Local connectivity of the Mandelbrot and Multibrot sets

References:

• Wikipedia
• arxiv/math/9902155

def multibrotSet (n : ℕ) : Set ℂ

The Multibrot set of power n is the set of all parameters c : ℂ for which 0 does not escape to infinity under repeated application of z ↦ z ^ n + c.

Equations

• multibrotSet n = {c : ℂ | ¬Filter.Tendsto (fun (k : ℕ) => (fun (z : ℂ) => z ^ n + c)^[k] 0) Filter.atTop (Bornology.cobounded ℂ)}

@[reducible, inline]

abbrev mandelbrotSet : Set ℂ

The Mandelbrot set is the special case of the multibrot set for n = 2. In other words, it is the set of all parameters c : ℂ for which 0 does not escape to infinity under repeated application of z ↦ z ^ 2 + c.

Equations

• mandelbrotSet = multibrotSet 2

theorem multibrotSet_eq {n : ℕ} (hn : 2 ≤ n) : multibrotSet n = {c : ℂ | ∀ (k : ℕ), ‖(fun (z : ℂ) => z ^ n + c)^[k] 0‖ ≤ 2 ^ (↑n - 1)⁻¹}

The multibrotSet n is equivalently the set of all parameters c for which the orbit of 0 under z ↦ z ^ n + c does not leave the closed disk of radius 2 ^ (n - 1)⁻¹ around the origin.

theorem mandelbrotSet_eq : mandelbrotSet = {c : ℂ | ∀ (k : ℕ), ‖(fun (z : ℂ) => z ^ 2 + c)^[k] 0‖ ≤ 2}

The mandelbrot set is equivalently the set of all parameters c for which the orbit of 0 under z ↦ z ^ 2 + c does not leave the closed disk of radius two around the origin.

theorem MLC : LocallyConnectedSpace ↑mandelbrotSet

The MLC conjecture, stating that the mandelbrot set is locally connected.

theorem MLC_general_exponent (n : ℕ) : LocallyConnectedSpace ↑(multibrotSet n)

A stronger version of the MLC conjecture, stating that all multibrots are locally connected. Note that we don't need to require 2 ≤ n because the conjecture holds in the trivial cases n = 0 and n = 1 too.