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''of a quadratic differential''
 
''of a quadratic differential''
  
A description of the behaviour as a whole of trajectories of a positive [[Quadratic differential|quadratic differential]] on a compact oriented [[Riemann surface|Riemann surface]] (cf. [[Quadratic differential|Quadratic differential]] for the definition of trajectory in this setting). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g0444601.png" /> be a compact oriented Riemann surface, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g0444602.png" /> be a positive quadratic differential on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g0444603.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g0444604.png" /> be the set of all zeros and simple poles of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g0444605.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g0444606.png" /> be the set of poles of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g0444607.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g0444608.png" />. The trajectories of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g0444609.png" /> form a family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446010.png" /> which has many of the properties of regular families of curves. This family of curves covers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446011.png" /> except for the points of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446012.png" />, i.e. through every point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446013.png" /> passes a unique element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446014.png" />. The behaviour of a trajectory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446015.png" /> in a neighbourhood of any point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446016.png" /> is described by the local structure of the trajectories of the quadratic differential (cf. [[Local structure of trajectories|Local structure of trajectories]]). In considering the global structure of the curves of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446017.png" /> at the points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446018.png" />, an important role is played by the following unions of trajectories. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446019.png" /> be the union of all trajectories of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446020.png" /> having limit end points at some point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446021.png" />; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446022.png" /> be the subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446023.png" /> that is the union of all the trajectories of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446024.png" /> which have one limit end point at a point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446025.png" /> and a second limit end point at a point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446026.png" />.
+
A description of the behaviour as a whole of trajectories of a positive [[Quadratic differential|quadratic differential]] on a compact oriented [[Riemann surface|Riemann surface]] (cf. [[Quadratic differential|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|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446027.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446028.png" /> is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446030.png" />-set with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446031.png" /> if each trajectory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446032.png" /> intersecting with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446033.png" /> is completely contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446034.png" />. The internal closure of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446035.png" /> is defined as the interior of the closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446036.png" /> and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446037.png" />. The internal closure of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446038.png" />-set is also an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446039.png" />-set. The terminal domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446040.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446041.png" /> is the largest connected open <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446042.png" />-set on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446043.png" /> with the following properties: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446044.png" /> contains no points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446045.png" />; 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446046.png" /> is filled with trajectories of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446047.png" />, each one of which has a limit end point in each one of the two possible directions at a given point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446048.png" />; and 3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446049.png" /> is conformally mapped by the function
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446050.png" /></td> </tr></table>
+
$$
 +
\zeta  = \int\limits [ Q ( z)]  ^ {1/2}  dz
 +
$$
  
onto the left or right half-plane of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446051.png" />-plane (depending on the choice of the branch of the root). It follows from the local structure of the trajectories of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446052.png" /> that the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446053.png" /> should be a pole of the differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446054.png" /> of order at least three.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446055.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446056.png" /> is the largest connected open <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446057.png" />-set on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446058.png" /> with the following properties: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446059.png" /> contains no points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446060.png" />; 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446061.png" /> is filled with the trajectories of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446062.png" />, each one of which has at one point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446063.png" /> a limit end point in one direction and at another point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446064.png" /> (which may coincide with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446065.png" />) a limit end point in the other direction; and 3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446066.png" /> is conformally mapped by the function
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446067.png" /></td> </tr></table>
+
$$
 +
\zeta  = \int\limits [ Q ( z)]  ^ {1/2}  dz
 +
$$
  
onto the strip <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446068.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446069.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446070.png" /> are finite real numbers and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446071.png" />. The points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446072.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446073.png" /> may be poles of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446074.png" /> of order two or larger.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446075.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446076.png" /> is the largest connected open <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446077.png" />-set on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446078.png" /> with the following properties: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446079.png" /> contains a unique double pole <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446080.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446081.png" />; 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446082.png" /> is filled with the trajectories of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446083.png" /> each one of which is a closed Jordan curve which separates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446084.png" /> from the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446085.png" />; and 3) if a purely-imaginary constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446086.png" /> has been suitably chosen, the function
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446087.png" /></td> </tr></table>
+
$$
 +
=   \mathop{\rm exp}  \left \{ c \int\limits [ Q ( z)]  ^ {1/2}  dz \right \} ,
 +
$$
  
