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Statement

For \(f_c(z)=z^2+c\) let \(K_c=\{z:\ (f_c^n(z))_n \text{ bounded}\}\) be the filled Julia set and \(J_c=\partial K_c\) the Julia set. Then exactly one of two cases holds:

  • If the critical orbit is bounded — equivalently \(0\in K_c\), equivalently \(c\in M\) (the Mandelbrot set) — then \(J_c\) is connected.
  • Otherwise (\(0\notin K_c\), \(c\notin M\)) \(J_c\) is totally disconnected: a Cantor set ("Cantor dust"), and \(K_c=J_c\).

Assumptions

  1. Quadratic family \(f_c(z)=z^2+c\); \(0\) is the unique finite critical point.
  2. The dichotomy is a statement about the topology of \(J_c\) as \(c\) varies.

Proof

  1. [asserted] (Fatou-Julia dichotomy for the quadratic family.) The connectedness of \(J_c\) is governed entirely by the fate of the critical orbit: \(J_c\) connected \(\iff\) the orbit of the critical point \(0\) is bounded. This is the central theorem of Fatou (1917-1920) and Julia (1918), as sharpened by Douady-Hubbard.
  2. [asserted] Boundedness of the critical orbit \((f_c^n(0))\) is by definition membership \(c\in M\). Hence \(J_c\) connected \(\iff c\in M\).
  3. [asserted] In the escaping case \(c\notin M\), the pullback of a large disc under \(f_c\) produces two disjoint preimages at every level; the nested intersection is a totally disconnected, perfect, compact set — a Cantor set — on which \(f_c\) acts as the full 2-shift (Böttcher/symbolic-dynamics argument, Douady-Hubbard).

Verification

  • Not machine-verified. This document records a named theorem; the dichotomy is asserted, not re-derived. No SymPy/NumPy computation can establish a topological connectedness statement for all \(c\).
  • The link between "critical orbit bounded" and "\(c\in M\)" is definitional and consistent with the Mandelbrot escape criterion, which is machine-verified. That connection is the only computationally checkable piece and it is checked there.
  • Asserted from: the Fatou-Julia theorem (Fatou 1917-1920; Julia 1918) and the Douady-Hubbard formalization of the quadratic family.

Historical source

Gaston Julia, "Memoire sur l'iteration des fonctions rationnelles," Journal de Mathematiques Pures et Appliquees (1918); Pierre Fatou, "Sur les equations fonctionnelles," Bulletin de la S.M.F. (1919-1920). The two developed the iteration theory of rational maps independently and near-simultaneously.

Related concepts

Citations

  1. Julia, G. "Memoire sur l'iteration des fonctions rationnelles." J. Math. Pures Appl. 8 (1918): 47-245.
  2. Fatou, P. "Sur les equations fonctionnelles." Bull. Soc. Math. France 47-48 (1919-1920).
  3. Douady, A., Hubbard, J. H. "Etude dynamique des polynomes complexes." Publ. Math. Orsay (1984-85).