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''of a quadratic differential''
 
''of a quadratic differential''
  
A description of the behaviour of the trajectories of a [[Quadratic differential|quadratic differential]] on an oriented [[Riemann surface|Riemann surface]] in a neighbourhood of any point of this surface. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l0602101.png" /> be an oriented Riemann surface and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l0602102.png" /> be a quadratic differential on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l0602103.png" />; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l0602104.png" /> be the set of all zeros and simple poles of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l0602105.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l0602106.png" /> be the set of all poles of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l0602107.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l0602108.png" />. The trajectories of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l0602109.png" /> form a regular family of curves on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021010.png" />. Under an extension of the concept of a regular family of curves this remains true on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021011.png" /> also. The behaviour of the trajectories in neighbourhoods of points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021012.png" /> is significantly more complicated. A complete description of the local structure of trajectories is given below.
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A description of the behaviour of the trajectories of a [[Quadratic differential|quadratic differential]] on an oriented [[Riemann surface|Riemann surface]] in a neighbourhood of any point of this surface. Let $R$ be an oriented Riemann surface and let $Q(z)\,dz^2$ be a quadratic differential on $R$; let $C$ be the set of all zeros and simple poles of $Q(z)\,dz^2$ and let $H$ be the set of all poles of $Q(z)\,dz^2$ of order $\geq2$. The trajectories of $Q(z)\,dz^2$ form a regular family of curves on $R\setminus(C\cup H)$. Under an extension of the concept of a regular family of curves this remains true on $R\setminus H$ also. The behaviour of the trajectories in neighbourhoods of points of $H$ is significantly more complicated. A complete description of the local structure of trajectories is given below.
  
a) For any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021013.png" /> there is a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021014.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021015.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021016.png" /> and a homeomorphic mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021017.png" /> onto the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021018.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021019.png" />) such that a maximal open arc of each trajectory in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021020.png" /> goes to a segment on which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021021.png" /> is constant. Consequently, through each point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021022.png" /> there passes a trajectory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021023.png" /> that is either an open arc or a Jordan curve on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021024.png" />.
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a) For any point $P\in R\setminus(C\cup H)$ there is a neighbourhood $N$ of $P$ on $R$ and a homeomorphic mapping of $N$ onto the disc $|w|<1$ ($w=u+iv$) such that a maximal open arc of each trajectory in $N$ goes to a segment on which $v$ is constant. Consequently, through each point of $R\setminus(C\cup H)$ there passes a trajectory of $Q(z)\,dz^2$ that is either an open arc or a Jordan curve on $R$.
  
b) For any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021025.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021026.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021027.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021028.png" /> is a zero and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021029.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021030.png" /> is a simple pole) there is a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021031.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021032.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021033.png" /> and a homeomorphic mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021034.png" /> onto the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021035.png" /> such that a maximal arc of each trajectory in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021036.png" /> goes to an open arc on which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021037.png" /> is constant. There are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021038.png" /> trajectories with ends at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021039.png" /> and with limiting tangential directions that make equal angles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021040.png" /> with each other.
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b) For any point $P\in C$ of order $\mu$ ($\mu>0$ if $P$ is a zero and $\mu=-1$ if $P$ is a simple pole) there is a neighbourhood $N$ of $P$ on $R$ and a homeomorphic mapping of $N$ onto the disc $|w|<1$ such that a maximal arc of each trajectory in $N$ goes to an open arc on which $\operatorname{Im}w^{(\mu+2)/2}$ is constant. There are $\mu+2$ trajectories with ends at $P$ and with limiting tangential directions that make equal angles $2\pi/(\mu+2)$ with each other.
  
c) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021041.png" /> be a pole of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021042.png" />. If a certain trajectory has an end at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021043.png" />, then it tends to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021044.png" /> along one of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021045.png" /> directions making equal angles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021046.png" />. There is a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021047.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021048.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021049.png" /> with the following properties: 1) every trajectory that passes through some point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021050.png" /> in each of the directions either tends to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021051.png" /> or leaves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021052.png" />; 2) there is a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021053.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021054.png" /> contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021055.png" /> and such that every trajectory that passes through some point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021056.png" /> tends to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021057.png" /> in at least one direction, remaining in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021058.png" />; 3) if some trajectory lies entirely in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021059.png" /> and therefore tends to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021060.png" /> in both directions, then the tangent to this trajectory as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021061.png" /> is approached in the corresponding direction tends to one of two adjacent limiting positions. The Jordan curve obtained by adjoining <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021062.png" /> to this trajectory bounds a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021063.png" /> containing points of the angle formed by the two adjacent limiting tangents. The tangent to any trajectory that has points in common with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021064.png" /> tends to these adjacent limiting positions as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021065.png" /> is approached in the two directions. By means of a suitable branch of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021066.png" /> the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021067.png" /> is mapped onto the half-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021068.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021069.png" /> is a real number); and 4) for every pair of adjacent limiting positions there is a trajectory having the properties described in 3).
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c) Let $P\in H$ be a pole of order $\mu>2$. If a certain trajectory has an end at $P$, then it tends to $P$ along one of $\mu-2$ directions making equal angles $2\pi/(\mu-2)$. There is a neighbourhood $N$ of $P$ on $R$ with the following properties: 1) every trajectory that passes through some point of $N$ in each of the directions either tends to $P$ or leaves $N$; 2) there is a neighbourhood $N^*$ of $P$ contained in $N$ and such that every trajectory that passes through some point of $N^*$ tends to $P$ in at least one direction, remaining in $N^*$; 3) if some trajectory lies entirely in $N$ and therefore tends to $P$ in both directions, then the tangent to this trajectory as $P$ is approached in the corresponding direction tends to one of two adjacent limiting positions. The Jordan curve obtained by adjoining $P$ to this trajectory bounds a domain $D$ containing points of the angle formed by the two adjacent limiting tangents. The tangent to any trajectory that has points in common with $D$ tends to these adjacent limiting positions as $P$ is approached in the two directions. By means of a suitable branch of the function $\zeta=\int[Q(z)]^{1/2}\,dz$ the domain $D$ is mapped onto the half-plane $\operatorname{Im}\zeta>c$ (where $c$ is a real number); and 4) for every pair of adjacent limiting positions there is a trajectory having the properties described in 3).
  
d) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021070.png" /> be a pole of order two and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021071.png" /> be the local parameter in terms of which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021072.png" /> is the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021073.png" />. Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021074.png" /> has (for some choice of the branch of the root) the following expansion in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021075.png" />:
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d) Let $P\in H$ be a pole of order two and let $z$ be the local parameter in terms of which $P$ is the point $z=0$. Suppose that $[Q(z)]^{-1/2}$ has (for some choice of the branch of the root) the following expansion in a neighbourhood of $z=0$:
  
<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/l/l060/l060210/l06021076.png" /></td> </tr></table>
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$$[Q(z)]^{-1/2}=(a+ib)z\{1+b_1z+b_2z^2+\dotsb\},$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021077.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021078.png" /> are real constants and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021079.png" /> are complex constants. The structure of the images of the trajectories of the differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021080.png" /> in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021081.png" />-plane is determined by which of the following three cases holds.
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where $a$ and $b$ are real constants and $b_1,b_2,\dots,$ are complex constants. The structure of the images of the trajectories of the differential $Q(z)\,dz^2$ in the $z$-plane is determined by which of the following three cases holds.
  
