Tips of Tongues in the Double Standard Family

TL;DR

This paper characterizes the local structure of parameter regions ('tips of tongues') where degree 2 circle maps have multiple zeros, showing they form regular cusps in parameter space.

Contribution

It provides a detailed local analysis of bifurcation points in degree 2 circle maps, revealing the cusp structure of the tips of tongues.

Findings

Tips of tongues are regular cusps in parameter space.

Zero of multiplicity 3 implies a cusp in the bifurcation diagram.

Local coordinates can be chosen to reveal the cusp structure.

Abstract

We answer a question raised by Misiurewicz and Rodrigues concerning the family of degree 2 circle maps $F_\lambda:\mathbb{R}/\mathbb{Z}\to \mathbb{R}/\mathbb{Z}$ defined by \[F_\lambda(x) := 2x + a+ \frac{b}{\pi} \sin(2\pi x){\quad\text{with}\quad} \lambda:=(a,b)\in \mathbb{R}/\mathbb{Z}\times (0,1).\] We prove that if $F_\lambda^{\circ n}-{\rm id}$ has a zero of multiplicity $3$ in $\mathbb{R}/\mathbb{Z}$, then there is a system of local coordinates $(\alpha,\beta):W\to \mathbb{R}^2$ defined in a neighborhood $W$ of $\lambda$, such that $\alpha(\lambda) =\beta(\lambda)=0$ and $F_\mu^{\circ n} - {\rm id}$ has a multiple zero with $\mu\in W$ if and only if $\beta^3(\mu) = \alpha^2(\mu)$. This shows that the tips of tongues are regular cusps.

Equations

$$F_{\lambda }(x):=2x+a+\frac{b}{\pi }\sin (2\pi x)\text{with}\lambda :=(a,b)\in \mathbb{R}/\mathbb{Z}\times (0,1).$$

$$F_{\lambda }(x):=2x+a+\frac{b}{\pi }\sin (2\pi x)\text{with}\lambda :=(a,b)\in \mathbb{R}/\mathbb{Z}\times [0,1].$$

$$\gamma:=\gamma_{0}\quad\text{and}\quad\dot{\gamma}:=\frac{d\gamma_{t}}{dt}\Big{|}_{t=0}.$$

$$F(\lambda ,z)=2z+a+\frac{b}{\pi }\sin (2\pi z)\text{with}\lambda :=(a,b)\in \Lambda .$$

$$F_{\lambda }(z):=F(\lambda ,z).$$

$$w\circ f={\mathrm{e}}^{2\pi \mathrm{i}a}{w}^2{\mathrm{e}}^{b(w-1/w)}$$

$$f^{\ast }(\mathrm{d}w)={\mathrm{e}}^{2\pi \mathrm{i}a}{\mathrm{e}}^{b(w-1/w)}(b{w}^2+2w+b)\mathrm{d}w.$$

$$w(c^{\pm })=\frac{-1\pm \sqrt{1-b^2}}{b}.$$

$$\tilde{f}(Z)=2Z+A+\frac{b}{\pi }\sin (2\pi Z)\text{so that}\mathbf{\pi }\circ \tilde{f}=f\circ \mathbf{\pi }.$$

$$\tilde{f}\circ \Gamma =2(X+\mathrm{i}Y)+A+\frac{b}{\pi }\sin (2\pi \Gamma ),\sin (2\pi \Gamma )=\frac{{\mathrm{e}}^{-2\pi Y}{\mathrm{e}}^{2\pi \mathrm{i}X}-{\mathrm{e}}^{2\pi Y}{\mathrm{e}}^{-2\pi \mathrm{i}X}}{2\mathrm{i}}$$

$$\tilde{f}\circ \Gamma (t)\sim 2X(t)-\frac{b}{4\pi \mathrm{i}}{\mathrm{e}}^{2\pi Y(t)}{\mathrm{e}}^{-2\pi \mathrm{i}X(t)}.$$

$$\bigl{|}\tilde{f}\circ\Gamma(t_{k})\bigr{|}\sim\frac{b}{4\pi}{\rm e}^{2\pi Y(t_{k})}\underset{k\to+\infty}{\longrightarrow}+\infty.$$

$$\tilde{f}\circ \Gamma (t_k)\sim 2X(t_k)+\mathrm{i}\frac{b}{4\pi }{\mathrm{e}}^{2\pi Y(t_k)}\underset{k\to +\infty }{⟶}\infty .$$

$$\zeta (\bar{z})=\bar{\zeta }(z)\text{and}\zeta \circ f^{\circ p}=\zeta +{\zeta }^3+\mathcal{O}({\zeta }^5).$$

$$\zeta \circ F_{\mu }^{\circ p}-\zeta =P_{\mu }(\zeta )\cdot g(\mu ,\zeta )$$

$$A(\lambda )=B(\lambda )=C(\lambda )=0,g(\lambda ,\zeta )=1+\mathcal{O}({\zeta }^2)$$

$$P_{\mu }(\zeta ):=A(\mu )+B(\mu )\zeta +C(\mu ){\zeta }^2+{\zeta }^3.$$

$$\zeta \circ F_{\mu }^{\circ n}-\zeta =P_{\mu }(\zeta )\cdot h(\mu ,\zeta )\text{with}h(\lambda ,\zeta )=m+\mathcal{O}({\zeta }^2)$$

$$\alpha :=\frac{C^3}{27}-\frac{BC}{6}+\frac{A}{2}\text{and}\beta :=\frac{C^2}{9}-\frac{B}{3}.$$

$$\mathrm{discriminant}(P_{\mu })=108{\beta }^3(\mu )-108{\alpha }^2(\mu ).$$

$$\mathrm{d}\alpha {|}_{{\mathrm{T}}_{\lambda }\Lambda }=\frac{1}{2}\mathrm{d}A{|}_{{\mathrm{T}}_{\lambda }\Lambda }\text{and}\mathrm{d}\beta {|}_{{\mathrm{T}}_{\lambda }\Lambda }=-\frac{1}{3}\mathrm{d}B{|}_{{\mathrm{T}}_{\lambda }\Lambda }.$$

$$\mathrm{D}f\circ {\vartheta }_v(z):={\mathrm{D}}_{\lambda ,z}F(v,0).$$

$${\zeta }_k:=\zeta \circ f^{\circ (p-k)}.$$

$$\mathrm{d}A(v)=\sum _{k=1}^p\mathrm{residue}(q_A\otimes {\vartheta }_v,x_k)\text{and}\mathrm{d}B(v)=\sum _{k=1}^p\mathrm{residue}(q_B\otimes {\vartheta }_v,x_k).$$

$$\vartheta_{v}(z):=({\mathrm{D}}_{z}f)^{-1}\bigl{(}{\mathrm{D}}_{\lambda,z}F(v,0)\bigr{)}\in{\rm T}_{z}{\mathbb{T}}.$$

$${\vartheta }_{v_a}=\frac{2\pi \mathrm{i}{\mathrm{e}}^{2\pi \mathrm{i}a}{w}^2{\mathrm{e}}^{b(w-1/w)}}{{\mathrm{e}}^{2\pi \mathrm{i}a}{\mathrm{e}}^{b(w-1/w)}(b{w}^2+2w+b)}\frac{\mathrm{d}}{\mathrm{d}w}=\frac{2\pi \mathrm{i}{w}^2}{b{w}^2+2w+b}\frac{\mathrm{d}}{\mathrm{d}w}$$

$${\vartheta }_{v_b}=\frac{{\mathrm{e}}^{2\pi \mathrm{i}a}{w}^2(w-1/w){\mathrm{e}}^{b(w-1/w)}}{{\mathrm{e}}^{2\pi \mathrm{i}a}{\mathrm{e}}^{b(w-1/w)}(b{w}^2+2w+b)}\frac{\mathrm{d}}{\mathrm{d}w}=\frac{w^3-w}{b{w}^2+2w+b}\frac{\mathrm{d}}{\mathrm{d}w}$$

$${\mathcal{T}}_f:=\left\{\frac{c_3{w}^3+c_2{w}^2+c_{1}w}{b{w}^2+2w+b}\frac{\mathrm{d}}{\mathrm{d}w};(c_1,c_2,c_3)\in {\mathbb{C}}^3\right\}.$$

$${\Theta }_f(v):={\vartheta }_v.$$

$$\tau :=w\frac{\mathrm{d}}{\mathrm{d}w}.$$

Full text

[TABLE]

