Julia Set

Filled Julia Set and Julia Set

Let $f_c(z)=z^2+c$ for a given complex parameter $c$, $z\in \mathbb{C}$ We classify the behavior of a point $z\in \mathbb{C}$ based on its orbit under $f_c$: $\{z,f_c(z),f_c^2(z),\dots \}$. Based on these orbits, we define two critical sets:

• Filled Julia Set ($K_c$): The set of all initial points whose orbits remain bounded. $K_c=\{z\in \mathbb{C}\mid \text{the orbit of }z\text{ is bounded}\}$
• Julia Set ($J_c$): The boundary of the filled Julia set. $J_c=\partial K_c$

Intuition $J_c$ acts as the boundary between escaping orbits (those heading to infinity) and trapped/bounded orbits. If $z\in J_c$, the behavior of orbits near $z$ is highly sensitive to initial conditions. Every open ball around a point in $J_c$ contains both bounded and unbounded orbits.

Escaping Radius

There exists a defined “escaping radius,” $R=\max \{|c|,2\}$. If at any point an orbit reaches $|z|>R$, it is guaranteed to escape to $\infty$. This implies that $K_c$ is always a bounded subset of $\mathbb{C}$.

A point $z^{\ast }$ is a fixed point if $f_c(z^{\ast })=z^{\ast }$. The stability of $z^{\ast }$ determines if nearby orbits are attracted to it or repelled away. To test for stability, we use the derivative (Lagrange multipliers): $\lambda =f_c^{\prime }(z^{\ast })=2z^{\ast }$

• If $|\lambda |<1$: $z^{\ast }$ is stable (attracting).
• If $|\lambda |>1$: $z^{\ast }$ is unstable (repelling). Crucial Theorem linking stability to $J_c$: The Julia set $J_c$ is the closure of the unstable periodic orbits of $f_c$.

Proposition

• Compact & Non-empty: $J_c$ is a closed and bounded (compact) subset of $\mathbb{C}$.
• Symmetric: $J_c$ is preserved under the mapping $z\to -z$.
• Perfect: Every point in $J_c$ is a limit point of $J_c$ (it has no isolated points). This implies $J_c$ is uncountable. (Using the fact that perfect non-empty set are uncountable)
• Invariant: $J_c$ is completely invariant under the function and its inverse: $f_c(J_c)=J_c$.

The Hausdorff dimension ${\dim }_H(J_c)$ is typically in the interval $(1,2)$, though it can sometimes be strictly $<1$ or exactly $2$.

Cantor Space

Let $(X,d)$ be a metric space. A subset $\mathcal{C}\subset X$ is a Cantor space if:

• $\mathcal{C}\ne \varnothing$
• $\mathcal{C}$ is compact (closed and bounded)
• $\mathcal{C}$ is perfect: For every $x\in \mathcal{C}$ and every $ϵ>0$, there exists $y\in \mathcal{C}$ such that $0<d(x,y)<ϵ$.
• $\mathcal{C}$ is totally disconnected: For any two distinct points $x,y\in \mathcal{C}$, there exist disjoint open sets $U,V$ such that $x\in U$, $y\in V$, and $\mathcal{C}\subset U\cup V$.

Theorem

Any Cantor space is homeomorphic to the Cantor set.

Theorem

A Julia set is either connected or a Cantor set.

Theorem

$J_c$ is connected iff the orbit of $0$ is bounded.