Lemniscates as Trajectories of Quadratic Differentials (I)

TL;DR

This paper explores polynomial and rational lemniscates as trajectories of quadratic differentials, providing new proofs for many classical results in complex analysis and geometric function theory.

Contribution

It introduces a novel approach by linking lemniscates to quadratic differentials, simplifying proofs of existing theorems.

Findings

Lemniscates can be characterized as trajectories of specific quadratic differentials.

Many classical lemniscate results are derived more straightforwardly.

The approach unifies geometric and analytical perspectives on lemniscates.

Abstract

In this note, we study polynomial and rational lemniscates as trajectories of related quadratic differentials. Many classic results can be then proved easily...

Figures

Equations

$${\varpi }_r\left(z\right)=-{\left(\frac{r^{\prime }\left(z\right)}{r\left(z\right)}\right)}^{2}d{z}^2=-{\left(\frac{p^{\prime }\left(z\right)q\left(z\right)-p\left(z\right)q^{\prime }\left(z\right)}{p\left(z\right)q\left(z\right)}\right)}^{2}d{z}^2.$$

$$\frac{r^{\prime }\left(z\right)}{r\left(z\right)}=\sum _{p\left(a\right)q\left(a\right)=0}\frac{m_a}{z-a},$$

$${\varpi }_r\left(z\right)=-\frac{m_a^2}{{\left(z-a\right)}^2}\left(1+\mathcal{O}(z-a)\right)d{z}^2,z\to a.$$

$$\mathrm{deg}\left(p^{\prime }q-p{q}^{\prime }\right)=\mathrm{deg}\left(pq\right)-1,$$

$${\varpi }_r\left(u\right)=-\frac{{\left(\mathrm{deg}\left(p\right)-\mathrm{deg}\left(q\right)\right)}^2}{u^2}\left(1+\mathcal{O}(u)\right)d{u}^2,u\to 0;$$

$$\mathrm{deg}\left(p^{\prime }q-p{q}^{\prime }\right)<\mathrm{deg}\left(pq\right)-2,$$

$$\mathrm{deg}\left(p^{\prime }q-p{q}^{\prime }\right)=\mathrm{deg}\left(pq\right)-2,$$

$${\varpi }_r\left(z\right)>0,$$

$$\Re {\int }^z\frac{r^{\prime }\left(t\right)}{r\left(t\right)}dt=\log \left|r\left(z\right)\right|=\text{const}.$$

$$t⟼\Im {\int }_0^t\frac{r^{\prime }\left(z\left(u\right)\right)}{r\left(z\left(u\right)\right)}{z}^{\prime }\left(u\right)du=\arg \left(r\left(z\left(t\right)\right)\right)-\arg \left(r\left(z\left(0\right)\right)\right)$$

$$\sum \left(1-\frac{\left(m_j+2\right)t_j}{2\pi }\right)=2+\sum n_i,$$

$${\Gamma }_{r,c}=\{z\in \mathbb{C}:|r(z)|=c\}$$

$$\begin{array}[c]{cc}f:\mathbb{C}\setminus\left\{\text{poles of }r\left(z\right)\right\}\longrightarrow\mathbb{R}&z\longmapsto\left|r\left(z\right)\right|\end{array}$$

$$\Re {\int }_{\gamma }\frac{r^{\prime }\left(t\right)}{r\left(t\right)}dt=0.$$

$$\left|w_i\right|=\left|w_j\right|=\max \left\{\left|w_k\right|:=r\left(z_k\right);k=1,...,m\right\}$$

$$0=2+\sum n_k,k=1,...,s.$$

$$0=2s+2\left(\mathrm{deg}\left(p^{\prime }q-p{q}^{\prime }\right)-\mathrm{deg}\left(pq\right)\right)=2s-2;$$

$${\Omega }_{-}$$

$${\Omega }_{+}$$

$$\psi :z⟼\exp \left(c{\int }^z\frac{p^{\prime }\left(t\right)}{p\left(t\right)}dt\right)$$

$$\psi \left(z\right)=\beta p{\left(z\right)}^c$$

$$c=\left\{\begin{array}[c]{c}\frac{1}{n},\text{ if }a=\infty\\ \frac{1}{\alpha},\text{ if }p\left(a\right)=0,\text{ }\end{array}\right.$$

