Conjectures about the Mandelbrot and Multibrot sets
The density of hyperbolicity conjecture for Multibrot sets, stating that the set of all parameters c for which fun z ↦ z ^ n + c has an attracting cycle is dense in multibrotSet n. Note that we need to require 2 ≤ n because the conjecture is trivial...
Mathematical statement
The density of hyperbolicity conjecture for Multibrot sets, stating that the set of all
parameters c for which fun z ↦ z ^ n + c has an attracting cycle is dense in multibrotSet n.
Note that we need to require 2 ≤ n because the conjecture is trivially false for n = 1.
Statement source: Wikipedia statement material
Statement terms: CC-BY-SA-4.0
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Statement artifacts, not proofs
These records expose exact Lean propositions and statement-only wrappers. Defining a proposition does not supply a proof of it. A placeholder-bearing target also contains no proof. Elaboration checks syntax and types; it does not certify that a formalization perfectly captures every nuance of the informal problem.
Pinned Lean formulation 1
density_of_hyperbolicity_general_exponent
theorem density_of_hyperbolicity_general_exponent {n : ℕ} (hn : 2 ≤ n) : multibrotSet n ⊆ closure {c | ∃ m z, IsAttractingCycle (fun z ↦ z ^ n + c) m z} := by sorry- Statement source
- Formal Conjectures
- Lean version
- v4.27.0
- Placeholder
- Present; no proof artifact
- Source evidence
- Pinned source index
- Fidelity review
- Community formulation
References