이색어려움

McMullen 2–3 Map

제곱이 점을 바깥으로 몰아내고 역삼차 항이 점을 원점의 극을 통해 보내는 유리 사상이에요.

FractalPark의 McMullen 2–3 렌더링: 구슬 같은 경계가 겹겹이 넓은 중앙 구멍을 둘러싸요.

개요

McMullen 2–3 사상은 보통 $f_\lambda(z)=z^n+\lambda/z^d$로 쓰는 계열에 속해요. FractalPark은 매개변수를 $c$라고 부르고 $n=2$, $d=3$으로 골랐어요. 처음에는 항이 하나 더 붙은 이차 사상처럼 보여요. 그런데 $z^3$이 분모에 나타나면서 규칙이 바뀌어요. 원점은 평범한 점이 아니라 극이에요.

이제 한 공식에 두 영역이 생겨요. 멀리서는 $z^2$이 지배하며 큰 값을 더 멀리 밀어내요. 0 근처에서는 $c/z^3$이 주도권을 잡아 점들을 극을 통해 무한대로 보내요. 그 사이에는 어느 쪽도 확실히 이기지 못하는 움직이는 경계가 있어요.

그 경계는 고리로 닫히거나, 둥근 세포로 부서지거나, 연결된 그물로 합쳐질 수 있어요. $c$와 임계점 궤도에 달려 있죠. 단 하나의 McMullen 윤곽은 존재하지 않아요.

수학

A quadratic term and a cubic pole

z_{n+1}=z_n^2+\frac{c}{z_n^3}

At each step, square the current complex value, then add $c/z^3$. When $|z|$ is large, the first term grows roughly like $|z|^2$, while the reciprocal term fades like $1/|z|^3$. Near zero the balance flips: the reciprocal term becomes enormous.

The same expression can be written as

$$ f_c(z)=\frac{z^5+c}{z^3}. $$

The map has a pole of order three at $z=0$, and—when $c\neq0$—is a rational map of degree five. The “2–3” names the exponents in the original expression, not the rational-map degree.

In the exact definition, the pole sends zero to infinity on the Riemann sphere. A numerical renderer cannot divide by zero, so FractalPark replaces values extremely close to the origin with a tiny nonzero value and guards the denominator. That is a computational safety rail, not a different definition.

역사

The family takes its name from Curtis T. McMullen. In Section 7 of his 1988 paper *Automorphisms of Rational Maps*, McMullen constructed rational maps whose Julia sets break into a Cantor set of Jordan curves—infinitely many disjoint loops arranged with Cantor-set structure. Later authors adopted “McMullen maps” for the family $z^n+\lambda/z^d$ associated with this construction. McMullen is now Cabot Professor of Mathematics at Harvard, where his personal academic homepage and publication list remain available.

The 2–3 choice is the smallest pair of exponents satisfying the characteristic inequality

$$ \frac{1}{n}+\frac{1}{d}<1, $$

because $1/2+1/3=5/6$. It also gives the lowest possible degree, $n+d=5$, in this part of the theory. The inequality alone does not guarantee that every parameter produces circles: the relevant critical orbits must also enter the appropriate escape region.

Work after McMullen mapped out that dependence in much greater detail. Robert Devaney, Daniel Look, and David Uminsky proved an “escape trichotomy” in 2005: when the free critical orbit escapes, the Julia set can be a Cantor set, a Cantor set of circles, or a Sierpiński curve, depending on how it reaches infinity.

시각적 특징

One large pale opening anchors the image. An uneven ring of rounded cells circles it; farther out, smaller cells make a fine lace. What looks like a smooth blue band at first resolves into a crowded rim.

The outer edge is gently faceted rather than round, and no two cells quite match. That belongs to this parameter, frame, and colouring—not every McMullen map. The pale palette favours relief and nested boundaries over an escape-speed gradient.

리믹스와 예제

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작품 보기

자주 묻는 질문

What do 2 and 3 mean in the name?

They name the two powers in $z^2+c/z^3$: the polynomial side squares $z$, while the reciprocal side divides by its cube. In the general McMullen family they are $n$ and $d$.

Do not read them as “degree two to degree three.” Combine the terms and the map is $(z^5+c)/z^3$, so for nonzero $c$ its rational-map degree is five.

What happens at z = 0?

The denominator $z^3$ vanishes, so zero is a pole, not an ordinary starting point. On the extended complex plane it maps to infinity. Nearby points are flung away with strength growing like $1/|z|^3$.

That pole is not a rendering accident; it is the feature that distinguishes this family from a polynomial such as $z^2+c$. FractalPark only adds a tiny numerical guard near zero so the shader can represent the same limiting behavior without performing an undefined division.

개요

수학