We prove that if $F_{\lambda}^{\circ n}-{\rm id}$ has a zero of multiplicity $3$ in ${\mathbb{R}}/{\mathbb{Z}}$, then there is a system of local coordinates $(\alpha,\beta):W\to{\mathbb{R}}^{2}$ defined in a neighborhood $W$ of $\lambda$, such that $\alpha(\lambda)=\beta(\lambda)=0$ and $F_{\mu}^{\circ n}-{\rm id}$ has a multiple zero with $\mu\in W$ if and only if $\beta^{3}(\mu)=\alpha^{2}(\mu)$. This shows that the tips of tongues are regular cusps.

Supported in part by the ANR grant Lambda ANR-13-BS01-0002 and by FRPDF allotment 2018-19 of Presidency University

Introduction

Following Misiurewicz and Rodrigues [MR07], we consider the family of circle maps $F_{\lambda}:{\mathbb{R}}/{\mathbb{Z}}\to{\mathbb{R}}/{\mathbb{Z}}$ defined by

[TABLE]

If $b\in[0,1/2)$, then $F_{\lambda}:{\mathbb{R}}/{\mathbb{Z}}\to{\mathbb{R}}/{\mathbb{Z}}$ is expanding and all periodic cycles of $F_{\lambda}$ in ${\mathbb{R}}/{\mathbb{Z}}$ are repelling. If $b\in[1/2,1]$, it may happen that $F_{\lambda}:{\mathbb{R}}/{\mathbb{Z}}\to{\mathbb{R}}/{\mathbb{Z}}$ has a non-repelling cycle. The multiplier of such a cycle belongs to $[0,1]$. There is at most one such cycle. Connected components of the open sets of parameters $\lambda\in(a,b)\in{\mathbb{R}}/{\mathbb{Z}}\times[0,1]$ for which $F_{\lambda}$ has an attracting cycle are called tongues (see [MR07] and [D10]). The period of the attracting cycle remains constant in each tongue, and is called the period of the tongue

Let $T$ be a tongue of period $p\geq 1$. The boundary of $T$ consists of two smooth curves which are graphs with respect to $b$ and intersect tangentially at the tip $\lambda_{T}\in{\mathbb{R}}/{\mathbb{Z}}\times(0,1)$ (see [MR07, MR08] and Figure 1). If $\lambda\in\partial T$ then $F_{\lambda}$ has a cycle of period $p$ and multiplier 1. On the one hand, if $\lambda\in\partial T{\smallsetminus}\{\lambda_{T}\}$, then points of the cycle are double zeros of $F_{\lambda}^{\circ p}-{\rm id}$. On the other hand, points of the cycle are triple zeros of $F_{\lambda_{T}}^{\circ p}-{\rm id}$.

There is a unique tongue of period $1$. Misiurewicz and Rodrigues [MR07] proved that the order of contact of its two boundary curves at the tip is $1/2$. In [MR08] they asked whether this property holds for all tongues of the family $F_{\lambda}$. In this article, we answer positively to this question. More precisely, we prove that near the tip of any tongue, the two boundary curves form an ordinary cusp.

Theorem 1**.**

Assume $F_{\lambda}^{\circ n}-{\rm id}$ has a zero of multiplicity $3$ in ${\mathbb{R}}/{\mathbb{Z}}$. Then there is a system of local coordinates $(\alpha,\beta):W\to{\mathbb{R}}^{2}$ defined a neighborhood $W$ of $\lambda$ in ${\mathbb{R}}/{\mathbb{Z}}\times(0,1)$, such that $\alpha(\lambda)=\beta(\lambda)=0$ and $F_{\mu}^{\circ n}-{\rm id}$ has a multiple zero with $\mu\in W$ if and only if $\beta^{3}(\mu)=\alpha^{2}(\mu)$.

Our proof relies on a transversality result due to Adam Epstein for families of finite type analytic maps, which itself relies on an injectivity result of a linear map acting on an appropriate space of quadratic differentials. In §1, we prove that the maps $F_{\lambda}$ are finite type analytic maps. In §2, we define the functions $\alpha$ and $\beta$. In §3, we identify the derivatives of those functions at $\lambda$. In §4, we state and prove the injectivity result. In §5, we prove that $(\alpha,\beta)$ is a system of local coordinates.

Some classical results on quadratic differentials are collected in Appendix A.

Notation

If $U$ is a complex manifold, we denote by ${\rm T}U$ the tangent bundle of $U$ and for $z\in U$, we denote by ${\rm T}_{z}U$ the tangent space to $U$ at $z$. If $\phi:U\to{\mathbb{C}}$ is a holomorphic function, we denote by ${\rm d}\phi:{\rm T}U\to{\mathbb{C}}$ the exterior derivative of $\phi$ (this is a holomorphic $1$-form on $U$). If $F:U\to V$ is a holomorphic map between complex manifolds $U$ and $V$, we denote by ${\mathrm{D}}F:{\rm T}U\to{\rm T}V$ the bundle map ${\rm T}_{z}U\ni v\mapsto{\mathrm{D}}_{z}F(v)\in{\rm T}_{F(z)}V$.

Assume $f:U\to V$ is a holomorphic map between Riemann surfaces. If $\omega$ is a holomorphic $1$-form on $V$, then $f^{*}\omega:=\omega\circ{\mathrm{D}}f$ is a holomorphic $1$-form on $U$. If $\vartheta$ is a holomorphic vector field on $U$, then there is a meromorphic vector field $f^{*}\vartheta$ on $U$ satisfying ${\mathrm{D}}f\circ f^{*}\vartheta=\vartheta\circ f$.

We will consider various holomorphic families $t\mapsto\gamma_{t}$ defined near [math] in ${\mathbb{C}}$. We will employ the notation

[TABLE]

1. Finite type analytic maps

The notion of finite type analytic maps originates in [E]. Let $f:{\mathbb{X}}\rightarrow{\mathbb{Y}}$ be an analytic map of complex 1-manifolds, possibly disconnected. An open set $V\subseteq{\mathbb{Y}}$ is evenly covered by $f$ if $f_{|U}:U\rightarrow V$ is a homeomorphism for each component $U$ of $f^{-1}(V)$; we say that $y\in{\mathbb{Y}}$ is a regular value for $f$ if some neighborhood $V\ni y$ is evenly covered, and a singular value for $f$ otherwise. Note that the set ${{\mathcal{S}}_{f}}$ of singular values is closed. Recall that $x\in{\mathbb{X}}$ is a critical point if the derivative of $f$ at $x$ vanishes, and then $f(x)\in{\mathbb{Y}}$ is a critical value. We say that $y\in{\mathbb{Y}}$ is an asymptotic value if $f$ approaches $y$ along some path tending to infinity relative to ${\mathbb{X}}$. It follows from elementary covering space theory that the critical values together with the asymptotic values form a dense subset of ${{\mathcal{S}}_{f}}$. In particular, every isolated point of ${{\mathcal{S}}_{f}}$ is a critical or asymptotic value.

An analytic map $f:{\mathbb{X}}\rightarrow{\mathbb{Y}}$ of complex $1$-manifolds is of finite type if

• •

$f$ is nowhere locally constant,

• •

$f$ has no isolated removable singularities,

• •

${\mathbb{Y}}$ is a finite union of compact Riemann surfaces, and

• •

${{\mathcal{S}}_{f}}$ is finite.