supplemented by the value zero at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446088.png" />, conformally maps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446089.png" /> onto a disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446090.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446091.png" /> is mapped to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446092.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446093.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446094.png" /> is the largest connected <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446095.png" />-set on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446096.png" /> with the following properties: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446097.png" /> does not contain any points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446098.png" />; 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g04446099.png" /> is filled with trajectories of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g044460100.png" /> each one of which is a closed Jordan curve; and 3) if a purely-imaginary constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g044460101.png" /> is suitably chosen, the function
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g044460102.png" /></td> </tr></table>
+
$$
 +
=   \mathop{\rm exp}  \left \{ c \int\limits [ Q ( z)]  ^ {1/2}  dz \right \}
 +
$$
  
conformally maps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g044460103.png" /> onto a circular annulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g044460104.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g044460105.png" />.
+
conformally maps $  D $
 +
onto a circular annulus $  r _ {1} < | w | \leq  r _ {2} $,
 +
0 < r _ {1} < r _ {2} $.
  
The dense domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g044460106.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g044460107.png" /> is the largest connected <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g044460108.png" />-set on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g044460109.png" /> with the following properties: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g044460110.png" /> does not contain any points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g044460111.png" />; and 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g044460112.png" /> is filled with trajectories of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g044460113.png" />, each one of which is everywhere-dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g044460114.png" />.
+
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 [[#References|[2]]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g044460115.png" /> be a compact oriented Riemann surface and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g044460116.png" /> be a positive quadratic differential on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g044460117.png" />, while excluding the following possible cases and all configurations obtainable from such cases by way of a [[Conformal mapping|conformal mapping]]: I. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g044460118.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g044460119.png" />-sphere, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g044460120.png" />; II. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g044460121.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g044460122.png" />-sphere, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g044460123.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g044460124.png" /> being positive and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g044460125.png" /> being a real number; and III. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g044460126.png" /> is a torus, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g044460127.png" /> is regular on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g044460128.png" />. Then 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g044460129.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g044460130.png" />, except for boundary components of the circular or annular domain which may coincide with a boundary components of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g044460131.png" />; for a strip-like domain two boundary elements issuing from points of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g044460132.png" /> subdivide the boundary into two parts, each one of which contains a point of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g044460133.png" />; 3) each pole of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g044460134.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g044460135.png" /> has a neighbourhood that can be covered by the internal closure of the union of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g044460136.png" /> terminal domains and a finite number (which may also be equal to zero) of strip-like domains; and 4) each pole of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g044460137.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g044460138.png" /> 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 basic structure theorem is valid [[#References|[2]]]. 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|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 [[#References|[1]]] in its original formulation immediately follows from this theorem: Under the conditions of the theorem the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g044460139.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g044460140.png" /> is empty for the quadratic differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g044460141.png" /> under consideration. The search for conditions under which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g044460142.png" /> is empty is also of interest in its own right. The following three-pole theorem provides an example of a quadratic differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g044460143.png" /> on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g044460144.png" />-sphere for which the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g044460145.png" /> is empty: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g044460146.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g044460147.png" />-sphere and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g044460148.png" /> is a quadratic differential on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g044460149.png" /> with at most three different poles, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044460/g044460150.png" /> is empty.
+
The statement of the basic structure theorem of J.A. Jenkins [[#References|[1]]] 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.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.A. Jenkins,  "Univalent functions and conformal mappings" , Springer  (1958)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.A. Jenkins,  "On the global structure of the trajectories of a positive quadratic differential"  ''Illinois J. Math.'' , '''4''' :  3  (1960)  pp. 405–412</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.A. Jenkins,  "Univalent functions and conformal mappings" , Springer  (1958)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.A. Jenkins,  "On the global structure of the trajectories of a positive quadratic differential"  ''Illinois J. Math.'' , '''4''' :  3  (1960)  pp. 405–412</TD></TR></table>

Latest revision as of 19:42, 5 June 2020


of a quadratic differential

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 [2]. 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 [1] 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.

References

[1] J.A. Jenkins, "Univalent functions and conformal mappings" , Springer (1958)
[2] J.A. Jenkins, "On the global structure of the trajectories of a positive quadratic differential" Illinois J. Math. , 4 : 3 (1960) pp. 405–412
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