# Global structure of trajectories

A description of the behaviour as a whole of trajectories of a positive quadratic differential on a compact oriented Riemann surface (cf. Quadratic differential for the definition of trajectory in this setting). Let $R$ be a compact oriented Riemann surface, let $Q( z) d z ^ {2}$ be a positive quadratic differential on $R$, let $C$ be the set of all zeros and simple poles of $Q( z) d z ^ {2}$, and let $H$ be the set of poles of $Q( z) d z ^ {2}$ of order $\geq 2$. The trajectories of $Q ( z) d z ^ {2}$ form a family $F$ which has many of the properties of regular families of curves. This family of curves covers $R$ except for the points of the set $C \cup H$, i.e. through every point of $R \setminus ( C \cup H)$ passes a unique element of $F$. The behaviour of a trajectory of $Q( z) d z ^ {2}$ in a neighbourhood of any point of $R$ is described by the local structure of the trajectories of the quadratic differential (cf. Local structure of trajectories). In considering the global structure of the curves of $F$ at the points of $R \setminus H$, an important role is played by the following unions of trajectories. Let $\Phi$ be the union of all trajectories of $Q( z) d z ^ {2}$ having limit end points at some point of $C$; let $\Lambda$ be the subset of $\Phi$ that is the union of all the trajectories of $Q( z) d z ^ {2}$ which have one limit end point at a point of $C$ and a second limit end point at a point of $C \cup H$.

A set $K$ on $R$ is called an $F$- set with respect to $Q( z) d z ^ {2}$ if each trajectory of $Q( z) d z ^ {2}$ intersecting with $K$ is completely contained in $K$. The internal closure of the set $K$ is defined as the interior of the closure $\overline{K}\;$ and is denoted by $\widehat{K}$. The internal closure of an $F$- set is also an $F$- set. The terminal domain $E$ with respect to $Q( z) d z ^ {2}$ is the largest connected open $F$- set on $R$ with the following properties: 1) $E$ contains no points of $C \cup H$; 2) $E$ is filled with trajectories of $Q( z) d z ^ {2}$, each one of which has a limit end point in each one of the two possible directions at a given point $A \in H$; and 3) $E$ is conformally mapped by the function

$$\zeta = \int\limits [ Q ( z)] ^ {1/2} dz$$

onto the left or right half-plane of the $\zeta$- plane (depending on the choice of the branch of the root). It follows from the local structure of the trajectories of $Q( z) d z ^ {2}$ that the point $A$ should be a pole of the differential $Q( z) d z ^ {2}$ of order at least three.

The strip-like domain $S$ with respect to $Q( z) d z ^ {2}$ is the largest connected open $F$- set on $R$ with the following properties: 1) $S$ contains no points of $C \cup H$; 2) $S$ is filled with the trajectories of $Q( z) d z ^ {2}$, each one of which has at one point $A \in H$ a limit end point in one direction and at another point $B \in H$( which may coincide with $A$) a limit end point in the other direction; and 3) $S$ is conformally mapped by the function

$$\zeta = \int\limits [ Q ( z)] ^ {1/2} dz$$

onto the strip $a < \mathop{\rm Im} \zeta < b$, where $a$ and $b$ are finite real numbers and $a < b$. The points $A$ and $B$ may be poles of $Q( z) d z ^ {2}$ of order two or larger.

The circular domain ${\mathcal C}$ with respect to $Q( z) d z ^ {2}$ is the largest connected open $F$- set on $R$ with the following properties: 1) ${\mathcal C}$ contains a unique double pole $A$ of $Q( z) d z ^ {2}$; 2) ${\mathcal C} \setminus A$ is filled with the trajectories of $Q( z) d z ^ {2}$ each one of which is a closed Jordan curve which separates $A$ from the boundary of ${\mathcal C}$; and 3) if a purely-imaginary constant $c$ has been suitably chosen, the function

$$w = \mathop{\rm exp} \left \{ c \int\limits [ Q ( z)] ^ {1/2} dz \right \} ,$$

supplemented by the value zero at $A$, conformally maps ${\mathcal C}$ onto a disc $| w | < R$, and $A$ is mapped to $w = 0$.