If ${\mathbb{Y}}$ is connected, we define ${\rm deg}f$ as the finite or infinite number {\rm{\rm card}}\bigl{(}f^{-1}(y)\bigr{)} which is independent of $y\in{\mathbb{Y}}{\smallsetminus}{{\mathcal{S}}_{f}}$. When $f:{\mathbb{X}}\to{\mathbb{Y}}$ is a finite type analytic map with ${\mathbb{X}}\subseteq{\mathbb{Y}}$, we say that $f$ is a finite type analytic map on ${\mathbb{Y}}$.

We first prove that the maps $F_{\lambda}$ extend to finite type analytic maps.

1.1. Preliminaries

Set ${\mathbb{T}}:={\mathbb{C}}/{\mathbb{Z}}$ and $\Lambda:={\mathbb{T}}\times{\mathbb{C}}^{*}$. Let $F:\Lambda\times{\mathbb{T}}\to{\mathbb{T}}$ be the holomorphic map defined by

[TABLE]

For $\lambda\in\Lambda$, let $F_{\lambda}:{\mathbb{T}}\to{\mathbb{T}}$ be the holomorphic map defined by

[TABLE]

It will be convenient to consider the global coordinate ${\mathbb{T}}\ni z\mapsto w:={\rm e}^{2\pi{\rm i}z}\in{\mathbb{C}}^{*}$. Note that $w:{\mathbb{T}}\to{\mathbb{C}}^{*}$ is an isomorphism. Thus, adding two points denoted $z=+{\rm i}\infty$ (or $w=0$) and $z=-{\rm i}\infty$ (or $w=\infty$), ${\mathbb{T}}$ may be compactified into a Riemann surface ${\widehat{{\mathbb{T}}}}$ isomorphic to the Riemann sphere.

We will prove that for $\lambda\in\Lambda$, the map $F_{\lambda}:{\mathbb{T}}\to{\widehat{{\mathbb{T}}}}$ is a finite type analytic map on ${\widehat{{\mathbb{T}}}}$.

1.2. The singular set

Fix $\lambda:=(a,b)\in\Lambda$ and set $f:=F_{a,b}:{\mathbb{T}}\to{\widehat{{\mathbb{T}}}}$. Note that

[TABLE]

and

[TABLE]

In particular, $f$ has two critical points counting multiplicities: the solutions of $bw^{2}+2w+b=0$, i.e., the points $c^{\pm}\in{\rm T}$ such that

[TABLE]

If $b\neq 1$, those are simple critical points of $f$. We denote by ${{\mathcal{C}}_{f}}\>=\{c^{+},c^{-}\}\subset{\mathbb{T}}$ the set of critical points of $f$ and by ${{\mathcal{V}}_{f}}:=f({{\mathcal{C}}_{f}})\subset{\mathbb{T}}$ the set of critical values of $f$.

Lemma 2**.**

The singular set ${{\mathcal{S}}_{f}}$ is equal to ${{\mathcal{V}}_{f}}\cup\{\pm{\rm i}\infty\}$.

Proof.

We already identified the set of critical values of $f$. Note that $\pm{\rm i}\infty$ are singular values since those points are omitted values. It is therefore enough to show that $f$ does not have any asymptotic value in ${\mathbb{T}}$.

If $v\in{\mathbb{T}}$ is an asymptotic value, then there exists a curve $\gamma:[0,1)\rightarrow{\mathbb{T}}$, such that $\gamma(t)\to\pm{\rm i}\infty$ and $f\circ\gamma(t)\to v$ as $t\to 1$. We assume that $\gamma(t)\rightarrow+{\rm i}\infty$. The proof for the case $\gamma(t)\rightarrow-{\rm i}\infty$ is analogous.

It is convenient to lift via the canonical covering $\boldsymbol{\pi}:{\mathbb{C}}\to{\mathbb{T}}:={\mathbb{C}}/{\mathbb{Z}}$. Choose $A\in{\mathbb{C}}$ such that $\boldsymbol{\pi}(A)=a$. Let $\tilde{f}:{\mathbb{C}}\to{\mathbb{C}}$ be defined by

[TABLE]

Let $\Gamma:[0,1)\to{\mathbb{C}}$ be a lift of $\gamma:[0,1)\to{\mathbb{T}}$, i.e., satisfying $\boldsymbol{\pi}\circ\Gamma=\gamma$. Then, $\tilde{f}\circ\Gamma$ is a lift of $f\circ\gamma$, thus $\tilde{f}\circ\Gamma(t)$ converges in ${\mathbb{C}}$ as $t\to 1$.

Set $X:={\rm Re}(\Gamma):[0,1)\to{\mathbb{R}}$ and $Y:={\rm Im}(\Gamma):[0,1)\to{\mathbb{R}}$. Then,

[TABLE]

and $Y(t)\to+\infty$ as $t\to 1$. It follows that as $t\to 1$,

[TABLE]

We can distinguish 2 cases. If there exists a sequence $\{t_{k}\}_{k\in{\mathbb{N}}}$ converging to $1$ with \bigl{\{}X(t_{k})\bigr{\}}_{k\in{\mathbb{N}}} bounded, then

[TABLE]

Otherwise, $X(t)\rightarrow\pm\infty$ as $t\rightarrow 1$ and there exists a sequence $\{t_{k}\}_{k\in{\mathbb{N}}}$ converging to $1$ with $X(t_{k})\in{\mathbb{Z}}$ for all $k\in{\mathbb{N}}$, so that

[TABLE]

In both cases, the sequence \bigl{\{}\tilde{f}\circ\Gamma(t_{k})\bigr{\}}_{k\in{\mathbb{N}}} cannot converge in ${\mathbb{C}}$. ∎

Corollary 3**.**

The map $f:{\mathbb{T}}\to{\widehat{{\mathbb{T}}}}$ is a finite type analytic map on ${\widehat{{\mathbb{T}}}}$. More precisely, $f:{\mathbb{T}}{\smallsetminus}f^{-1}({{\mathcal{V}}_{f}})\to{\mathbb{T}}{\smallsetminus}{{\mathcal{V}}_{f}}$ is a covering map.

2. Splitting triple zeros

In the remainder of the article, we fix a parameter $\lambda:=(a,b)\in{\mathbb{R}}/{\mathbb{Z}}\times(0,1)$ such that $F_{\lambda}^{\circ n}-{\rm id}$ has a triple zero $x\in{\mathbb{R}}/{\mathbb{Z}}$. We set $f:=F_{\lambda}:{\mathbb{T}}\to{\mathbb{T}}$. The point $x$ is periodic for $f$ with period $p$ dividing $n$. For $k\geq 0$, we set $x_{k}:=f^{\circ k}(x)$ and we denote by $\left<x\right>:=\{x_{0},x_{1},\ldots,x_{p-1}\}$ the cycle of $x$.

Since $f:{\mathbb{R}}/{\mathbb{Z}}\to{\mathbb{R}}/{\mathbb{Z}}$ preserves the orientation, the multiplier of $f^{\circ p}$ at $x$ is necessarily $1$ and there is a local coordinate $\zeta:({\mathbb{T}},x)\to({\mathbb{C}},0)$ vanishing at $x$ satisfying

[TABLE]

According to the Weierstrass Preparation Theorem, there exist a neighborhood $W_{1}\subset\Lambda$ of $\lambda$, a neighborhood $W_{2}\subset{\mathbb{C}}$ of [math] and analytic functions $A:W_{1}\to{\mathbb{C}}$, $B:W_{1}\to{\mathbb{C}}$, $C:W_{1}\to{\mathbb{C}}$ and $g:W_{1}\times W_{2}\to{\mathbb{C}}$ such that for $\mu\in W_{1}$,

[TABLE]

with

[TABLE]

and

[TABLE]

The polynomial $P_{\lambda}$ has a zero of multiplicity $3$ at [math], and as $\mu$ varies in $W_{1}$, this zero splits in three zeros (counting multiplicities) of $P_{\mu}$. When $\mu\in{\mathbb{R}}/{\mathbb{Z}}\times(0,1)$, the map $F_{\mu}^{\circ n}-{\rm id}$ commutes with $z\mapsto\bar{z}$, so that the polynomial $P_{\mu}$ has real coefficients. For such a parameter $\mu$, a multiple zero of $P_{\mu}$ is necessarily real.

