a)

A compact oriented surface $B$, a finite subset $B^{sing}\subset B$.

b)

A ${\bf Z}$-affine structure on $Y=B^{sm}=B\setminus B^{sing}$ with the standard singularities near each $b\in B^{sing}$.

c)

A set ${\cal L}$ of lines. With each line $l\in{\cal L}$ there is an associated continuous map $f_{l}:(0,+\infty)\to Y$. We assume that ${\cal L}$ is decomposed into a disjoint union of two subsets ${\cal L}={\cal L}_{in}\sqcup{\cal L}_{com}$. Lines belonging to ${\cal L}_{in}$ are called initial, while those in ${\cal L}_{com}$ are called composite. We assume that for any $l\in{\cal L}$ there exists a continuous extension $f_{l}:[0,+\infty)\to B$ such that $f_{l}(0)\in B^{sing}$ if $l\in{\cal L}_{in}$ and $f_{l}(0)\in Y=B^{sm}$ if $l\in{\cal L}_{com}$.

d)

A collection of covariantly constant nowhere vanishing integer-valued $1$-forms $\alpha_{l}\in\Gamma((0,+\infty),f_{l}^{\ast}((T^{\ast})^{\bf Z}),l\in{\cal L}$. We assume that for $l\in{\cal L}_{in}$ in the standard coordinates $(x,y)$ near singular point $f_{l}(0)$ we have: $f_{l}(t)=(0,t)$ or $f_{l}(t)=(0,-t)$ for all sufficiently small $t>0$, and $\alpha_{l}(t)=\pm f_{l}^{\ast}(dy)$.

e)

A map ${\cal L}\to{\cal L}\times{\cal L},\,\,\,l\mapsto(p_{left}(l),p_{right}(l))$ (the letter $p$ stands for “parent”: one can think about these lines as “generating $l$ in a collision”).

Notice that since the form $dy$ is invariant with respect to the monodromy, the condition in d) is coordinate-independent. The covector $\alpha_{l}(t)$ will be called a direction covector of $l$ at time $t$. It gives rise to a half-plane

$$P_{l,t}^{(0)}=\{v\in T_{f_{l}(t)}Y|\langle\alpha_{l}(t),v\rangle>0\}\,\,.$$

To every $l\in{\cal L}_{in}$ we assign a pair $\left(f_{l}(0),sgn\left(\alpha_{l}(0)\right)\right)\in B^{sing}\times\{\pm 1\}$, where $sgn(\alpha_{l}(0))$ is a choice of sign in $\pm f_{l}^{\ast}(dy)$ (see data d) in the previous subsection). In this way we obtain a map $r:{\cal L}_{in}\to B^{sing}\times\{\pm 1\}$.

Axiom 1.

Map $r$ is one-to-one.

Let $U\subset Y$ be a simply-connected domain, and line $l$ intersects $U$. Let $I\subset{\bf R}_{+}$ be an interval such that $f_{l}(I)\subset U$. Then there exists a covariantly constant closed non-zero $1$-form $\beta_{U}$ in $U$ (with constant integer coefficients), such that $f_{l}^{\ast}(\beta_{U})=\alpha_{l}$, when both sides are restricted to $I$.

Axiom 2.

For any $t_{1},t_{2}\in I$ one has

$$\int_{f_{l}(t_{1})}^{f_{l}(t_{2})}\beta_{U}=t_{2}-t_{1}\,\,.$$

Let $l_{1},l_{2}\in{\cal L}$, $t_{1},t_{2}>0$ satisfy the condition $f_{l_{1}}(t_{1})=f_{l_{2}}(t_{2})=x\in Y$. In this case we say that lines $l_{1}$ and $l_{2}$ have a collision at $x$ at the times $t_{1}$ and $t_{2}$ respectively.

Axiom 3.

Under the above assumptions there are only two possibilities:

3a) either $l_{1}=l_{2}$ and $t_{1}=t_{2}$, or

3b) covector $\alpha_{l_{1}}(t_{1})$ is not proportional to $\alpha_{l_{2}}(t_{2})$. Then we may assume that $\alpha_{l_{1}}(t_{1})\wedge\alpha_{l_{2}}(t_{2})>0$. Under these conditions we require that for any coprime positive integers $n_{1},n_{2}$ there exists a unique line $l\in{\cal L}$ such that $l_{1}=p_{left(l)},l_{2}=p_{right}(l)$, $f_{l}(0)=x$ and $\alpha_{l}(0)=n_{1}\alpha_{l_{1}}(t_{1})+n_{2}\alpha_{l_{2}}(t_{2})$.

