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# 数学代写|黎曼曲面代写Riemann surface代考|The Wolff lemma

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## 数学代写|黎曼曲面代写Riemann surface代考|The Wolff lemma

We now discuss the Wolff lemma, the second boundary version of the Schwarz lemma; in the next section, we shall see some of its consequences, mainly on the structure of the automorphism group of hyperbolic Riemann surfaces, even though for us the main applications will be in the study of the dynamics of holomorphic self-maps, as we shall see in the next chapter.

The original Schwarz lemma said something about functions $f \in \operatorname{Hol}(\mathbb{D}, \mathbb{D})$ with a fixed point in $\mathbb{D}$. We now assume instead as hypothesis that $f$ has no fixed points in $\mathbb{D}$. It turns out that then it exists a point $\tau \in \partial \mathbb{D}$ such that $f$ sends every horocycle centered in $\tau$ into itself, exactly as a function with a fixed point $z_0 \in \mathbb{D}$ sends every Poincaré ball centered in $z_0$ into itself. This is the content of the Wolff lemma.

Theorem 2.5.1 (Wolff lemma, 1926). Let $f \in \operatorname{Hol}(\mathbb{D}, \mathbb{D})$ be without fixed points. Then there is a unique $\tau \in \partial \mathbb{D}$ such that for all $z \in \mathbb{D}$,
$$\frac{|\tau-f(z)|^2}{1-|f(z)|^2} \leq \frac{|\tau-z|^2}{1-|z|^2},$$
i.e.,
$$f(E(\tau, R)) \subseteq E(\tau, R)$$
for all $R>0$. Moreover, the equality in (2.65) holds at one point (and hence everywhere) if and only if $f$ is a parabolic automorphism of $\mathbb{D}$ leaving $\tau$ fixed.

## 数学代写|黎曼曲面代写Riemann surface代考|The automorphism group of hyperbolic Riemann surfaces

The main applications of the Wolff lemma are in dynamics, as we shall see in the next chapter. Here, we shall instead describe a different application of the Wolff and Julia lemmas, with remarkable consequences for the study of Riemann surfaces.

In Proposition 1.4.12, we saw that two automorphisms of $\mathbb{D}$ commute if and only if they have the same fixed points. We shall now prove a first extension of that result.
Theorem 2.6.1. Let $y \in \operatorname{Aut}(\mathbb{D})$ be hyperbolic, and $f \in \operatorname{Hol}(\mathbb{D}, \mathbb{D})$ be such that
$$f \circ \gamma=y \circ f$$
Then either $f$ is a hyperbolic automorphism of $\mathbb{D}$ with the same fixed points as $y$ or $f \equiv \mathrm{id}_{\mathbb{D}}$.

Proof. Assume $f \neq \mathrm{id}_{\mathbb{D}}$; in particular, $f$ cannot have more than one fixed point (Corollary 1.1.14). If $f$ has a fixed point $z_0 \in \mathbb{D}$, then by (2.70)
$$f\left(\gamma\left(z_0\right)\right)=y\left(f\left(z_0\right)\right)=\gamma\left(z_0\right)$$
i. e., $\gamma\left(z_0\right)=z_0$, impossible. Hence $f$ is fixed point free and we can apply the Wolff lemma to get a point $\tau \in \partial \mathbb{D}$ satisfying (2.66). However, $\gamma(\tau)$ still satisfies (2.66), by (2.70) and Proposition 2.1.5; therefore, the uniqueness part of Wolff lemma implies $\gamma(\tau)=\tau$.

## 数学代写|黎曼曲面代写Riemann surface代考|The Wolff lemma

$$\frac{|\tau-f(z)|^2}{1-|f(z)|^2} \leq \frac{|\tau-z|^2}{1-|z|^2},$$

$$f(E(\tau, R)) \subseteq E(\tau, R)$$

## 数学代写|黎曼曲面代写Riemann surface代考|The automorphism group of hyperbolic Riemann surfaces

$$f \circ \gamma=y \circ f$$

$$f\left(\gamma\left(z_0\right)\right)=y\left(f\left(z_0\right)\right)=\gamma\left(z_0\right)$$

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