For any $\mu\in\Lambda$, the function $\zeta\circ F_{\mu}^{\circ p}-\zeta$ vanishes at the periodic points of $F_{\mu}$ of period dividing $p$, and so, divides $\zeta\circ F_{\mu}^{\circ n}-\zeta$ which vanishes at the periodic points of period dividing $n$. In addition, if $n=mp$, then $\zeta\circ f^{\circ n}-\zeta=m\zeta^{3}+{\mathcal{O}}(\zeta^{5})$. So, there is an analytic function $h:W_{1}\times W_{2}\to{\mathbb{C}}$ such that for $\mu\in W_{1}$,

[TABLE]

Since $f$ only has two critical points in ${\mathbb{T}}$, it has a single non-repelling cycle, that is, the cycle $\left<x\right>$. All other cycles of $f$ in ${\mathbb{R}}/{\mathbb{Z}}$ are repelling. Shrinking $W_{1}$ is necessary, it follows that for $\mu\in W_{1}$, the function $\zeta\circ F_{\mu}^{\circ n}-\zeta$ has a multiple zero in ${\mathbb{R}}/{\mathbb{Z}}$ if and only if the polynomial $P_{\mu}$ has a multiple zero in ${\mathbb{R}}/{\mathbb{Z}}$. According to the previous discussion, this is the case if and only if $P_{\mu}$ has a multiple zero.

Let $\alpha:W_{1}\to{\mathbb{C}}$ and $\beta:W_{1}\to{\mathbb{C}}$ be defined by

[TABLE]

Then,

[TABLE]

So, if $\mu\in W_{1}$, the polynomial $P_{\mu}$ has a multiple zero if and only if $\beta^{3}(\mu)=\alpha^{2}(\mu)$.

In order to prove Theorem 1, it is therefore enough to show that $(\alpha,\beta)$ is a system of local coordinates near $\lambda$. For this purpose, we shall show that the restrictions of ${\rm d}\alpha$ and ${\rm d}\beta$ to ${\rm T}_{\lambda}\Lambda$ are linearly independent. Since $A$, $B$ and $C$ vanish at $\lambda$,

[TABLE]

It is therefore enough to show that the forms ${\rm d}A|_{{\rm T}_{\lambda}\Lambda}$ and ${\rm d}B|_{{\rm T}_{\lambda}\Lambda}$ are linearly independent.

3. Identifying the derivatives

Here, we identify ${\rm d}A(v)$ and ${\rm d}B(v)$ for $v\in{\rm T}_{\lambda}\Lambda$. First, to each $v\in{\rm T}_{\lambda}\Lambda$, we shall associate a meromorphic vector field $\vartheta_{v}$ on ${\mathbb{T}}$ having simple poles along ${{\mathcal{C}}_{f}}\cup\{\pm{\rm i}\infty\}$, such that for all $z\in{\mathbb{T}}{\smallsetminus}{{\mathcal{C}}_{f}}$,

[TABLE]

Second, for $k\in[1,p]$, let $\zeta_{k}:({\mathbb{T}},x_{k})\to({\mathbb{C}},0)$ be the local coordinate vanishing at $x_{k}$ defined by

[TABLE]

Our identification goes as follows.

Proposition 4**.**

Let $q_{A}$ and $q_{B}$ be quadratic differentials, defined and meromorphic near $\left<x\right>$, such that $q_{A}-({\rm d}\zeta_{k})^{2}/\zeta_{k}$ and $q_{B}-({\rm d}\zeta_{k})^{2}/\zeta_{k}^{2}$ are holomorphic at $x_{k}$ for all $k\in[1,p]$. Then, for all $v\in{\rm T}_{\lambda}\Lambda$,

[TABLE]

In the remaining parts of this section we prove Proposition 4.

3.1. Meromorphic vector fields

Assume $v\in{\rm T}_{\lambda}\Lambda$ and $z\in{\mathbb{T}}{\smallsetminus}{{\mathcal{C}}_{f}}$. Then, the derivative ${\mathrm{D}}_{z}f:{\rm T}_{z}{\mathbb{T}}\to{\rm T}_{f(z)}{\mathbb{T}}$ is an isomorphism and ${\mathrm{D}}_{\lambda,z}F(v,0)\in{\rm T}_{f(z)}{\mathbb{T}}$. Let $\vartheta_{v}$ be the vector field defined on ${\mathbb{T}}{\smallsetminus}{{\mathcal{C}}_{f}}$ by

[TABLE]

Lemma 5**.**

For all $v\in{\rm T}_{\lambda}\Lambda$, the vector field $\vartheta_{v}$ is holomorphic on ${\mathbb{T}}{\smallsetminus}{{\mathcal{C}}_{f}}$, meromorphic on ${\widehat{{\mathbb{T}}}}$, vanishes at $z=\pm{\rm i}\infty$ and has at worst simple poles along ${{\mathcal{C}}_{f}}$.

Proof.

The map $v\mapsto\vartheta_{v}$ is linear. So, it is enough to prove the result for $v_{a}:={\rm d}/{\rm d}a$ and $v_{b}:={\rm d}/{\rm d}b$. We have

[TABLE]

and

[TABLE]

Those two vector fields have the required properties. ∎

Denote by ${{\mathcal{T}}_{f}}$ the space of meromorphic vector fields on ${\widehat{{\mathbb{T}}}}$ which are holomorphic on ${\mathbb{T}}{\smallsetminus}{{\mathcal{C}}_{f}}$, vanish at $\pm{\rm i}\infty$ and have at worst simple poles along ${{\mathcal{C}}_{f}}$. In other words,

[TABLE]

Let $\Theta_{f}:{\rm T}_{\lambda}\Lambda\to{{\mathcal{T}}_{f}}$ be the linear map defined by

[TABLE]

Let $\tau\in{{\mathcal{T}}_{f}}$ be the radial vector field

[TABLE]

Note that $\tau-f^{*}\tau$ belongs to ${{\mathcal{T}}_{f}}$. Indeed,

[TABLE]

Lemma 6**.**

The space ${{\mathcal{T}}_{f}}$ is the direct sum of the image of $\Theta_{f}$ and the line spanned by $\tau-f^{*}\tau$:

[TABLE]

Proof.

The dimension of ${{\mathcal{T}}_{f}}$ is $3$. Thus, it is enough to show that the three vector fields $\vartheta_{v_{a}}$, $\vartheta_{v_{b}}$ and $\tau-f^{*}\tau$ are linearly independent. Equivalently, it is enough to show that the three functions

[TABLE]

are linearly independent. This is true since $b\neq 0$. ∎

Assume now $v\in{\rm T}_{\lambda}\Lambda$ and let $t\mapsto\lambda_{t}\in\Lambda$ be a curve such that $\dot{\lambda}=v$. Let $t\mapsto f_{t}$ be the family of maps defined by

[TABLE]

Then, for each $z\in{\mathbb{T}}$,

[TABLE]

Lemma 7**.**

For all $k\geq 1$,

[TABLE]

Proof.