Case I: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021082.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021083.png" />. For sufficiently small <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021084.png" /> the image of each trajectory that intersects the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021085.png" /> tends to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021086.png" /> in one direction, and leaves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021087.png" /> in the other direction. Both the modulus and the argument of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021088.png" /> vary monotonically on the image of the trajectory in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021089.png" />. Each image of a trajectory twists around the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021090.png" /> and behaves asymptotically like a [[Logarithmic spiral|logarithmic spiral]].
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Case I: $a\neq0$, $b\neq0$. For sufficiently small $\alpha>0$ the image of each trajectory that intersects the disc $|z|<\alpha$ tends to $z=0$ in one direction, and leaves $|z|<\alpha$ in the other direction. Both the modulus and the argument of $z$ vary monotonically on the image of the trajectory in $|z|<\alpha$. Each image of a trajectory twists around the point $z=0$ and behaves asymptotically like a [[Logarithmic spiral|logarithmic spiral]].
  
Case II: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021091.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021092.png" />. For sufficiently small <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021093.png" /> the image of every trajectory that intersects the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021094.png" /> tends to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021095.png" /> in one direction and leaves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021096.png" /> in the other direction. The modulus of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021097.png" /> varies monotonically on the image of the trajectory in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021098.png" />. Different images of trajectories have different limiting directions at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l06021099.png" />.
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Case II: $a\neq0$, $b=0$. For sufficiently small $\alpha>0$ the image of every trajectory that intersects the disc $|z|<\alpha$ tends to $z=0$ in one direction and leaves $|z|<\alpha$ in the other direction. The modulus of $z$ varies monotonically on the image of the trajectory in $|z|<\alpha$. Different images of trajectories have different limiting directions at the point $z=0$.
  
Case III: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l060210100.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l060210101.png" />. For each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l060210102.png" /> there is a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l060210103.png" /> such that for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l060210104.png" /> the image of a trajectory that intersects the circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l060210105.png" /> is a Jordan curve lying in the circular annulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060210/l060210106.png" />.
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Case III: $a=0$, $b\neq0$. For each $\epsilon>0$ there is a number $\alpha(\epsilon)>0$ such that for $0<\alpha\leq\alpha(\epsilon)$ the image of a trajectory that intersects the circle $|z|=\alpha$ is a Jordan curve lying in the circular annulus $\alpha(1+\epsilon)^{-1}<|z|<\alpha(1+\epsilon)$.
  
 
====References====
 
====References====

Latest revision as of 21:47, 1 January 2019

of a quadratic differential

A description of the behaviour of the trajectories of a quadratic differential on an oriented Riemann surface in a neighbourhood of any point of this surface. Let $R$ be an oriented Riemann surface and let $Q(z)\,dz^2$ be a quadratic differential on $R$; let $C$ be the set of all zeros and simple poles of $Q(z)\,dz^2$ and let $H$ be the set of all poles of $Q(z)\,dz^2$ of order $\geq2$. The trajectories of $Q(z)\,dz^2$ form a regular family of curves on $R\setminus(C\cup H)$. Under an extension of the concept of a regular family of curves this remains true on $R\setminus H$ also. The behaviour of the trajectories in neighbourhoods of points of $H$ is significantly more complicated. A complete description of the local structure of trajectories is given below.

a) For any point $P\in R\setminus(C\cup H)$ there is a neighbourhood $N$ of $P$ on $R$ and a homeomorphic mapping of $N$ onto the disc $|w|<1$ ($w=u+iv$) such that a maximal open arc of each trajectory in $N$ goes to a segment on which $v$ is constant. Consequently, through each point of $R\setminus(C\cup H)$ there passes a trajectory of $Q(z)\,dz^2$ that is either an open arc or a Jordan curve on $R$.

b) For any point $P\in C$ of order $\mu$ ($\mu>0$ if $P$ is a zero and $\mu=-1$ if $P$ is a simple pole) there is a neighbourhood $N$ of $P$ on $R$ and a homeomorphic mapping of $N$ onto the disc $|w|<1$ such that a maximal arc of each trajectory in $N$ goes to an open arc on which $\operatorname{Im}w^{(\mu+2)/2}$ is constant. There are $\mu+2$ trajectories with ends at $P$ and with limiting tangential directions that make equal angles $2\pi/(\mu+2)$ with each other.