$$z\longmapsto\left\{\begin{array}[c]{c}p\left(z\right)^{\frac{1}{n}},\text{ if }a=\infty,\\ p\left(z\right)^{\frac{1}{\alpha}},\text{ if }p\left(a\right)=0.\end{array}\right.$$

$$\left\{\begin{array}[c]{c}\phi_{+}^{-1}\left(z\right)=p\left(z\right)^{\frac{1}{n}},\text{ if }a=\infty,\\ \phi_{-}^{-1}\left(z\right)=p\left(z\right)^{\frac{1}{\alpha}},\text{ if }p\left(a\right)=0.\end{array}\right..$$

$$k(z)=B(z{)}^{1/n},$$

$$B(z)=e^{i\theta }\prod _{k=1}^n\frac{z-a_k}{1-\overline{a_k}z}$$

$$p\left(z\right)=\text{vskip}12.0ptplus4.0ptminus4.0pt{\left(z-a\right)}^{\alpha }{p}_1\left(z\right),\alpha \in {\mathbb{N}}^{\ast };p_1\left(z\right)=\prod _{i=1}^{n-\alpha }\text{vskip}12.0ptplus4.0ptminus4.0pt\left(z-a_i\right),p_1\left(a\right)\ne 0.$$

$$z⟼\frac{p\circ {\varphi }_{+}\left(z\right)}{{\prod }_{i=1}^{n-\alpha }\frac{z-{\varphi }_{+}^{-1}\left(a_i\right)}{1-\overline{{\varphi }_{+}^{-1}\left(a_i\right)}z}};\left|z\right|\ge 1$$

$$p\circ {\varphi }_{+}\left(z\right)=e^{i\theta }{z}^n\prod _{i=1}^{n-\alpha }\frac{z-{\varphi }_{+}^{-1}\left(a_i\right)}{1-\overline{{\varphi }_{+}^{-1}\left(a_i\right)}z};\left|z\right|\ge 1,$$

$$k^{-1}(z)=z^{\frac{n}{\alpha }}{B}_1{\left(z\right)}^{\frac{1}{\alpha }}.$$

$$B_1\left(z\right)=e^{i\theta }\prod _{i=1}^{n-\alpha }\frac{z-{\varphi }_{+}^{-1}\left(a_i\right)}{1-\overline{{\varphi }_{+}^{-1}\left(a_i\right)}z}.$$

Full text

2010 Mathematics subject classification: 30C10, 30C15, 34E05.

Keywords and phrases: Quadratic differentials. Lemniscates.Fingerprints.

1

A quadratic differential

Given a rational function $r\left(z\right)=\frac{p\left(z\right)}{q\left(z\right)},$ where $p\left(z\right)$ and $q\left(z\right)$ are two co-prime complex polynomials, we consider the quadratic differential on the Riemann sphere $\widehat{\mathbb{C}}$ :

[TABLE]

Finite critical points and *infinite critical points *of $\varpi_{r}$ are respectively its zero’s and poles; all other points of $\widehat{\mathbb{C}}$ are called regular points of $\varpi_{r}.$

It is obvious that the partial fraction decomposition of $\frac{r^{\prime}\left(z\right)}{r\left(z\right)}$ is as follows :

[TABLE]

where $m_{a}\in\mathbb{Z}^{\ast}$ is the multiplicity of the zero $a$ of $p\left(z\right)q\left(z\right).$ We deduce that

[TABLE]

In other words, the zero’s of $p$ and $q$ are poles of order $2$ of $\varpi_{r}$ with negative residue.

If

[TABLE]

(in particular, if $\deg\left(p\right)\neq\deg\left(q\right)$), then, with the parametrization $u=1/z$, we get

[TABLE]

thus, $\infty$ is another double pole* *of $\varpi_{r}$ with negative residue. If

[TABLE]

then $\infty$ is zero of $\varpi_{r}$ with multiplicity greater than $1.$ In the case

[TABLE]

$\infty$ is a regular point.

Horizontal trajectories (or just trajectories) of the quadratic differential $\varpi_{r}$ are the zero loci of the equation

[TABLE]

or equivalently

[TABLE]

If $z\left(t\right),t\in\mathbb{R}$ is a horizontal trajectory, then the function

[TABLE]

is monotone.

The vertical (or, orthogonal) trajectories are obtained by replacing $\Im$ by $\Re$ in equation (3). The horizontal and vertical trajectories of the quadratic differential $\varpi_{r}$ produce two pairwise orthogonal foliations of the Riemann sphere $\widehat{\mathbb{C}}$.