The proof follows from an elementary induction on $k\geq 1$ using the following fact: if $h_{t}=g_{t}\circ f_{t}$ with $\dot{f}={\mathrm{D}}f\circ\vartheta$ and $\dot{g}={\mathrm{D}}g\circ\tau$, then

[TABLE]

Note that the poles of $\vartheta_{v}^{n}$ are the critical points of $f$ and their iterated preimages (up to order $n-1$). The two critical points of $f$ are in ${\mathbb{T}}{\smallsetminus}{\mathbb{R}}/{\mathbb{Z}}$, and so are all their preimages. Therefore, $\vartheta_{v}^{n}$ is holomorphic in a neighborhood of ${\mathbb{R}}/{\mathbb{Z}}$. In particular, it is holomorphic near the parabolic periodic point $x\in{\mathbb{R}}/{\mathbb{Z}}$.

3.2. Polar parts of quadratic differentials

Our identification of the derivatives ${\rm d}A|_{{\rm T}_{\lambda}\Lambda}$ and ${\rm d}B|_{{\rm T}_{\lambda}\Lambda}$ relies on the use of quadratic differentials (see Appendix A for basics regarding quadratic differentials). Recall that $\zeta:({\mathbb{T}},x)\to({\mathbb{C}},0)$ is a local coordinate vanishing at $x$ such that

[TABLE]

We shall use the quadratic differential $({\rm d}\zeta)^{2}/\zeta$ and $({\rm d}\zeta)^{2}/\zeta^{2}$ which are defined and meromorphic near $x$ in ${\mathbb{T}}$.

Following §A.7, if $Z\subset{\rm T}$ is a finite set, if $q$ is a quadratic differential, defined and meromorphic near $Z$ and if $\vartheta$ is a vector field, defined and meromorphic near $Z$, we shall use the notation

[TABLE]

If $q$ has at worst simple poles along $Z$ and if $\theta$ is defined on $Z$ with $\theta(z)\in{\rm T}_{z}{\mathbb{T}}$ for $z\in Z$, we shall use the notation

[TABLE]

where $\vartheta$ is any vector field, defined and holomorphic near $Z$, with $\vartheta(z)=\theta(z)$ for $z\in Z$. The result does not depend on the choice of extension.

Lemma 8**.**

For all $v\in{\rm T}_{\lambda}\Lambda$,

[TABLE]

Proof.

According to Equations (2), (3) and (4),

[TABLE]

Taking the derivative with respect to $t$ and evaluating at $t=0$ yields

[TABLE]

According to Equation (1),

[TABLE]

As a consequence

[TABLE]

Thus,

[TABLE]

and similarly

[TABLE]

Rather than working near $x$ with the vector field $\vartheta_{v}^{p}$, it will be convenient to work along the cycle $\left<x\right>$ with the vector field $\vartheta_{v}$. Recall that for $k\in[1,p]$, the local coordinate $\zeta_{k}:({\mathbb{T}},x_{k})\to({\mathbb{C}},0)$ vanishes at $x_{k}$ and is defined by

[TABLE]

Lemma 9**.**

For all $k\in{\mathbb{Z}}/p{\mathbb{Z}}$,

[TABLE]

are holomorphic near $x_{k}$.

Proof.

If $k\in[1,p-1]$, then $\zeta_{k}=\zeta_{k+1}\circ f$, so that

[TABLE]

If $k=p$, then $\zeta_{p}=\zeta$ and \zeta_{1}\circ f=\zeta\circ f^{\circ p}=\bigl{(}1+{\mathcal{O}}(\zeta_{p}^{2})\bigr{)}\zeta_{p}. As a consequence, f^{*}({\rm d}\zeta_{1})=\bigl{(}1+{\mathcal{O}}(\zeta_{p}^{2})\bigr{)}{\rm d}\zeta_{p},

[TABLE]

Proof of Proposition 4.

Recall that $\zeta_{p}=\zeta$. According to the previous lemma, for all $k\in{\mathbb{Z}}/p{\mathbb{Z}}$,

[TABLE]

is holomorphic near $x$. By assumption, $q_{A}-({\rm d}\zeta_{k})^{2}/\zeta_{k}$ is holomorphic at $x_{k}$. It follows that $(f^{\circ k})^{*}q_{A}-({\rm d}\zeta)^{2}/\zeta$ is holomorphic near $x$.

Since $(f^{\circ k})^{*}\vartheta_{v}$ is holomorphic near $x$, we therefore have

[TABLE]

As a consequence

[TABLE]

This proves Proposition 4 for ${\rm d}A$. The proof for ${\rm d}B$ is similar. ∎

4. Injectivity of $\nabla_{f}$

In order to prove Theorem 1, we need to use the global properties of the map $f$. Up to now, we only used the local properties near the cycle. For this purpose, it is important that the quadratic differentials $q_{A}$ and $q_{B}$ which appear in Proposition 4 are globally meromorphic on ${\widehat{{\mathbb{T}}}}$. Here, we define such quadratic differentials $q_{A}$ and $q_{B}$ and we prove that the linear map

[TABLE]

is well defined and injective on ${\rm Vect}(q_{A},q_{B})$.

4.1. A space of quadratic differentials

Denote by ${\mathcal{Q}}({\mathbb{T}})$ the space of meromorphic quadratic differentials on ${\widehat{{\mathbb{T}}}}$ which have at worst simple poles at $z=\pm{\rm i}\infty$. Given $Z\subset{\mathbb{T}}$, denote by ${\mathcal{Q}}({\mathbb{T}};Z)\subset{\mathcal{Q}}({\mathbb{T}})$ the subspace of quadratic differentials which are holomorphic outside $Z$. Finally, we denote by ${\mathcal{Q}}^{1}({\mathbb{T}};Z)\subset{\mathcal{Q}}({\mathbb{T}};Z)$ the subspace of quadratic differentials having at worst simple poles.

Lemma 10**.**

Any polar part of quadratic differential along $\left<x\right>$ may be realized as the polar part of a quadratic differential in {\mathcal{Q}}\bigl{(}{\mathbb{T}};\left<x\right>\cup\{c^{+}\}\bigr{)} having at worst a simple pole at $c^{+}$.

Proof.

For all $k\in[0,p-1]$, the quadratic differentials

[TABLE]

belong to {\mathcal{Q}}\bigl{(}{\mathbb{T}};\left<x\right>\cup\{c^{+}\}\bigr{)}. The first has a simple pole at $x_{k}$, the second has a double pole at $x_{k}$, and the third has a pole of order $j\geq 3$ at $x_{k}$. Thus, they generate the space of polar parts at $x_{k}$. ∎

From now on, we assume that q_{A}\in{\mathcal{Q}}\bigl{(}{\mathbb{T}};\left<x\right>\cup\{c^{+}\}\bigr{)} and q_{B}\in{\mathcal{Q}}\bigl{(}{\mathbb{T}};\left<x\right>\cup\{c^{+}\}\bigr{)} have at worst simple poles at $c^{+}$ and that $q_{A}-({\rm d}\zeta_{k})^{2}/\zeta_{k}$ and $q_{B}-({\rm d}\zeta_{k})^{2}/\zeta_{k}^{2}$ are holomorphic at $x_{k}$ for all $k\in[0,p-1]$. We set

[TABLE]

4.2. Pairing quadratic differentials in ${{\mathcal{Q}}_{f}}$ with vector fields in ${{\mathcal{T}}_{f}}$

Recall that

[TABLE]

Lemma 11**.**

For all $q\in{{\mathcal{Q}}_{f}}$,

[TABLE]

Proof.

