McMullen 2–3 Map

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.

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.

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.