A trajectory passing through a critical point of $\varpi_{r}$ is called critical trajectory. In particular, if it starts and ends at a finite critical point, it is called finite critical trajectory, otherwise, we call it an infinite critical trajectory.

If two different trajectories are not disjoint, then their intersection must be a zero of the quadratic differential.

The closure of the set of finite and infinite critical trajectories is called the critical graph of $\varpi_{r},$ we denote it by $\Gamma_{r}.$

The local and global structures of the trajectories is well known (more details about the theory of quadratic differentials can be found in [5],[3], or [6]), in particular :

• •

At any regular point, horizontal (resp. vertical) trajectories look locally as simple analytic arcs passing through this point, and through every regular point of $\varpi_{p}$ passes a uniquely determined horizontal (resp. vertical) trajectory of $\varpi_{p};$ these horizontal and vertical trajectories are locally orthogonal at this point.

• •

From each zero with multiplicity $m$ of $\varpi_{r},$ there emanate $m+2$ critical trajectories spacing under equal angle $2\pi/(m+2)$.

• •

Any double pole has a neighborhood such that, all trajectories inside it take a loop-shape encircling the pole or a radial form diverging to the pole, respectively if the residue is negative or positive.

• •

A trajectory in the large can be, either a closed curve not passing through any critical point (closed trajectory), or an arc connecting two critical points, or an arc that has no limit along at least one of its directions (recurrent trajectory).

The set $\widehat{\mathbb{C}}\setminus\Gamma_{r}$ consists of a finite number of domains called the domain configurations of $\varpi_{r}.$ For a general quadratic differential on a $\widehat{\mathbb{C}}$, there are five kind of domain configuration, see [3, Theorem3.5]. Since all the infinite critical points of $\varpi_{r}$ are poles of order $2$ with negative residues, then there are three possible domain configurations:

• •

the Circle domain : It is swept by closed trajectories and contains exactly one double pole. Its boundary is a closed critical trajectory. For a suitably chosen real constant $c$ and some real number $r>0,$ the function $z\longmapsto r\exp\left(c\int^{z}\frac{p^{\prime}\left(t\right)}{p\left(t\right)}dt\right)$ is a conformal map from the circle domain $D$ onto the unit circle; it extends continuously to the boundary $\partial D,$ and sends the double pole to the origin.

• •

the Ring domain: It is swept by closed trajectories. Its boundary consists of two connected components. For a suitably chosen real constant $c$ and some real numbers $0<r_{1}<r_{2},$ the function $z\longmapsto\exp\left(c\int^{z}\frac{p^{\prime}\left(t\right)}{p\left(t\right)}dt\right)$ is a conformal map from the circle domain $D$ onto the annulus $\left\{z:r_{1}<\left|z\right|<r_{2}\right\}$ and it extends continuously to the boundary $\partial D.$

• •

the *Dense domain : *It is swept by recurrent critical trajectory i.e., the interior of its closure is non-empty. Jenkins Three-pole Theorem (see [5, Theorem 15.2]) asserts that a quadratic differential on the Riemann sphere with at most three poles cannot have recurrent trajectories. In general, the non-existence of such trajectories is not guaranteed, but here, following the idea of level function of Baryshnikov and Shapiro (see [1]), the quadratic differential $\varpi_{r}$ excludes the dense domain, as we will see in Proposition 4.

A very helpful tool that will be used in our investigation is the Teichmüller lemma (see [5, Theorem 14.1]).

Definition 1

A domain in $\widehat{\mathbb{C}}$ bounded only by segments of horizontal and/or vertical trajectories of $\varpi_{r}$ (and their endpoints) is called $\varpi_{r}$-polygon.