Assume $q\in{{\mathcal{Q}}_{f}}$. According to Lemma 9, $q-f^{*}q$ is holomorphic near $\left<x\right>$. Since $\tau$ is also holomorphic near $\left<x\right>$, and since $f$ is a local isomorphism near $\left<x\right>$,

[TABLE]

4.3. Pushing forward quadratic differentials in ${\mathcal{Q}}_{f}$

According to Corollary 3, $f:{\mathbb{T}}{\smallsetminus}f^{-1}({{\mathcal{V}}_{f}})\to{\mathbb{T}}{\smallsetminus}{{\mathcal{V}}_{f}}$ is a covering map. Here, we show that for all $q\in{\mathcal{Q}}$, the following series defines a meromorphic quadratic differential on ${\widehat{{\mathbb{T}}}}$:

[TABLE]

The (minor) difficulty is that the degree of the covering map is not finite, and that $q$ may fail to be integrable on ${\widehat{{\mathbb{T}}}}$ since it may have multiple poles along $\left<x\right>$. So, we cannot apply directly the results presented in Appendix A. The reason why the series in Equation (5) converges is that $q$ is locally integrable near the essential singularities of $f$, i.e., the points $\pm{\rm i}\infty$.

Lemma 12**.**

If $q\in{{\mathcal{Q}}_{f}}$, the series in Equation (5) converges locally uniformly in {\mathbb{T}}{\smallsetminus}\bigl{(}{{\mathcal{V}}_{f}}\cup\left<x\right>\bigr{)}. Its sum $f_{*}q$ is a meromorphic quadratic differential on ${\widehat{{\mathbb{T}}}}$.

Proof.

Assume $q\in{{\mathcal{Q}}_{f}}$. Let V\subset{\mathbb{T}}{\smallsetminus}\bigl{(}{{\mathcal{V}}_{f}}\cup\left<x\right>\bigr{)} is compactly contained in ${\widehat{{\mathbb{T}}}}{\smallsetminus}\left<x\right>$. Then, $U:=f^{-1}(V)$ is compactly contained in ${\widehat{{\mathbb{T}}}}{\smallsetminus}\left<x\right>$. In particular, $q$ is integrable on $U$. In addition, $f:U\to V$ is a covering map. It follows that the series in Equation (5) converges uniformly on $V$ and that $f_{*}q$ is integrable on $V$. This shows $f_{*}q$ is holomorphic on {\mathbb{T}}{\smallsetminus}\bigl{(}{{\mathcal{V}}_{f}}\cup\left<x\right>\bigr{)} and has at worst simple poles at $\pm{\rm i}\infty$ and on ${{\mathcal{V}}_{f}}$.

To see that $f_{*}q$ is meromorphic near $x_{k}$, $k\in[1,p]$, let $V\subset{\mathbb{T}}{\smallsetminus}{{\mathcal{V}}_{f}}$ be a topological disk containing $x_{k}$. Then, $U:=f^{-1}(V)$ is the disjoint union of a topological disk $U^{\prime}$ containing $x_{k-1}$ and a open set $U^{\prime\prime}$ compactly contained in ${\widehat{{\mathbb{T}}}}{\smallsetminus}\left<x\right>$. Then, $(f|_{U^{\prime\prime}})_{*}q$ is holomorphic. In addition, $f:U^{\prime}\to U$ is an isomorphism so that $(f|_{U^{\prime}})_{*}q$ – and thus $f_{*}q=(f|_{U^{\prime}})_{*}q+(f|_{U^{\prime}})_{*}q$ – is meromorphic near $x_{k}$. ∎

We may therefore consider the linear map

[TABLE]

It will be convenient to set

[TABLE]

Lemma 13**.**

We have the inclusion

[TABLE]

Proof.

Assume $q\in{{\mathcal{Q}}_{f}}$. As mentioned in the proof of the previous lemma, $f_{*}q$ is holomorphic on {\mathbb{T}}{\smallsetminus}\bigl{(}{{\mathcal{V}}_{f}}\cup\left<x\right>\bigr{)} and has at worst simple poles at $\pm{\rm i}\infty$ and on ${{\mathcal{V}}_{f}}$. In addition, for $k\in[1,p]$, the polar part of $f_{*}q$ at $x_{k}$ is equal to the polar part of $g^{*}q$ where $g$ is the inverse branch of $f$ sending $x_{k}$ to $x_{k-1}$. According to Lemma 9, $q-f_{*}q$ is therefore holomorphic near $\left<x\right>$. It follows that $q-f_{*}q\in{\mathcal{Q}}^{1}({\mathbb{T}};Y)$. ∎

4.4. Injectivity of $\nabla_{f}$

An observation due to Adam Epstein is that the linear map $\nabla_{f}$ is injective on ${{\mathcal{Q}}_{f}}$, and that this is the key to the proof of Theorem 1.

Proposition 14**.**

The linear map $\nabla_{f}:{{\mathcal{Q}}_{f}}\to{\mathcal{Q}}({\mathbb{T}})$ is injective.

Proof.

We must prove that $f_{*}q\neq q$ for $q\in{{\mathcal{Q}}_{f}}{\smallsetminus}\{0\}$. If $q$ were integrable on ${\mathbb{T}}$, the result would follow immediately from Proposition 21, since we would have $\|f_{*}q\|_{L^{1}({\mathbb{T}})}<\|q\|_{L^{1}({\mathbb{T}})}$. Since $q$ may have double poles near $\left<x\right>$, it may fail to be integrable on ${\mathbb{T}}$. In that case, we may proceed as follows.

Assume $q\in{{\mathcal{Q}}_{f}}{\smallsetminus}\{0\}$. For ${\varepsilon}>0$ small, let $V_{\varepsilon}\subset{\rm T}$ be the union of topological disks

[TABLE]

Set $U_{\varepsilon}:=f^{-1}(V_{\varepsilon})\subset{\mathbb{T}}$. Then, $\left<x\right>\subset U_{\varepsilon}$ and so, $q$ is integrable on ${\mathbb{T}}{\smallsetminus}U_{\varepsilon}$. As a consequence,

[TABLE]

Similarly, for ${\varepsilon}^{\prime}<{\varepsilon}$,

[TABLE]

As a consequence, the function

[TABLE]

is positive and decreasing. In particular, it has a positive limit. Note that

[TABLE]

We deduce from the following lemma that $f_{*}q\neq q$. ∎

Lemma 15**.**

For any $q\in{{\mathcal{Q}}_{f}}$,

[TABLE]

Proof.

Since $\zeta\circ f^{\circ p}=\zeta+{\mathcal{O}}(\zeta^{3})$, there is a constant $\kappa_{1}$ such that

[TABLE]

Since $q$ has at worst a double pole at $x$, there is a constant $\kappa_{2}$ such that for $|\zeta|$ small enough

[TABLE]

Note that for ${\varepsilon}>0$ small enough,

[TABLE]

Thus,

[TABLE]

5. Linear independence

We may now complete the proof that ${\rm d}A|_{{\rm T}_{\lambda}\Lambda}$ and ${\rm d}B|_{{\rm T}_{\lambda}\Lambda}$ are linearly independent. According to Proposition 4, for all $v\in{\rm T}_{\lambda}\Lambda$,

[TABLE]

According to Lemma 6,

[TABLE]

Showing that ${\rm d}A|_{{\rm T}_{\lambda}\Lambda}$ and ${\rm d}B|_{{\rm T}_{\lambda}\Lambda}$ are linearly independent therefore amounts to proving that for all $q\in{{\mathcal{Q}}_{f}}{\smallsetminus}\{0\}$, there exists $\vartheta\in{{\mathcal{T}}_{f}}$ such that $\left<q,\vartheta\right>_{\left<x\right>}\neq 0$.

5.1. Guiding vector fields

Set $Z:={{\mathcal{C}}_{f}}\cup{{\mathcal{V}}_{f}}$ and denote by ${\rm T}Z$ the space of maps $\xi:Z\to{\rm T}{\mathbb{T}}$ satisfying $\xi(z)\in{\rm T}_{z}{\mathbb{T}}$ for all $z\in{\mathbb{Z}}$.

