# 数学代写|复分析作业代写Complex function代考|MATH3711

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## 数学代写|复分析作业代写Complex function代考|Convergence Properties of Biholomorphic Codes

Possibly, one could think that an endless elongation of the biholomorphic encoding rule, by using increasingly longer lengths in the backward compositions of biholomorphic mappings, will shield the encoding process, making it more difficult to decode. And conversely, one would think that increasingly longer forward compositions of biholomorphic mappings would make the decoding process more feasible.

In this section, we will give a series of examples, for several ordinary cases, through which it will be clearly seen that the arbitrary choice of finite length for the biholomorphic encoding rules, as well as the arbitrary choice of the domains of definition and the different intrinsic forms of the biholomorphic mappings participating in the biholomorphic codification chains, create an “insurmountable” computational protect of the codification, in the sense that they constitute absolutely essential conditions that cannot substituted or even be approached by large and simple encoding rules. And similarly, the attempts for biholomorphic decoding should be specific, of finite length, with different domains of definition and with different intrinsic forms of biholomorphic mappings in the decoding rules.

To this end, suppose a biholomorphic code $(\mathcal{E}, \mathcal{D})$ on a non-empty domain $\Omega \subset$ $\mathbb{C}$ consists in a given sequence
$$\left(\left(F^{(M)}=f_M \circ f_{M-1} \circ \ldots \circ f_1 \circ f_0\right),\left(G^{(M)}=g_0 \circ g_1 \circ \ldots \circ g_{M-1} \circ g_M\right)\right)_{M, N \in \mathbb{N}_0}$$
of biholomorphic ciphers on $\Omega$. An application of the Contraction Theorem for holomorphic functions (see, for instance, [9]) shows that

Theorem 1 (Contraction of Biholomorphic Encoding Rules) If for any $M \in \mathbb{N}0$, the codification links $f_0, f_1, \ldots, f_M$ are all equal to the same function $f \in \bar{A}(\Omega)$ having bounded range $f(\Omega)$ in $\Omega$, then the biholomorphic encoding rule $$F^{(M)}(z)=f^M(z)=(\underbrace{f \circ f \circ \ldots \circ f \circ f}{M+1 \text { times }})(z)$$
trivializes in $\Omega$ as its length $M+1$ grows illimitably, in the sense that
$$F^{(M)}(z) \underset{M \rightarrow \infty}{\rightarrow \rightarrow} A,$$

## 数学代写|复分析作业代写Complex function代考|Dynamic Properties of Biholomorphic Codes

Certainly, it is of interest, for our intentions, to study the recurring points into biholomorphic codification chains $F=\ldots \circ f_M \circ \ldots \circ f_1 \circ f_0$ which can entirely be covered by segments from successive sub-chains of the $F$, each of which maps a set into itself. Observe that if $f_0=f_1=\ldots=f_M=\ldots=f$, then the study is limited just on the study of the iterations and dynamics of the biholomorphic mapping $f$.
To this end, we must look for characteristic properties of points that are kept fixed through the parts $f_{n_{j+1}-1} \circ \ldots \circ f_{n_j+1} \circ f_{n_j}$ of a biholomorphic codification chain $F$ on an open domain $\Omega \subset \mathbb{C}$.
Notation 1 We will use the symbol
$$F_n^{(M)}$$
to denote the segment of the biholomorphic codification chain $F$ that involves the composition of all functions which are at the part of $F$ that starts from the function in the $(n+1)^{t h}$ place of $F$ and ends with the function in the $(M+1)^{t h}$ place of $F$. It is clear that
$$F_n^{(M)}=f_M \circ \ldots \circ f_{n+1} \circ f_n .$$
Thus, for instance, we may write
$$F_3^{(8)}=f_8 \circ \ldots \circ f_4 \circ f_3 \text { and } F_{n_j}^{\left(n_{j+1}-1\right)}=f_{n_{j+1}-1} \circ \ldots \circ f_{n_j+1} \circ f_{n_j} .$$
In particular, we have
$$F_M^{(M)}(p)=f_M(p), F_M^{(M \mid 1)}(p)=\left(f_{M+1} \circ f_M\right)(p), \ldots, F_M^{M+n}(p)=\left(f_{M+n} \circ \ldots \circ f_{M+1} \circ f_M\right)(p) .$$
If $n>0$ and $M=\infty$, then we set
$$F_n^{(\infty)}=\ldots f_M \circ \ldots, f_{n+1} \circ f_n$$
and we say that $F^{(\infty)}$ is a truncated codification chain with late start in position $n$. For example, the segment
$$F_6^{(\infty)}=\ldots f_8 \circ f_7 \circ f_6$$
of $F$ is a truncated codification chain with late start in position 6 .

# 复分析代写

## 数学代写|复分析作业代写Complex function代考|Convergence Properties of Biholomorphic Codes

$$\left(\left(F^{(M)}=f_M \circ f_{M-1} \circ \ldots \circ f_1 \circ f_0\right),\left(G^{(M)}=g_0 \circ g_1\right.\right.$$

$$F^{(M)}(z)=f^M(z)=(\underbrace{f \circ f \circ \ldots \circ f \circ f} M+1 \text { times })(z)$$

$$F^{(M)}(z) \underset{M \rightarrow \infty}{\rightarrow} \rightarrow$$

## 数学代写|复分析作业代写Complex function代考|Dynamic Properties of Biholomorphic Codes

$F=\ldots \circ f_M \circ \ldots \circ f_1 \circ f_0$ 它可以完全被来自连续子链的段覆盖 $F$ ，每个都将一个集合映射到自身。观察如果
$f_0=f_1=\ldots=f_M=\ldots=f_1$ ，那么研究仅限于双全纯映射的迭 代和动力学研究 $f$.

$f_{n_{j+1}-1} \circ \ldots \circ f_{n_j+1} \circ f_{n_j}$ 双全纯编码链 $F$ 在一个开放的领域
$\Omega \subset \mathbb{C}$.

$$F_n^{(M)}$$

$$F_n^{(M)}=f_M \circ \ldots \circ f_{n+1} \circ f_n$$

$$F_3^{(8)}=f_8 \circ \ldots \circ f_4 \circ f_3 \text { and } F_{n_j}^{\left(n_{j+1}-1\right)}=f_{n_{j+1}-1} \circ \ldots \circ f_{n_j+1} \circ$$

$$F_M^{(M)}(p)=f_M(p), F_M^{(M \mid 1)}(p)=\left(f_{M+1} \circ f_M\right)(p), \ldots, F_M^{M+n}(p)$$

$$F_n^{(\infty)}=\ldots f_M \circ \ldots, f_{n+1} \circ f_n$$

$$F_6^{(\infty)}=\ldots f_8 \circ f_7 \circ f_6$$

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