---
type: Fractal Proof
title: Mandelbrot Escape-Radius Criterion
description: Any orbit of f_c(z) = z^2 + c that exceeds modulus 2 (with |z| at least |c|) diverges to infinity, making radius 2 an exact bailout.
resource: https://tyson-swetnam.github.io/fractal-notebooks/okf/proofs/mandelbrot-escape/
tags: [complex-dynamics, complex-analysis, methods]
result: "|z|>2 and |z|>=|c| implies |f_c^n(0)| -> infinity; bailout radius = 2"
concept: ../concepts/complex-dynamics/mandelbrot-set.md
historical_source: Benoit B. Mandelbrot, "Fractal aspects of the iteration of z -> lambda z(1-z)" (1980); set popularized 1978-1980.
verification_status: verified
sources:
  - id: history-chapter
    resource: https://tyson-swetnam.github.io/fractal-notebooks/foundations/history/
    title: "Chapter 1: History of Fractal Mathematics"
generated:
  by: human:tswetnam
  at: "2026-07-13T00:00:00Z"
verified:
  - by: human:tswetnam
    at: "2026-07-13T00:00:00Z"
status: stable
---

# Statement

Let $f_c(z)=z^2+c$ with $c\in\mathbb{C}$, and consider the orbit
$z_0=0,\; z_{n+1}=f_c(z_n)$. If at any step $|z_n|>2$ and $|z_n|\ge|c|$, then
$|z_n|\to\infty$ monotonically, so $c$ is **not** in the Mandelbrot set
$M=\{c:\ (f_c^n(0))_n \text{ is bounded}\}$. Consequently a bailout radius of
$R=2$ is exact: escape past modulus 2 guarantees divergence.

# Assumptions

1. $c\in\mathbb{C}$ and the standard quadratic family $f_c(z)=z^2+c$.
2. Escape criterion invoked at a step where $|z_n|>2$. For the seed $z_0=0$ one
   has $z_1=c$, so once $|z_n|>2$ the auxiliary bound $|z_n|\ge|c|$ holds along
   the orbit (each iterate's modulus dominates $|c|$ once past radius 2); the
   lemma is stated with both hypotheses to make the algebra self-contained.

# Proof

1. **[verified]** Reverse triangle inequality:
   $|z^2+c|\ge |z|^2-|c|$. With $|z|=m$ and $|c|\le m$ this gives
   $|f_c(z)|\ge m^2-m = m(m-1)$.
2. **[verified]** We want $|f_c(z)|>|z|$, i.e. $m(m-1)>m$, i.e.
   $m(m-2)>0$. Since $m>0$, this holds **iff $m>2$**. SymPy factoring gives
   $m(m-1)-m = m(m-2)$ and `solve(m*(m-2)>0) = (2 < m)`.
3. **[verified]** Hence for $m=|z_n|>2$ there is a fixed
   $\lambda = m-1 > 1$ with $|z_{n+1}|\ge \lambda|z_n|$, and since the next
   modulus is again $>2$ the bound iterates: $|z_{n+k}|\ge\lambda^{k}|z_n|\to\infty$.
4. **[asserted]** Because divergence of the orbit is exactly the complement of
   boundedness, $c\notin M$. This is the definition of $M$ together with the
   escape-time algorithm's correctness (standard, e.g. Douady-Hubbard).

# Verification

- **Symbolic (SymPy):** `factor(m*(m-1) - m) = m*(m - 2)` and
  `solve(m*(m-2) > 0, m)` returns $2<m$ — confirms the threshold is exactly 2.
- **Numerical (NumPy):** 98,703 random trials drawing $z,c$ uniformly in
  $[-5,5]^2\subset\mathbb{C}$; of these **48,715** satisfied the hypotheses
  $|z|>2,\ |z|\ge|c|$, and **0** violated the conclusion $|z^2+c|>|z|$
  (0 violations).
- **Asserted:** Step 4 (escape $\Rightarrow c\notin M$) is definitional; no
  named deep theorem is required.

# Historical source

Mandelbrot introduced the set now bearing his name in work circa 1978-1980;
the radius-2 escape criterion is standard in the Douady-Hubbard theory of the
quadratic family.

# Related concepts

- [Mandelbrot set](../concepts/complex-dynamics/mandelbrot-set.md)
- [Julia set](../concepts/complex-dynamics/julia-set.md)

# Citations

1. Mandelbrot, B. B. "Fractal aspects of the iteration of $z\mapsto\lambda z(1-z)$ for complex $\lambda$ and $z$." *Annals of the New York Academy of Sciences* 357 (1980): 249-259.
2. Douady, A., Hubbard, J. H. "Etude dynamique des polynomes complexes." *Publ. Math. Orsay* (1984-85).