### 数学代写|复杂系统与重整化代写Complex Systems and Reengineering代考|COM3523

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• Foundations of Data Science 数据科学基础

## 数学代写|复杂系统与重整化代写Complex Systems and Reengineering代考|Complex dynamics

This work presents a study of renormalization of quadratic polynomials and a rapid introduction to techniques in complex dynamics.
Around 1920 Fatou and Julia initiated the theory of iterated rational maps
$$f: \widehat{\mathbb{C}} \rightarrow \widehat{\mathbb{C}}$$
on the Riemann sphere. More recently methods of geometric function theory, quasiconformal mappings and hyperbolic geometry have contributed to the depth and scope of research in the field. The intricate structure of the family of quadratic polynomials was revealed by work of Douady and Hubbard [DH1], [Dou1]; analogies between rational maps and Kleinian groups surfaced with Sullivan’s proof of the no wandering domains theorem [Sul3] and continue to inform both subjects [Mc2].

It can be a subtle problem to understand a high iterate of a rational map $f$ of degree $d>1$. There is tension between expanding features of $f$-such as the fact that its degree tends to infinity under iteration – and contracting features, such as the presence of critical points. The best understood maps are those for which the critical points tend to attracting cycles. For such a map, the tension is resolved by the concentration of expansion in the Julia set or chaotic locus of the map, and the presence of contraction on the rest of the sphere.

The central goal of this work is to understand a high iterate of a quadratic polynomial. The special case we consider is that of an infinitely renormalizable polynomial $f(z)=z^2+c$.

For such a polynomial, the expanding and contracting properties lie in a delicate balance; for example, the critical point $z=0$ belongs to the Julia set and its forward orbit is recurrent. Moreover high iterates of $f$ can be renormalized or rescaled to yield new dynamical systems of the same general shape as the original map $f$.

This repetition of form at infinitely many scales provides the basic framework for our study. Under additional geometric hypotheses, we will show that the renormalized dynamical systems range in a compact family. Compactness is established by combining universal estimates for the hyperbolic geometry of surfaces with distortion theorems for holomorphic maps.

With this information in hand, we establish quasiconformal rigidity of the original polynomial $f$. Rigidity of $f$ supports conjectures about the behavior of a generic complex dynamical system, as described in the next section.

## 数学代写|复杂系统与重整化代写Complex Systems and Reengineering代考|Central conjectures

We now summarize the main problems which motivate our work.
Let $f: \widehat{\mathbb{C}} \rightarrow \widehat{\mathbb{C}}$ be a rational map of the Riemann sphere to itself of degree $d>1$. The map $f$ is hyperbolic if its critical points tend to attracting periodic cycles under iteration. Within all rational maps, the hyperbolic ones are among the best behaved; for example, when $f$ is hyperbolic there is a finite set $A \subset \widehat{\mathbb{C}}$ which attracts all points in an open, full-measure subset of the sphere (see $\S 3.4$ ).

One of the central problems in conformal dynamics is the following:

Conjecture $1.1$ (Density of hyperbolicity) The set of hyperbolic rational maps is open and dense in the space Rat $_d$ of all rational maps of degree $d$.

Openness of hyperbolic maps is known, but density is not. In some form this conjecture goes back to Fatou (see $\S 4.1$ ).

Much study has been devoted to special families of rational maps, particularly quadratic polynomials. Every quadratic polynomial $f$ is conjugate to one of the form $f_c(z)=z^2+c$ for a unique $c \in \mathbb{C}$. Even this simple family of rational maps exhibits a full spectrum of dynamical behavior, reflecting many of the difficulties of the general case. Still unresolved is:

Conjecture 1.2 The set of $c$ for which $z^2+c$ is hyperbolic forms ип ореп dense subsel of lhe complex plane.

The Mandelbrot set $M$ is the set of $c$ such that under iteration, $f_c^n(0)$ does not tend to infinity; here $z=0$ is the unique critical point of $f_c$ in $\mathbb{C}$. A component $U$ of the interior of $M$ is hyperbolic if $f_c$ is hyperbolic for some $c$ in $U$. It is known that the maps $f_c$ enjoy a type of structural stability as $c$ varies in any component of $\mathbb{C}-\partial M$; in particular, if $U$ is hyperbolic, $f_c$ is hyperbolic for every $c$ in $U$ (see $\S 4)$. It is clear that $f_c$ is hyperbolic when $c$ is not in $M$, because the critical point tends to the superattracting fixed point at infinity. Thus an equivalent formulation of Conjecture $1.2$ is:

Conjecture 1.3 Every component of the interior of the Mandelbrot set is hyperbolic.

An approach to these conjectures is developed in [MSS] and [McS], using quasiconformal mappings. This approach has the advantage of shifting the focus from a family of maps to the dynamics of a single map, and leads to the following:

Conjecture $1.4$ (No invariant line fields) A rational map $f$ carries no invariant line field on its Julia set, except when $f$ is double covered by an integral torus endomorphism.

# 复杂系统与重整化代写

## 数学代写|复杂系统与重整化代写Complex Systems and Reengineering代考|Complex dynamics

1920 年前后，Fatou 和 Julia 提出了迭代有理映射理论

F:C^→C^

## 数学代写|复杂系统与重整化代写Complex Systems and Reengineering代考|Central conjectures

[MSS] 和 [McS] 中使用拟共形映射开发了这些猜想的方法。这种方法的优点是将焦点从一系列地图转移到单个地图的动态，并导致以下结果：

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