Lemma 16**.**

For any $\vartheta\in{{\mathcal{T}}_{f}}$, there exists a unique $\xi_{\vartheta}\in{\rm T}Z$ such that for any vector field $\xi$, defined and holomorphic near $Z$ with $\xi(z)=\xi_{\vartheta}(z)$ for $z\in Z$, the vector field $\vartheta+\xi-f^{*}\xi$ is holomorphic and vanishes along ${{\mathcal{C}}_{f}}$.

Proof.

Fix $\vartheta\in{{\mathcal{T}}_{f}}$. Let us first prove the uniqueness of $\xi_{\vartheta}\in{\rm T}Z$. Assume $\xi_{1}$ and $\xi_{2}$ are two vector fields, defined and holomorphic near $Z$, such that $\vartheta+\xi_{1}-f^{*}\xi_{1}$ and $\vartheta+\xi_{2}-f^{*}\xi_{2}$ are holomorphic near ${{\mathcal{C}}_{f}}$. Then, $(\xi_{1}-\xi_{2})-f^{*}(\xi_{1}-\xi_{2})$ is holomorphic and vanishes along ${{\mathcal{C}}_{f}}$. As a consequence, ${\mathrm{D}}f\circ(\xi_{1}-\xi_{2})-(\xi_{1}-\xi_{2})\circ f$ vanishes on ${{\mathcal{C}}_{f}}$. Since ${\mathrm{D}}f\circ(\xi_{1}-\xi_{2})$ vanishes on ${{\mathcal{C}}_{f}}$, this forces $\xi_{1}-\xi_{2}$ to vanish on ${{\mathcal{V}}_{f}}$. In that case, $f^{*}(\xi_{1}-\xi_{2})$ vanishes on ${{\mathcal{C}}_{f}}$ and so, $\xi_{1}-\xi_{2}$ vanishes on ${{\mathcal{C}}_{f}}$. This shows the uniqueness of $\xi_{\vartheta}\in{\rm T}Z$.

This also proves that if $\vartheta+\xi-f^{*}\xi$ is holomorphic and vanishes along ${{\mathcal{C}}_{f}}$ for some vector field $\xi$, defined and holomorphic near $Z$ with $\xi(z)=\xi_{\vartheta}(z)$ for $z\in Z$, then $\vartheta+\xi-f^{*}\xi$ is holomorphic and vanishes along ${{\mathcal{C}}_{f}}$ for any vector field $\xi$, defined and holomorphic near $Z$ with $\xi(z)=\xi_{\vartheta}(z)$ for $z\in Z$.

Let us now prove the existence of $\xi_{\vartheta}\in{\rm T}Z$. Note that ${\mathrm{D}}f\circ\vartheta$ is a map from ${\mathbb{T}}{\smallsetminus}{{\mathcal{C}}_{f}}$ to the tangent bundle ${\rm T}{\mathbb{T}}$. Note that it is not a vector field since for $z\in{\mathbb{T}}$, the vector ${\mathrm{D}}f\circ\vartheta(z)$ belongs to ${\rm T}_{f(z)}{\mathbb{T}}$. However, since $\vartheta$ has at worst simple poles along ${{\mathcal{C}}_{f}}$ and since ${\mathrm{D}}f$ vanishes on ${{\mathcal{C}}_{f}}$, the map ${\mathrm{D}}f\circ\vartheta$ extends holomorphically to ${\mathbb{T}}$. Set

[TABLE]

Next, let $\xi$ be any vector field, defined and holomorphic near ${{\mathcal{V}}_{f}}$, coinciding with $\xi_{\vartheta}$ on ${{\mathcal{V}}_{f}}$. Then, $\vartheta-f^{*}\xi$ is holomorphic near ${{\mathcal{C}}_{f}}$ and we may set

[TABLE]

Recall that $Y:=\{c^{+}\}\cup{{\mathcal{V}}_{f}}\subset Z$. It will be convenient to consider the linear map $\Xi_{f}:{{\mathcal{T}}_{f}}\to{\rm T}Y$ defined by

[TABLE]

Lemma 17**.**

The map $\Xi_{f}:{{\mathcal{T}}_{f}}\to{\rm T}Y$ is an isomorphism.

Proof.

Since the dimensions of ${{\mathcal{T}}_{f}}$ and ${\rm T}Y$ are both equal to three, it is enough to show that the map is injective. Assume $\vartheta\in{{\mathcal{T}}_{f}}$ and $\xi_{\vartheta}$ vanishes on $\{c^{+}\}\cup{{\mathcal{V}}_{f}}$. Let $\xi$ be a vector field, defined and holomorphic near $Z$, which coincides with $\xi_{\vartheta}$ on $Z$. We may assume that $\xi$ identically vanishes near $\{c^{+}\}\cup{{\mathcal{V}}_{f}}$. Then, $\vartheta+f^{*}\xi-\xi=\vartheta-\xi$ is holomorphic and vanishes on ${{\mathcal{C}}_{f}}$. This shows that $\vartheta$ is holomorphic near ${{\mathcal{C}}_{f}}$ and vanishes at $c^{+}$. As a consequence, $\vartheta$ is globally holomorphic on ${\widehat{{\mathbb{T}}}}$, and vanishes at three points: $c^{+}$, $+{\rm i}\infty$ and $-{\rm i}\infty$. So, $\vartheta=0$. ∎

5.2. From the cycle to the critical set

We may now transfer the local computations done near the cycle $\left<x\right>$ to local computations done near the critical set $Y$.

Lemma 18**.**

For all $\vartheta\in{{\mathcal{T}}_{f}}$ and all $q\in{{\mathcal{Q}}_{f}}$,

[TABLE]

Proof.

Let $\xi$ be a vector field, defined and holomorphic near $Z$, coinciding with $\xi_{\vartheta}:=\Xi_{f}(\vartheta)$ on $Z$. Then, $\vartheta+\xi-f^{*}\xi$ is holomorphic near ${{\mathcal{C}}_{f}}$. In addition, since $\nabla_{f}q$ is holomorphic near $c^{-}$,

[TABLE]

In the second line, we used the fact that the only poles of $q\otimes f^{*}\xi$ in $f^{-1}({{\mathcal{V}}_{f}})$ belong to ${{\mathcal{C}}_{f}}$. For the last equality, we used the fact that $q\otimes\vartheta$ is a globally meromorphic $1$-form on ${\widehat{{\mathbb{T}}}}$, whose poles are contained in ${{\mathcal{C}}_{f}}\cup\left<x\right>$, and that the sum of all residues of a globally meromorphic $1$-form on a compact Riemann surface is [math]. ∎

5.3. Completion of the proof

Assume by contradiction that ${\rm d}A|_{{\rm T}_{\lambda}\Lambda}$ and ${\rm d}B|_{{\rm T}_{\lambda}\Lambda}$ are not linearly independent. Then, there is a $q\in{{\mathcal{Q}}_{f}}{\smallsetminus}\{0\}$ such that for all $\vartheta\in{{\mathcal{T}}_{f}}$,

[TABLE]

According to Lemma 17, the map $\Xi_{f}:{{\mathcal{T}}_{f}}\to{\rm T}Y$ is an isomorphism. In particular, it is surjective. It follows that for all $\xi\in{\rm T}Y$,

[TABLE]

As a consequence, $\nabla_{f}q$ is holomorphic near $Y$ and thus, has at most three simple poles at $c^{-}$, $+{\rm i}\infty$ and $-{\rm i}\infty$. A non zero quadratic differential on ${\widehat{{\mathbb{T}}}}$ has at least four poles, counting multiplicities. Thus, $\nabla_{f}q=0$.

According to Proposition 14, the map $\nabla_{f}:{{\mathcal{Q}}_{f}}\to{\mathcal{Q}}^{1}({\mathbb{T}};Y)$ is injective. It follows that $q=0$. Contradiction.