Lemma 2** (Teichműller)**

Let $\Omega$ be a $\varpi_{r}$-polygon, and let $z_{j}$ be the critical points on the boundary $\partial\Omega$ of $\Omega,$ and let $t_{j}$ be the corresponding interior angles with vertices at $z_{j},$ respectively . Then

[TABLE]

where $m_{j}$ are the multiplicities of $z_{j},$ and $n_{i}$ are the multiplicities of critical points of $\varpi_{r}$ inside $\Omega.\vskip 12.0pt plus 4.0pt minus 4.0pt$

2 Lemniscates

We use the notations of [2]. Let us denote $n=\deg r=\max\left(\deg p,\deg q\right)>0$. For $c>0,$ the set

[TABLE]

is called rational lemniscate of degree $n.$ For more details, see [4]. From the point of view of the theory of quadratic differentials, each connected component of the lemniscate $\Gamma_{r,c}$ coincides with a horizontal trajectory of $\varpi_{r}=-\left(\frac{r^{\prime}\left(z\right)}{r\left(z\right)}\right)^{2}dz^{2},$ as we have seen in equation (3). The lemniscate $\Gamma_{r,c}$ is entirely determined by the knowledge of the critical graph $\Gamma_{r}$ (which is the union of the lemniscates $\Gamma_{r,\left|r\left(a\right)\right|},$ for all zero’s $a$ of $\varpi_{r}$) of the quadratic differential of $\varpi_{r}.$ In particular, if we denote by $n_{z}$ and $n_{p}$ respectively the number of zero’s and poles $r\left(z\right)$ in $\widehat{\mathbb{C}},$ then, from the local behavior of the trajectories, we see that, for $c\rightarrow 0^{+}$, the lemniscate $\Gamma_{r,c}$ is formed by exactly $n_{z}$ disjoint closed curves each of them encircles a zero of $r\left(z\right)$, while for $c\rightarrow+\infty,$ $\Gamma_{r,c}$ is formed by exactly $n_{p}$ disjoint closed curves each of them encircles a pole of $r\left(z\right)$. If $\deg\left(p^{\prime}q-pq^{\prime}\right)<\deg\left(pq\right)-2,$ then, $\infty$ is a zero of $\varpi_{r}$ of multiplicity $m\geq 2,$ and there are $m+2$ critical trajectories emerging from $\infty$ dividing in a symmetric way the complement of some zero centred ball into $m+2$ connected components. See Figure 1. In the rest of this note, we assume that $\infty$ is a double pole, i.e., $\deg\left(p^{\prime}q-pq^{\prime}\right)=\deg\left(pq\right)-1.$

Definition 3

A quadratic differential on $\widehat{\mathbb{C}}$ is called Strebel if the complement to the union of its closed trajectories has vanishing area.

Proposition 4

The quadratic differential $\varpi_{r}$ is Strebel.

Proof. Since the critical points of $\varpi_{r}$ are only zero’s and double poles with negative residues, it is sufficient to prove that $\varpi_{r}$ has no recurrent trajectory. The function

[TABLE]

is continuous, and constant on each horizontal trajectory of $\varpi_{r}.$ If $\varpi_{r}$ has a recurrent trajectory, then, its domain configuration contains a dense domain $D.$ Thus, the function $f$ must be constant on $D,$ which is clearly impossible by analyticity of the rational function $z\longmapsto r\left(z\right).$

A necessary condition for the existence of a finite critical trajectory connecting two finite critical points of $\varpi_{r}$ is the existence of a Jordan arc $\gamma$ connecting them, such that

[TABLE]

Unfortunately, this condition is not sufficient in general, as it can be shown easily for the case of $r\left(z\right)=\left(z^{2}-1\right)\left(z^{2}-4\right);$ see Figure 2.

However, a more sufficient condition will be shown by the following Proposition

Proposition 5

Let us denote $z_{1},...,z_{m}$ the finite critical points of $\varpi_{r}.$ If

[TABLE]

for some $1$ $\leq i<j\leq m,$ then, there exists a finite critical trajectory joining $z_{i}$ and $z_{j}.$ In particular, the critical graph $\Gamma_{r}$ is connected, if and only if $\left|w_{1}\right|=\cdot\cdot\cdot=\left|w_{m}\right|.$

Proof. If no finite critical trajectory joins $z_{i}$ and $z_{j},$ then a lemniscate $\Gamma_{r,c},$ for some $c>\left|w_{i}\right|,$ is not connected : $\Gamma_{r,c}$ is a disjoint union of $s\geq 2$ loops $L_{1},...,L_{s},$ each of them encircles a part of the critical graph $\Gamma_{r}.$ Looking at each of these loops as a $\varpi_{r}$-polygon and applying Lemma 2, we get :

[TABLE]

Making the sum of all equalities in (7), and taking into account our assumption that $\left(\deg\left(p^{\prime}q-pq^{\prime}\right)=\deg\left(pq\right)\right)-1,$ we get

[TABLE]

a contradiction. The second point is a mere consequence.