In other words, $l_{1}$ and $l_{2}$ are “parents of $l$”, and the direction covector of $l$ at the intersection point is a primitive integral linear combination of those for $l_{1}$ and $l_{2}$ (see Figure 4).

Axiom 4.

For every line $l\in{\cal L}_{com}$ there exist $l_{1}$ and $l_{2}$ such that they satisfy the condition 3b).

Axiom 5.

For any $x\in Y$ there are no more than two pairs $(l,t)\in{\cal L}\times(0,+\infty)$ such that $x=f_{l}(t)$. In other words, there are no more than two lines intersecting at a point in $Y$.

Let $l_{1},l_{2},t_{1},t_{2},x$ mean the same as in the Axiom 3, and assume that $\alpha_{l_{1}}(t_{1})\wedge\alpha_{l_{2}}(t_{2})>0$. Let us consider the set ${\cal L}_{(x)}$ of germs of all $l\in{\cal L}_{com}$ starting at $x$ (i.e. such that $f_{l}(0)=x$).

Axiom 6.

For any finite subset ${\cal L}^{\prime}\subset{\cal L}_{(x)}$ there is an orientation preserving homeomorphism of a neighborhood of $x$ onto a neighborhood of $(0,0)\in{\bf R}^{2}$ such that:

6a) Germs of oriented curves which are images of $l_{1}$ and $l_{2}$ get transformed into the germs at $(0,0)$ of coordinate axes $(x,0)$ and $(0,y)$ respectively.

6b) Germ of the image of $l\in{\cal L}^{\prime}$ gets transformed into the germ of the ray $\{(n_{1}t,n_{2}t)\,|\,t>0\}$ where $\alpha_{l}(0)=n_{1}\alpha_{l_{1}}(t_{1})+n_{2}\alpha_{l_{2}}(t_{2})$.

Figure 5 illustrates this axiom.

Axiom 7.

Let $p_{i}$ denotes either $p_{left}$ or $p_{right}$. Then for any $l\in{\cal L}$ there exists $N\geq 1$ such that if the line $p_{1}(p_{2}(\dots p_{N}(l)\dots)$ is well-defined then it belongs to ${\cal L}_{in}$.

This axiom says that any composed line $l\in{\cal L}_{com}$ appears as a result of finitely many collisions. The tree of ancestors of a given line form a tree embedded in $B$, see Figure 6.

Here we offer a construction of the set of lines satisfying the above axioms.

Let us use the standard ${\bf R}^{2}$ as a model around each $b\in B^{sing}$ in order to fix a structure of smooth manifold on the whole surface $B$. Let $\widetilde{Y}$ denotes the covering of $Y$ such that the fiber over $y\in Y$ is $(T_{y}^{\ast}Y)^{{\bf Z}}\setminus\{0\}$.

Let us fix a generic smooth metric on $B$. By the pull-back it gives a metric on $\widetilde{Y}$. Notice that there is a canonical closed $1$-form $\beta$ on $\widetilde{Y}$ such that $\beta_{|(y,\mu)}=\mu$, where $y\in Y,\,\,\mu\in(T_{y}^{\ast}Y)^{{\bf Z}}$. Using the metric we obtain dual to $\beta$ gradient vector field $v$ on $\widetilde{Y}$.

For any $s\in B^{sing}$ and a choice of 1-form $\alpha(0)=\pm dy$ in local coordinates, we take the unique integral line of $v$ starting at $(s,\alpha(0))$. Set ${\cal L}_{in}$ will be the set of all lines obtained in this way. Each line $l\in{\cal L}_{in}$ carries a covariantly constant closed $1$-form $\alpha_{l}$. Using Axiom 2 as a definition, we obtain a canonical parametrization of each line by the time parameter $t$. Since the metric is generic, a line cannot return to a point in $B^{sing}$.

Then we proceed inductively. If two already constructed lines $l_{1},l_{2}\in{\cal L}$ meet at $x\in Y$ we produce a new integral line $l$ of $v$ with the direction covector satisfying the condition 3b) for any pair of coprime positive integers $n_{1},n_{2}$. In this way we construct a set of lines ${\cal L}$ satisfying all the axioms. The only non-trivial thing to check is that for each line values of the parameter $t$ are in one-to-one correspondence with the interval $(0,+\infty)$. In order to see this we observe that the length of each line is infinite. Indeed, an integral curve of $v$ cannot have a limiting point in $Y$ (since the flow generated by $v$ is smooth, and the lengths of tangent vectors are bounded from below because of the integrality of $1$-forms).

We conclude that there exists a set ${\cal L}$ of lines satisfying Axioms 1-7.