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# 数学代写|黎曼曲面代写Riemann surface代考|Random iteration of small perturbations

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## 数学代写|黎曼曲面代写Riemann surface代考|Random iteration of small perturbations

In this section, we shall discuss the behavior of iterated function systems generated by functions close enough to a given self-map $F$; in particular, we would like to understand whether the dynamics of the iterated function systems mimics the dynamics of the sequence of iterates of $F$.

Recalling Theorem 3.3.2, we see that we have three cases to consider: when $F$ has an attracting fixed point, when $F$ is a periodic or pseudoperiodic automorphism, and when the sequence $\left{F^k\right}$ is compactly divergent.
In the first case, we have a fairly complete result.
Theorem 3.7.1. Let $X$ be a hyperbolic Riemann surface and let $F \in \operatorname{Hol}(X, X)$ be with an attracting fixed point $z_0 \in X$. Then:
(i) there exists a neighborhoodU of $F$ in $\operatorname{Hol}(X, X)$ such that every right iterated function system generated by $\left{f_v\right} \subset \mathcal{U}$ converges to a constant in $X$;
(ii) if $\left{f_v\right} \subset \operatorname{Hol}(X, X)$ is a sequence converging to $F$, then the left iterated function system generated by $\left{f_v\right}$ converges to $z_0$.

## 数学代写|黎曼曲面代写Riemann surface代考|Discrete dynamics on the unit disk

The previous chapter was mostly devoted to the study of dynamics on general hyperbolic Riemann surfaces, even though we did prove at least one important theorem regarding the dynamics on the unit disk $\mathbb{D}$, namely the Wolff-Denjoy theorem. In this chapter, we shall instead concentrate on the dynamics in $\mathbb{D}$, obtaining deep and detailed results.

We shall encounter two main interrelated themes: the study of how the orbits approach the Wolff point and the study of the possible models (in the sense of Definition 3.5.2) a holomorphic self-map can have. When the Wolff point of $f \in \operatorname{Hol}(\mathbb{D}, \mathbb{D})$ is in $\mathbb{D}$, then the theory is relatively simple. If the derivative $f^{\prime}(\tau)$ at the Wolff point satisfies $0<\left|f^{\prime}(\tau)\right|<1$ (attracting elliptic case), then the orbits essentially behave like the orbits of the linear map $z \mapsto f^{\prime}(\tau) z$ in $\mathbb{D}$, which is the model for our $f$. When instead $f^{\prime}(\tau)=0$ (superattracting elliptic case) the model is given by a power map, and even though the intertwining map in general is not defined on the whole of $\mathbb{D}$, we shall anyway be able to understand how the orbits approach the Wolff point.

When the Wolff point $\tau$ belongs to the boundary, the situation becomes more complicated-and more interesting. In the hyperbolic case, i. e., when the angular derivative $f^{\prime}(\tau)$ at the Wolff point satisfies $0<f^{\prime}(\tau)<1$, it turns out that the orbits converge to the Wolff point nontangentially with a precise slope; furthermore, the model will be again given by the multiplication by $f^{\prime}(\tau)$ but on the upper half-plane, not in $\mathbb{D}$. When $f$ is parabolic, i.e., $f^{\prime}(\tau)=1$, it turns out that we have two different cases to consider. When the hyperbolic step, that is the limit of the Poincaré distance between two consecutive points in an orbit, is positive, then we shall see that the orbits approach the Wolff point tangentially; moreover, there are two possible models, both on $\mathbb{H}^{+}$but one given by $w \mapsto w+1$ and the other by $w \mapsto w-1$. When instead the hyperbolic step is zero, then there is only one model, given again by $w \mapsto w+1$ but this time on $\mathbb{C}$; furthermore, there are examples where the orbits converge tangentially to the Wolff point and examples where the orbits converge nontangentially to the Wolff point. To understand when a given parabolic map has positive or zero hyperbolic step and, in the latter case, decide how the orbits converge to the Wolff point is a problem not yet completely solved.

## 数学代写|黎曼曲面代写Riemann surface代考|Random iteration of small perturbations

(i)在$\operatorname{Hol}(X, X)$中存在$F$的邻域du，使得$\left{f_v\right} \subset \mathcal{U}$生成的每一个右迭代函数系统收敛于$X$中的一个常数;
(ii)如果$\left{f_v\right} \subset \operatorname{Hol}(X, X)$是收敛于$F$的序列，则$\left{f_v\right}$生成的左迭代函数系统收敛于$z_0$。

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