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# 数学代写|黎曼曲面代写Riemann surface代考|Continuous dynamics on Riemann surfaces

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## 数学代写|黎曼曲面代写Riemann surface代考|Continuous dynamics on Riemann surfaces

In this chapter, we shall look at dynamics from another point of view. Let $X$ be a Riemann surface; then $\operatorname{Hol}(X, X)$, endowed, as usual, with the compact-open topology, is a topological semigroup with identity, i. e., the operation given by the composition $(f, g) \mapsto f \circ g$ is continuous, associative, and has an identity. From this point of view, a semigroup homomorphism $\Phi: \mathbb{N} \rightarrow \operatorname{Hol}(X, X)$ is the same thing as the sequence of iterates of the single function $\Phi(1)$. In other words, in the previous chapters we have actually studied semigroup homomorphisms of $\mathbb{N}$ into $\operatorname{Hol}(X, X)$.

From this point of view, a natural generalization of the sequence of iterates is a one-parameter semigroup, i. e., a continuous semigroup homomorphism $\Phi: \mathbb{R}^{+} \rightarrow$ $\operatorname{Hol}(X, X)$. In this chapter, we shall thoroughly study these objects, aiming toward a complete classification. This will be possible because on Riemann surfaces with nonAbelian fundamental group every one-parameter semigroup $\Phi$ is trivial, i. e., $\Phi_t=\mathrm{id}_X$ for all $t \geq 0$. Furthermore, the one-parameter semigroups on other Riemann surfaces different from the disk can be classified (Section 5.3); so the main problem is the description of one-parameter semigroups on $\mathbb{D}$.

We shall actually provide several different descriptions of one-parameter semigroups on $\mathbb{D}$, useful in different contexts. We shall show how to relate one-parameter semigroups to Cauchy problems and ordinary differential equations, proving that a semigroup is completely determined by a holomorphic function $F: \mathbb{D} \rightarrow \mathbb{C}$, its infinitesimal generator. We shall give both a differential characterization and a completely explicit description of infinitesimal generators. Finally, we shall show how to replace $\mathbb{D}$ by another simply connected domain (in essentially a unique way) so to express a generic one-parameter semigroup in a particularly simple form; in a sense we shall transfer the analytic intricacies of one-parameter semigroups in a geometrically simple domain as $\mathbb{D}$ to the geometrical intricacies of a domain of definition for analytically very simple one-parameter semigroups, expressed in terms of affine maps.

## 数学代写|黎曼曲面代写Riemann surface代考|Algebraic semigroup homomorphisms

In this section, we collect some well-known facts about algebraic semigroups homomorphism of $\mathbb{R}^{+}$into other groups or semigroups that we shall need later. In this section, as operation on $\mathbb{R}^{+}$we shall always consider the sum, that makes $\mathbb{R}^{+}$in a semigroup but of course not a group. Moreover, we shall put $\mathbb{R}^{+}=(0,+\infty)$, so that $\left(\mathbb{R}^{+}, \cdot\right)$ is a topological group.

Definition 5.1.1. Let $G$ be a semigroup with identity element $e$. A function $\Phi: \mathbb{R}^{+} \rightarrow G$ is a semigroup homomorphism if $\Phi(0)=e$ and $\Phi(t+s)=\Phi(t) \circ \Phi(s)$ for all $t, s \geq 0$, where – denotes the operation in $G$. In the sequel, we shall often write $\Phi_t$ instead of $\Phi(t)$.
Lemma 5.1.2. Let $G$ be a group. Then:
(i) every semigroup homomorphism $\Phi: \mathbb{R}^{+} \rightarrow G$ can be extended in a unique way to a group homomorphism $\tilde{\Phi}: \mathbb{R} \rightarrow G$; in particular, if $G$ is a topological group and $\Phi$ is continuous, then $\Phi$ is continuous too;
(ii) if $G$ is finite, then every semigroup homomorphism $\Phi: \mathbb{R}^{+} \rightarrow G$ is trivial.
Proof. (i) The (unique) extension is obviously given by
$$\tilde{\Phi}(t)= \begin{cases}\Phi(t) & \text { if } t \geq 0 ; \ {[\Phi(-t)]^{-1}} & \text { if } t \leq 0,\end{cases}$$
where $[\cdot]^{-1}$ denotes the inverse operator in $G$. The continuity of $\tilde{\Phi}$ follows immediately from the continuity of the group operations.

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