod_{\to}^{(k)}$ is a bijection. Let $h$ be an element of $G/G^{\geq k+1}$ and $\overline{h}$ its image in $G/G^{\geq k}$. By the induction assumption there exist unique $\overline{h}_{i}\in G_{\lambda_{i}}/G_{\lambda_{i}}^{\geq k+1},1\leq i\leq N_{k+1}$ such that $\overline{h}_{1}\dots\overline{h}_{N_{k+1}}=\overline{h}$. Let $h_{i},1\leq i\leq N_{k+1}$ be any liftings of $\overline{h}_{i}$ to $G_{i}/G_{i}^{\geq k}$. Then $h_{1}\dots h_{N_{k+1}}=h\pmod{G^{\geq k}}$, hence $c:=h_{1}\dots h_{N_{k+1}}h^{-1}$ belongs to $G^{\geq k}/G^{\geq k+1}\subset Center(G/G^{\geq k+1})$. The last inclusion holds because $[{\bf g},{\bf g}^{\geq k}]=[{\bf g}^{\geq 1},{\bf g}^{\geq k}]\subset{\bf g}^{\geq k+1}$.

Next we observe that the isomorphism of abelian Lie algebras

$$\bigoplus_{1\leq i\leq{N_{k+1}}}{\bf g}_{\lambda_{i}}^{\geq k}/{\bf g}_{\lambda_{i}}^{\geq k+1}\simeq{\bf g}^{\geq k}/{\bf g}^{\geq k+1}$$

implies an isomorphism of the corresponding abelian groups

$$\prod_{1\leq i\leq{N_{k+1}}}G_{i}^{\geq k}/G_{i}^{\geq k+1}\simeq G^{\geq k}/G^{\geq k+1}\,\,.$$

Hence we can write uniquely $c=c_{1}\dots c_{N_{k+1}}$, where $c_{i}\in G^{\geq k}/G^{\geq k+1}\subset Center(G/G^{\geq k+1})$. It follows that $\prod_{\to}^{(k+1)}\left((h_{i}c_{i}^{-1})\right)=h$. Also it is now clear that this decomposition of $h$ is unique. This concludes the proof. $\blacksquare$

For $l\in{\cal L}$ we will define an order function

$$ord_{l}\in\Gamma((0,+\infty),f_{l}^{\ast}(Aff_{{\bf Z},Y}))$$

(its meaning will become clear later) by the following inductive procedure:

1. 1.

   Let $l\in{\cal L}_{in}$ and $t>0$ be sufficiently small. Then in the standard affine coordinates near $s=f_{l}(0)$ one has $\alpha_{l}=\pm f_{l}^{\ast}(dy)$. We define $ord_{l}=\pm f_{l}^{\ast}(y)$. Then $d(ord_{l})=\alpha_{l}$, and we can extend uniquely $ord_{l}$ for all $t\in(0,+\infty)$.

2. 2.

   Let $l\in{\cal L}_{com}$ and $l_{1},l_{2}$ be parents of $l$. In the notation of Axiom 3 we have $f_{l_{1}}(t_{1})=f_{l_{2}}(t_{2})=f_{l}(0)$ and $\alpha_{l}(0)=n_{1}\alpha_{l_{1}}(t_{1})+n_{2}\alpha_{l_{2}}(t_{2})$. Then we define $ord_{l}(0):=n_{1}ord_{l_{1}}(t_{1})+n_{2}ord_{l_{2}}(t_{2})$. Again, using the condition $d(ord_{l})=\alpha_{l}$ and the knowledge of $ord_{l}(0)$ we can extend $ord_{l}$ for $t>0$.

Notice that $ord_{l}(t)$ can be thought of as affine function on the tangent space $T_{f_{l}(t)}Y$ (in the induced integral affine structure). In particular, we have a half-plane $P_{l,t}\subset T_{f_{l}(t)}Y$ defined by the inequality $ord_{l}(t)>0$. The family of half-planes $P_{l,t}$ is covariantly constant with respect to $\nabla^{aff}$.

Each half-plane $P_{l,t}$ contains $0\in T_{f_{l}(t)}Y$ strictly in its interior. Recall that at the end of Section 9.1 we defined another half-plane $P_{l,t}^{(0)}\subset T_{f_{l}(t)}Y$. It is easy to see that $P_{l,t}^{(0)}$ is the half-plane parallel to $P_{l,t}$ such that $0\in T_{f_{l}(t)}Y$ is on the boundary of $P_{l,t}^{(0)}$.

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.