This completes the proof of Theorem 1.

Appendix A Quadratic differentials

A.1. Meromorphic quadratic differentials

A quadratic differential on a Riemann surface $U$ is a section of the square of the cotangent bundle ${\rm T}^{*}U\otimes{\rm T}^{*}U$. We shall usually think of a quadratic differential $q$ as a field of quadratic forms. In particular, if $\vartheta$ is a vector field on $U$ and $\phi$ is a function on $U$, then $q(\vartheta)$ is a function on $U$ and $q(\phi\vartheta)=\phi^{2}q(\vartheta)$.

If $\zeta:U\to{\mathbb{C}}$ is a coordinate, we shall use the notation $({\rm d}\zeta)^{2}={\rm d}\zeta\otimes{\rm d}\zeta$ - not be confused with $1$-form ${\rm d}(\zeta^{2})$. Then, a quadratic differential $q$ on $U$ is of the form $q=\phi\ ({\rm d}\zeta)^{2}$ for some function $\phi$. We say that $q$ is meromorphic on $U$ if $\phi$ is meromorphic on $U$. In that case, the order of $q$ at a point $x\in U$ is ${\rm ord}_{x}q:={\rm ord}_{x}\phi$, i.e., [math] if $\phi$ is holomorphic and does not vanish at $x$, $k\geq 1$ if $\phi$ has a zero of multiplicity $k$ at $x$, and $-k\leq-1$ if $\phi$ has a pole of multiplicity $k$ at $x$.

A.2. Pullback

The derivative ${\mathrm{D}}f:{\rm T}U\to{\rm T}V$ of a holomorphic map $f:U\to V$ naturally induces a pullback map $f^{*}$ from quadratic differentials on $V$ to quadratic differentials on $U$:

[TABLE]

Lemma 19**.**

If $f:(U,x)\to(V,y)$ is holomorphic at $x$, and $q$ is meromorphic at $y=f(x)$, then

[TABLE]

Proof.

Choose local coordinates $z:(U,x)\to({\mathbb{C}},0)$ and $w:(V,y)\to({\mathbb{C}},0)$ such that $w\circ f=z^{k}$, with $k:={\rm deg}_{x}f$. If $q=\phi\ ({{\rm d}w})^{2}$, then $f^{*}q=\phi\circ f\cdot(kz^{k-1}{{\rm d}z})^{2}$. Thus,

[TABLE]

A.3. Pushforward for finite degree covering maps

Assume $f:U\to V$ is a finite degree covering map. If $q$ is a quadratic differential on $U$, we define a quadratic differential $f_{*}q$ on $V$ by

[TABLE]

If $q$ is holomorphic on $U$, then $f_{*}q$ is holomorphic on $V$.

Lemma 20**.**

Assume $U:=\widehat{U}{\smallsetminus}\{x\}$ and $V:=\widehat{V}{\smallsetminus}\{y\}$ are punctured disks, $f:U\to V$ is a covering map ramifying at $x$ with local degree ${\rm deg}_{x}f$ and $q$ is meromorphic at $x$. Then, $f_{*}q$ has at worst a pole at $y$ and

[TABLE]

Proof.

The group of deck transformations of $f:U\to V$ is a cyclic group of order ${\rm deg}_{x}f$. Note that

[TABLE]

and ${\rm ord}_{x}h^{*}q={\rm ord}_{x}q$ for all deck transformations $h$, so that

[TABLE]

Then,

[TABLE]

A.4. Integrable quadratic differentials

If $q$ is a quadratic differential on $U$, we denote by $|q|$ the positive $(1,1)$-form on $U$ defined by

[TABLE]

If $\zeta:U\to{\mathbb{C}}$ is a coordinate and $q=\phi\ ({\rm d}\zeta)^{2}$, then

[TABLE]

We shall say that $q$ is integrable on $U$ if

[TABLE]

Note that $q$ is integrable in a neighborhood of a pole if and only if the pole is simple. If $f:U\to V$ is an isomorphism and $q$ is an integrable quadratic differential on $V$, then $f^{*}q$ is integrable on $U$ and $\|f^{*}q\|_{L^{1}(U)}=\|q\|_{L^{1}(V)}$.

A.5. Pushforward for infinite degree covering maps

Assume $f:U\to V$ is an infinite degree covering map. If $q$ is an integrable quadratic differential on $U$, we may still define

[TABLE]

Indeed, the series converges in $L^{1}_{\rm loc}$ since if $V^{\prime}\subset V$ is a topological disk, so that the inverse branches $g:V^{\prime}\to U$ of $f$ are defined on $V^{\prime}$, and if $U^{\prime}:=f^{-1}(V^{\prime})$, then

[TABLE]

The limit of a sequence of holomorphic functions converging in $L^{1}_{\rm loc}$ is itself holomorphic. It follows that if $q$ is holomorphic on $U$, then $f_{*}q$ is holomorphic on $V$.

A.6. The Contraction Principle

Proposition 21**.**

Let $f:U\to V$ be a covering map and let $q$ be a holomorphic integrable quadratic differential on $U$. Then, $\|f_{*}q\|_{L^{1}(V)}\leq\|q\|_{L^{1}(U)}$ and equality holds if and only if either $q=0$, or the degree of $f$ is finite and $f^{*}(f_{*}q)={\rm deg}(f)\cdot q$.

Proof.

The proof is an immediate application of the triangle inequality: for any topological disk $V^{\prime}\subset V$, we have

[TABLE]

where the sums range over the inverse branches $g:V^{\prime}\to U$ of $f$. It follows that

[TABLE]

with equality if and only if for all inverse branches $g$ of $f$, we have $g^{*}q=\psi_{g}\ f_{*}q$ for some function $\psi_{g}:V^{\prime}\to[0,1]$ satisfying $\sum_{g}\psi_{g}=1$. Setting \phi\bigl{(}g(y)\bigr{)}:=\psi_{g}(y), we see that $q=\phi\ f^{*}(f_{*}q)$ for some function $\phi:U\to[0,1]$. Since $q$ and $f^{*}(f_{*}q)$ are holomorphic, either $q=0$, or the function $\phi$ is constant, let us say equal to $c\in[0,1]$. Since $\sum_{g}\psi_{g}=1$, we have that ${\rm deg}(f)\cdot c=1$, which forces the degree of $f$ to be finite with $f^{*}(f_{*}q)={\rm deg}(f)\cdot q$. ∎

A.7. Pairing quadratic differentials and vector fields

If $q$ is a quadratic differential on $U$ and $\vartheta$ is a vector field on $U$, we may consider the $1$-form $q\otimes\vartheta$ defined on $U$ by its action on vector fields $\tau$:

[TABLE]

Note that if $q=\phi\ ({\rm d}\zeta)^{2}$ and $\vartheta=\psi\ {\rm d}/{\rm d}\zeta$, then $q\otimes\vartheta=\phi\psi\ {\rm d}\zeta.$

If $x\in U$, and if $\vartheta$ and $q$ are meromorphic on $U$, we set

[TABLE]

If $q$ has at worst a simple pole at $x$, then $\left<q,\vartheta\right>_{x}$ only depends on $\theta:=\vartheta(0)$, and we use the notation

[TABLE]

Lemma 22**.**

Let $U:=\widehat{U}{\smallsetminus}\{x\}$ and $V:=\widehat{V}{\smallsetminus}\{y\}$ be punctured disks, let $f:U\to V$ be a covering map ramifying at $x$, let $q$ be a meromorphic quadratic differential on $\widehat{U}$ and let $\vartheta$ be a meromorphic vector field on $U$. Then

[TABLE]

Proof.

Let $\gamma\subset V$ be a loop around $y$ with basepoint $a$. Then

[TABLE]

where the sum ranges over the inverse branches $g$ of $f$ defined on $\gamma{\smallsetminus}\{a\}$. ∎