The numbers $w_{1}=r\left(z_{1}\right),...,w_{m}=r\left(z_{m}\right)$ are called the non-vanishing critical values of $r\left(z\right).$

3

Fingerprints of polynomial lemniscates

Here following a brief mention of the case of polynomial lemniscates $\Gamma_{p,1}$. Let us denote by

[TABLE]

The maximum modulus theorem asserts that $\Omega_{+}$ is a connected open subset containing a neighborhood of $\infty$ in $\widehat{\mathbb{C}}$.

Definition 6

*A lemniscate $\Gamma_{p,1}$ of degree $n$ is *proper if it is smooth ($p^{\prime}\left(z\right)\neq 0$ on $\Gamma_{p,1}$) and connected.

Let $z_{1,}...,z_{s},$ $s\leq n-1$ be the zero’s (repeated according to their multiplicity) of $\varpi_{p}.$ The non-vanishing critical values for $p\left(z\right)$ are the values $w_{1}=p\left(z_{1}\right),...,w_{s}=p\left(z_{s}\right).$ For a smooth lemniscate $\Gamma_{p,1}$ of degree $n$, the following characterizes the property of being proper through the critical values :

Proposition 7

Assume that the lemniscate $\Gamma_{p,1}$ is smooth. Then, $\Gamma_{p,1}$ is proper if and only if all the critical values $w_{1},...,w_{s}$ satisfy $|w_{k}|<1.$

Proof. Proof of this Proposition can be found in [2]. We provide here a more evident proof relying on quadratic differentials theory. The smoothness of $\Gamma_{p,1}$ implies that it is not a critical trajectory. Suppose that $|w_{k}|>1$ for some $k\in\left\{1,...,s\right\},$ and consider two critical trajectories emerging from $z_{k}$ that form a loop $\gamma$. This loop cannot intersect $\Gamma_{p,1},$ and $\gamma\cap\Omega_{-}\neq\emptyset$ since $\gamma$ contains a pole in its interior; a contradiction. The other point is clear.

Note that the interior $\Omega_{-}$ of a proper lemniscate of degree $n$ (or, for a general smooth lemniscate, each component of ) is also simply connected, since its complement is connected.

Let $\gamma$ be a $\mathcal{C}^{\infty}$ Jordan curve in $\mathbb{C};$ by a Jordan theorem, $\gamma$ splits $\widehat{\mathbb{C}}$ into a bounded and an unbounded simply connected components $D_{-}$ and $D_{+}.$ The Riemann mapping theorem asserts that there exist two conformal maps $\phi_{-}:\Delta\longrightarrow$ $D_{-},$ and $\phi_{+}:\widehat{\mathbb{C}}\setminus\overline{\Delta}\longrightarrow$ $D_{+},$ where $\Delta$ is the unit disk. The map $\phi_{+}$ is uniquely determined by the normalization $\phi_{+}\left(\infty\right)=\infty$ and $\phi_{+}\left(\infty\right)>0.$ It is well-known that $\phi_{-}$ and $\phi_{+}$ extend to $\mathcal{C}^{\infty}$-diffeomorphisms on the closure of their respective domain. The fingerprint of $\gamma$ is the map $k$ $:=$ $\phi_{+}^{-1}\circ\phi_{-}:S^{1}\longrightarrow S^{1}$ from the unit circle $S^{1}$ to itself. Note that $k$ is uniquely determined by up to post-composition with an automorphism of $D$ onto itself. Moreover, the fingerprint $k$ is invariant under translations and scalings of the curve $\gamma.$

3.1 Lemniscates in a Circle Domain

Let $a$ be a double pole of $\varpi_{p}$ ( $a=\infty$ or $p\left(a\right)=0$ ). Jenkins Theorem on the Configuration Domains of the quadratic differential $\varpi_{p}$ asserts that there exists a connected neighborhood $\mathcal{U}_{a}$ of $a$ (a Circle Domain of $\varpi_{p}$) bounded by finite critical trajectories of $\varpi_{p},$ such that all trajectories of $\varpi_{p}$ (lemniscates of $p$) inside $\mathcal{U}_{a}$ are closed smooth curves encircling $a.$ Moreover, for a suitably chosen non-vanishing real constant $c,$ the function