The annular domain $D$ with respect to $Q( z) d z ^ {2}$ is the largest connected $F$- set on $R$ with the following properties: 1) $D$ does not contain any points of $C \cup H$; 2) $D$ is filled with trajectories of $Q( z) d z ^ {2}$ each one of which is a closed Jordan curve; and 3) if a purely-imaginary constant $c$ is suitably chosen, the function

$$w = \mathop{\rm exp} \left \{ c \int\limits [ Q ( z)] ^ {1/2} dz \right \}$$

conformally maps $D$ onto a circular annulus $r _ {1} < | w | \leq r _ {2}$, $0 < r _ {1} < r _ {2}$.

The dense domain ${\mathcal F}$ with respect to $Q( z) d z ^ {2}$ is the largest connected $F$- set on $R$ with the following properties: 1) ${\mathcal F}$ does not contain any points of $H$; and 2) ${\mathcal F} \setminus C$ is filled with trajectories of $Q( z) d z ^ {2}$, each one of which is everywhere-dense in ${\mathcal F}$.

The basic structure theorem is valid . Let $R$ be a compact oriented Riemann surface and let $Q( z) d z ^ {2}$ be a positive quadratic differential on $R$, while excluding the following possible cases and all configurations obtainable from such cases by way of a conformal mapping: I. $R$ is a $z$- sphere, $Q ( z) d z ^ {2} = d z ^ {2}$; II. $R$ is a $z$- sphere, $Q( z) d z ^ {2} = K e ^ {i \alpha } d z / z ^ {2}$, $K$ being positive and $\alpha$ being a real number; and III. $R$ is a torus, and $Q( z) d z ^ {2}$ is regular on $R$. Then 1) $R \setminus \overline \Lambda \;$ consists of a finite number of terminal, strip-like, annular, and dense domains; 2) each such domain is bounded by a finite number of trajectories together with points at which the latter meet; each boundary component of such a domain contains a point of $C$, except for boundary components of the circular or annular domain which may coincide with a boundary components of $R$; for a strip-like domain two boundary elements issuing from points of the set $H$ subdivide the boundary into two parts, each one of which contains a point of the set $C$; 3) each pole of $Q( z) d z ^ {2}$ of order $m > 2$ has a neighbourhood that can be covered by the internal closure of the union of $m - 2$ terminal domains and a finite number (which may also be equal to zero) of strip-like domains; and 4) each pole of $Q( z) d z ^ {2}$ of order $m = 2$ has a neighbourhood that can be covered by the internal closure of the union of a finite number of strip-like domains, or has a neighbourhood contained in a circular domain.

The statement of the basic structure theorem of J.A. Jenkins  in its original formulation immediately follows from this theorem: Under the conditions of the theorem the set $R \setminus \overline \Phi \;$ consists of a finite number of terminal, strip-like, circular, and annular domains. In a number of studies in the theory of univalent functions, main stress is laid on proving the fact that the set $\widehat \Phi$ is empty for the quadratic differential $Q( z) d z ^ {2}$ under consideration. The search for conditions under which $\widehat \Phi$ is empty is also of interest in its own right. The following three-pole theorem provides an example of a quadratic differential $Q( z) d z ^ {2}$ on the $z$- sphere for which the set $\widehat \Phi$ is empty: If $R$ is the $z$- sphere and $Q( z) d z ^ {2}$ is a quadratic differential on $R$ with at most three different poles, then $\widehat \Phi$ is empty.

How to Cite This Entry:
Global structure of trajectories. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Global_structure_of_trajectories&oldid=47099
This article was adapted from an original article by G.V. Kuz'mina (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article