Denote by $W_{\cal L}:=\cup_{l\in{\cal L}}f_{l}([0,+\infty))$ the set of all points of all lines. It has measure zero. Let $p$ be a point of $Y$. We consider two convex neighborhoods $U\subset U^{\prime}$ of $p$ such that $U$ is relatively compact in $U^{\prime}$.

For any two points $x,y$ belonging to $U\setminus W_{\cal L}$, and a path $\gamma$ joining $x$ and $y$ in $U$, we would like to define an infinite ordered product $i_{x,y}^{\gamma}$ of transformations $\varphi_{L}^{\pm 1}$, where factors correspond to the intersection points of $\gamma$ with all possible pieces $L$ relative to $(U,U^{\prime})$. Factors in the infinite product are ordered according to the time parameter of $\gamma$, the sign corresponds to the mutual position of orientations of $\gamma$ and a piece $L$ at the intersection point.

In order to give a precise meaning to the infinite product the neighborhood $U$ of $p$ should be sufficiently small. Then we will have an analytic continuation of symplectomorphisms $\varphi_{L}$ to $U$, and the convergence of the infinite product. We are also going to prove that the product is independent of the choice of path $\gamma$. In order to achieve these goals it suffices to assume:

C1

for any $l,t$ such $f_{l}(t)\in U$ the set $(\exp_{f_{l}(t)})^{-1}(U)$ is contained in $P_{l,t}\,$;

C2

for any $C\in{\bf R}$ there is only a finite number of pieces $L$ of lines in $U$ such that $\inf_{x\in U}ord_{L}(x)<C$.

Proof. Condition C1 implies that all transformations $\varphi_{L}$ admit an analytic continuation to $U$. Let us introduce a decreasing filtration by positive real numbers $Symp^{\geq r}(U),\,\,r\in{\bf R},r\geq 0$ on group $Symp(U)$ by the formula

$$\left\{g\in Symp(U)\,|\,\,\log|\xi^{\prime}/\xi-1|,\log|\eta^{\prime}/\eta-1|<-r\,\mbox{ where }(\xi^{\prime},\eta^{\prime})=g((\xi,\eta))\,\right\}$$

This is a complete filtration, and condition C2 implies that in any quotient $Symp(U)/Symp^{\geq r}(U)$ only a finite number of elements $\varphi_{L}$ are non-trivial. Therefore we can define the product in the quotient group.

In order to prove independence of $\gamma$, we consider the quotient group $Symp(U)/Symp^{\geq r}(U)$, and the finite $1$-dimensional CW-complex (graph) consisting of finitely many pieces $L$, such that $\varphi_{L}\neq 1$ in the quotient. For each vertex $v$ of the graph there is a natural cyclic order on the edges incident to $v$. The product $\varphi_{v}=\prod_{L}\varphi_{L}^{\pm 1}$ taken in the cyclic order over the set of edges incident to $v$ is equal to $id$ (this follows from the construction of $\varphi_{l}$ via factorizations). Since $U$ is simply-connected, we conclude that the image of $i_{x,y}^{\gamma}$ in $Symp(U)/Symp^{\geq r}(U)$ does not depend on $\gamma$. Using completeness of the filtration we see that $i_{x,y}:=i_{x,y}^{\gamma}$ does not depend on $\gamma$. Proof of the identity $i_{x,y}i_{y,z}=i_{x,z}$ is similar. $\blacksquare$

Proof. Assumption A1 implies that the result near any singular point $s\in B^{sing}\,$, as there are only two lines near $s$. If we are far from $B^{sing}$ then obviously A2 implies C1.

In order to check C2 we prove the following lemma

Proof: We proceed by induction in “complexity of the line”. Let $\delta\in{\bf R}_{>0}$ be the infimum of $ord_{l}(t)(f_{l}(t))$ where $l\in{\cal L}_{in}$ has a collision at time $t$. This number is strictly positive because the number of initial lines is finite, and by A1 there is no collisions at small times. Observe that the value of $ord_{l}(0)$ at the beginning of any composite line $l$ is greater or equal to the sum $ord_{l_{1}}(t_{1})+ord_{l_{2}}(t_{2})$. Therefore the inequality in the lemma implies that the number of collisions is bounded from above by $C/\delta$. Also we have an upper bound on integer coefficients $(n_{1},n_{2})$ in each collision (see Axiom $\bf 3b)$ in Section 9.2). Let us observe that the length of each edge of the ansector tree of $l$ is also bounded from above by $A\,ord_{l}$, for some absolute constant $A>0$. Hence we have only finitely many possibilities for intersections. $\blacksquare$

For point $p\in Y$ which is far from $B^{sing}$ we chose as $U$ a neighborhood of radius $\epsilon^{\prime}\ll\epsilon$ where $\epsilon>0$ is constant from Assumption A2. Then for any point of a line $f_{l}(t)\in U$ we will have the inclusion

$$U\subset\exp_{f_{l}(t)}\left(\frac{1}{2}P_{l,t}\right)\,\,.$$

This implies that $ord_{L}$ in $U$ for the corresponding piece $L$ is bounded below by

$$\frac{1}{2}ord_{l}(t)(f_{l}(t))\,\,.$$

Since (by the last lemma) there exists only a finite number of pieces $L$ intersecting such $U$, we obtain convergence condition C2. $\blacksquare$