[TABLE]

is a conformal map from $\mathcal{U}_{a}$ onto a certain disk centered in $z=0.$ A more obvious form of it, is

[TABLE]

for some complex number $\beta.$ Baring in mind that $\psi$ is univalent near $a$, we get

[TABLE]

where $\alpha$ is the multiplicity of $a$ if $p\left(a\right)=0.$ It follows that the function

[TABLE]

is a conformal map from $\mathcal{U}_{a}$ onto a certain disk $\Delta_{a}$ centered in $z=0.$ We may assume for the sake of simplicity that $\Delta_{a}$ with a radius $R>1.$ For the given lemniscate $\Gamma_{p,1}$ in $\mathcal{U}_{a}$ (see Figure 3 ), it is straightforward that the function $z\longmapsto p\left(z\right)^{\frac{1}{\alpha}}$ maps $\Omega_{-}$ conformally onto the unit disk $\Delta.$ Thus,

[TABLE]

In the first case, we notice that $\Gamma_{p,1}$ is proper if and only if $a=\infty;$ the next Theorem gives its fingerprint.

Theorem 8** (Ebenfelt, Khavinson and Shapiro )**

The fingerprint $k$ $:S^{1}\longrightarrow S^{1}$ of a proper lemniscate $\Gamma_{p,1}$ of the polynomial $p\left(z\right)=\prod_{k=1}^{n}\left(z-\varsigma_{k}\right)$ is given by

[TABLE]

where $B$ is the Blaschke product of degree $n$

[TABLE]

for some real number $\theta$, and $a_{k}=\phi_{-}\left(\varsigma_{k}\right),k=1,,,n.$

In the case $p\left(a\right)=0$, let

[TABLE]

With the normalization $\phi_{+}\left(z\right)\rightarrow\infty$ as $z\rightarrow\infty,$ the function

[TABLE]

is holomorphic in $\mathbb{C}\setminus\overline{\Delta}$, does not vanish there, is continuous in $\mathbb{C}\setminus\Delta,$ and has modulus one on $\partial\Delta=S^{1}$. We deduce the existence of $\theta\in\mathbb{R}$ such that

[TABLE]

which proves the

Theorem 9

Let $\Gamma_{p,1}$ be a smooth connected lemniscate such that $z=a$ is the only zero of $p$ in $\Omega_{-}$ . The fingerprint $k$ $:S^{1}\longrightarrow S^{1}$ of $\Gamma_{p,1}$ is given by

[TABLE]

where $B_{1}\left(z\right)$ is the Blaschke product

[TABLE]

3.2 Lemniscates in a Ring Domain

In the following, let $\mathcal{U}$ be a Ring Domain of the quadratic differential $\varpi_{p}.$ It is bounded by two lemniscates $\Gamma_{p,r}$ and $\Gamma_{p,R}.$ We may assume that

[TABLE]

For the sake of simplicity, we may assume that $p$ has exactly two different zeros $a$ and $b$ in the bounded domain of $\mathbb{C}$ with boundary $\Gamma_{p,r}.$

[TABLE]

We consider the lemniscate $\Gamma_{p,1}$ of $p$ in $\mathcal{U}$ (see Figure 4 ).

Since the function

[TABLE]

is holomorphic in $\Delta$, is continuous in $\overline{\Delta},$ has $\phi_{-}^{-1}\left(a\right)$ and $\phi_{-}^{-1}\left(b\right)$ as unique zeros (with multiplicities $\alpha$ and $\beta$) in $\Delta,$ and has modulus one on $\partial\Delta$. We deduce that there exists $\theta_{1}\in\mathbb{R}$ such that

[TABLE]

Reasoning like in the previous subsection on $\phi_{+}\left(z\right),$ we get for some $\theta_{2}\in\mathbb{R}$

[TABLE]

Combining the last two equalities for $\left|z\right|=1,$ we obtain the following

Theorem 10

Let $\Gamma_{p,1}$ be a smooth connected lemniscate such that $\Omega_{-}$ contains exactly two different zeros $a$ and $b$ of $p$ with respective multiplicities $\alpha$ and $\beta.$The fingerprint $k$ $:S^{1}\longrightarrow S^{1}$ of $\Gamma_{p,1}$ satisfies the functional equation

[TABLE]

where $A$ and $B$ are the Blaschke products given by

[TABLE]

[TABLE]