For $x\in Y$ let $U\subset U^{\prime}$ be convex neighborhoods of $x$ such that $U$ is relatively compact in $U^{\prime}$. Let $l\in{\cal L}$. Then there is a natural embedding $f_{l}^{-1}(U)\to f_{l}^{-1}(U^{\prime})$.

In plain words a piece $L$ of $l$ is an equivalence class of a connected interval of $l\cap U$. Two connected intervals are equivalent if they are contained in a larger connected interval of $l\cap U^{\prime}$. The sole purpose of the introduction of the notion of a piece is to avoid some pathology. Namely, for any pair $(U,U^{\prime})$ as above, any $l\in{\cal L}$ and any $T\in{\bf R}_{>0}$, there is only a finite number of pieces of $l$ in $(U,U^{\prime})$ which have points with time parameter $t\in(0,T)$.

Let $L$ be a piece of $l$ defined by a pair $(U,U^{\prime})$. Then one can define an affine function $ord_{L}\in Aff_{{\bf Z},Y}(U^{\prime})$ in the following way. Let $t>0$ be such that $f_{l}(t)$ belongs to $L$. Since $U^{\prime}$ is convex, there is a unique continuation of $ord_{l}(t)$ to $U^{\prime}$. This is an affine function which does not depend on the choice of $t$. We will denote it by $ord_{L}$.

For any germ of a symplectomorphism $\varphi\in Symp_{p}$ at a point $p\in Y$ we define its convergence region as the maximal convex subset $\Omega(\varphi)\subset T_{p}Y$ such that the pullback $\exp_{p}^{\ast}(\varphi)$ extends to $\Omega(\varphi)$. Since the definition of $\varphi_{l}$ (and hence its convergence region) is covariant with respect to the affine connection we have the following result:

It is clear that one can define convergence regions for symplectomorphisms associated with pieces of lines, and a similar property holds for them.