c) Let $P\in H$ be a pole of order $\mu>2$. If a certain trajectory has an end at $P$, then it tends to $P$ along one of $\mu-2$ directions making equal angles $2\pi/(\mu-2)$. There is a neighbourhood $N$ of $P$ on $R$ with the following properties: 1) every trajectory that passes through some point of $N$ in each of the directions either tends to $P$ or leaves $N$; 2) there is a neighbourhood $N^*$ of $P$ contained in $N$ and such that every trajectory that passes through some point of $N^*$ tends to $P$ in at least one direction, remaining in $N^*$; 3) if some trajectory lies entirely in $N$ and therefore tends to $P$ in both directions, then the tangent to this trajectory as $P$ is approached in the corresponding direction tends to one of two adjacent limiting positions. The Jordan curve obtained by adjoining $P$ to this trajectory bounds a domain $D$ containing points of the angle formed by the two adjacent limiting tangents. The tangent to any trajectory that has points in common with $D$ tends to these adjacent limiting positions as $P$ is approached in the two directions. By means of a suitable branch of the function $\zeta=\int[Q(z)]^{1/2}\,dz$ the domain $D$ is mapped onto the half-plane $\operatorname{Im}\zeta>c$ (where $c$ is a real number); and 4) for every pair of adjacent limiting positions there is a trajectory having the properties described in 3).

d) Let $P\in H$ be a pole of order two and let $z$ be the local parameter in terms of which $P$ is the point $z=0$. Suppose that $[Q(z)]^{-1/2}$ has (for some choice of the branch of the root) the following expansion in a neighbourhood of $z=0$:

$$[Q(z)]^{-1/2}=(a+ib)z\{1+b_1z+b_2z^2+\dotsb\},$$

where $a$ and $b$ are real constants and $b_1,b_2,\dots,$ are complex constants. The structure of the images of the trajectories of the differential $Q(z)\,dz^2$ in the $z$-plane is determined by which of the following three cases holds.

Case I: $a\neq0$, $b\neq0$. For sufficiently small $\alpha>0$ the image of each trajectory that intersects the disc $|z|<\alpha$ tends to $z=0$ in one direction, and leaves $|z|<\alpha$ in the other direction. Both the modulus and the argument of $z$ vary monotonically on the image of the trajectory in $|z|<\alpha$. Each image of a trajectory twists around the point $z=0$ and behaves asymptotically like a logarithmic spiral.

Case II: $a\neq0$, $b=0$. For sufficiently small $\alpha>0$ the image of every trajectory that intersects the disc $|z|<\alpha$ tends to $z=0$ in one direction and leaves $|z|<\alpha$ in the other direction. The modulus of $z$ varies monotonically on the image of the trajectory in $|z|<\alpha$. Different images of trajectories have different limiting directions at the point $z=0$.

Case III: $a=0$, $b\neq0$. For each $\epsilon>0$ there is a number $\alpha(\epsilon)>0$ such that for $0<\alpha\leq\alpha(\epsilon)$ the image of a trajectory that intersects the circle $|z|=\alpha$ is a Jordan curve lying in the circular annulus $\alpha(1+\epsilon)^{-1}<|z|<\alpha(1+\epsilon)$.

References

[1] J.A. Jenkins, "Univalent functions and conformal mapping" , Springer (1958)


Comments

Cf. Quadratic differential for the notion of the trajectory of a quadratic differential.

This description is taken essentially from section 3.2 of [1]. For a detailed treatment of quadratic differentials see also [a1].

For the global structure see Global structure of trajectories.

References

[a1] K. Strebel, "Quadratic differentials" , Springer (1984) (Translated from German)
[a2] F.P. Gardiner, "Teichmüller theory and quadratic differentials" , Wiley (1987)
How to Cite This Entry:
Local structure of trajectories. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Local_structure_of_trajectories&oldid=11783
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