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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

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).