Julia Sets, One Rule and Many Starting Points

Definition Fix a complex number $c$ and the rule $f_c(z) = z^2 + c$. Starting from a point $z_0$ of the complex plane, the rule gives $z_1 = f_c(z_0)$, then $z_2 = f_c(z_1)$, and so on, and this sequence is the orbit of $z_0$. The orbit is bounded when some number $B$ satisfies $\lvert z_n \rvert \le B$ for every $n$. The filled Julia set $K_c$ is the set of starting points whose orbits are bounded. The Julia set $J_c$ is the boundary of $K_c$, the set of points $p$ such that every disk centered at $p$, however small, contains points of $K_c$ and points outside $K_c$. The filled Julia set contains its boundary, so the points of $J_c$ have bounded orbits, yet arbitrarily close to each of them there are starting points that escape. So the answer to the question of which starting points stay bounded is the set $K_c$, and the intricate part of any picture of it is its edge $J_c$. For $c = 0$ the filled Julia set is the closed unit disk and the Julia set is the unit circle, while for other values of $c$ the Julia set can be far more intricate, although the rule still has only a squaring and an addition. For the rules $z^2 + c$, a classical theorem says that $J_c$ is either in one piece or a dust of separate points, and The Mandelbrot Set, a Map of Julia Worlds explains which. That two-way choice is a theorem about this quadratic family, and it does not hold for every rule. Julia sets are named after Gaston Julia, who studied repeated rules given by polynomials and fractions of polynomials in a long memoir of 1918, as Pierre Fatou did in the same years.

Definition Quadratic iteration is the repeated application of a rule $f_c(z) = z^2 + c$, where $c$ is a fixed complex number called the parameter. Each step first squares the current number, which squares its modulus and doubles its argument, and then adds $c$, which slides the result by the arrow from $0$ to $c$. The same $c$ is used at every step and for every starting point. All the rules $z^2 + c$ together form the quadratic family. This family covers every quadratic rule $z \mapsto az^2 + bz + d$ with $a \neq 0$, where the arrow $\mapsto$ reads “is replaced by”. Relabeling the plane by $w = az + \tfrac{b}{2}$ changes such a rule into $w \mapsto w^2 + c$ with $c = ad + \tfrac{b}{2} - \tfrac{b^2}{4}$, so studying the rules $z^2 + c$ studies all quadratic rules at once.

Definition The orbit of a starting point $z_0$ under a rule $f$ is the sequence $z_0, z_1, z_2, \dots$ in which $z_{n+1} = f(z_n)$ for every $n \ge 0$. It is bounded when all its points lie within some fixed distance of $0$, and it escapes when $\lvert z_n \rvert$ grows beyond every bound and tends to infinity. For a rule $z^2 + c$ every orbit does one or the other, by the escape radius card below. A fixed point is a start with $f(z_0) = z_0$, and an orbit is a cycle of length $k$ when $k$ is the smallest positive whole number with $z_k = z_0$. Bounded does not mean settled. A bounded orbit may approach a point, run around a cycle, or wander forever without repeating, as some starts on the unit circle do for $c = 0$.

Definition For a fixed complex number $c$, the filled Julia set $K_c$ is the set of starting points $z_0$ whose orbits under $z^2 + c$ are bounded. By the escape radius card below, a start belongs to $K_c$ exactly when every point of its orbit satisfies $\lvert z_n \rvert \le \max(2, \lvert c \rvert)$, so $K_c$ lies inside that closed disk. The set $K_c$ contains its boundary, and that boundary is the Julia set $J_c$. When $K_c$ has an inside, made of points with a whole small disk around them in $K_c$, the Julia set is only its edge. When $K_c$ has no inside, the two sets are equal. That happens for $c = -2$, where $K_{-2}$ is the segment of the real axis from $-2$ to $2$. Every start in that segment stays in it, since $z^2$ is then between $0$ and $4$, and every other start escapes. Membership in $K_c$ is a statement about every step of an orbit, so no finite computation proves it by running the rule, although an exact argument, such as finding a cycle, can.

Definition For the rule $z^2 + c$, the number

the larger of $2$ and $\lvert c \rvert$, is an escape radius. If some point of an orbit satisfies $\lvert z_n \rvert > R$, strictly, then the orbit escapes, and from then on every point is farther from $0$ than the one before. The reason is that adding $c$ can bring a point at most $\lvert c \rvert$ closer to $0$. For a point $z$ with $r = \lvert z \rvert > R$ this gives

since $\lvert c \rvert \le R < r$. As $r > 2$, the factor $r - 1$ is greater than $1$, so the next point is farther out than $r$ and again beyond $R$. If $\lambda = r - 1$ for the first point beyond $R$, every later step multiplies the distance from $0$ by more than $\lambda$, so the distances grow at least as fast as the powers of $\lambda$, which grow without bound. The strict inequality matters. For $c = -2$ the radius is $2$, and the start $2$ is a fixed point, since $2^2 - 2 = 2$, so a point exactly at distance $R$ need not escape.