Mandelbrot.lean`

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

15 declarations, 6 contain sorry, 0 declared as axioms

Conjectures about the Mandelbrot and Multibrot sets

This file adds three conjectures about the Mandelbrot and Multibrot sets:

the MLC conjecture, stating that these sets are locally connected
the density of hyperbolicity conjecture, stating that parameters with attracting cycles are dense in the Mandelbrot and Multibrot sets
the conjecture that the boundaries of these sets have zero area. The first two conjectures are related in that the former implies the latter.

References:

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/Subsets/FC100OpenSet1.lean

def Mandelbrot.multibrotSet 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 line 43 source
Source documentation
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.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Set, Complex
Proof uses: Set.ofPred, Not, Filter.Tendsto, Nat.iterate, HAdd.hAdd, HPow.hPow, OfNat.ofNat, Filter.atTop, Bornology.cobounded
16 instances, coercions and auxiliary constants left out.
```
def multibrotSet (n : ℕ) : Set ℂ :=
…
```
abbrev Mandelbrot.mandelbrotSet The Mandelbrot set is the special case of the multibrot set for n = 2 line 49 source
Source documentation
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.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Complex
Proof uses: Mandelbrot.multibrotSet, OfNat.ofNat, Nat
1 instances, coercions and auxiliary constants left out.
```
abbrev mandelbrotSet := multibrotSet 2
…
```
Mentions: Mandelbrot.multibrotSet
theorem Mandelbrot.multibrotSet_eq Documented declaration: multibrotSet_eq line 54 source
Source documentation
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.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Eq, Set, Complex, Mandelbrot.multibrotSet, Set.ofPred, Real, Norm.norm, Nat.iterate, HAdd.hAdd, HPow.hPow, Inv.inv, HSub.hSub
Proof uses: LT.lt, LT.lt.trans_le, one_lt_two, Eq.mpr, id, congrArg, rfl, Real.rpow_pos_of_pos, Mathlib.Meta.Positivity.pos_of_isNat, inferInstance, PartialOrder, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Mathlib.Meta.NormNum.instAtLeastTwo, Bool, Nat.ble, of_eq_true, Eq.trans, True, congrFun' and 177 more
249 instances, coercions and auxiliary constants left out.

Tactic trace: 63 steps
Open to load the checked before-and-after goal states.
```
theorem multibrotSet_eq {n : ℕ} (hn : 2 ≤ n) :
    multibrotSet n = {c | ∀ k, ‖(fun z ↦ z ^ n + c)^[k] 0‖ ≤ 2 ^ (n - 1 : ℝ)⁻¹} := by
…
```
Mentions: Mandelbrot.multibrotSet, c
theorem Mandelbrot.mandelbrotSet_eq 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 line 106 source
Source documentation
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.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Set, Complex, Mandelbrot.mandelbrotSet, Set.ofPred, Nat, LE.le, Real, Norm.norm, Nat.iterate, HAdd.hAdd, HPow.hPow, OfNat.ofNat
Proof uses: Eq.mp, Mandelbrot.multibrotSet, Inv.inv, HSub.hSub, congrArg, funext, forall_congr, Eq.trans, Mathlib.Meta.NormNum.isNat_eq_true, Mathlib.Meta.NormNum.instAddMonoidWithOne, Mathlib.Meta.NormNum.IsInt.to_isNat, Mathlib.Meta.NormNum.isInt_sub, Int.ofNat, Eq.refl, Mathlib.Meta.NormNum.IsNat.to_isInt, Mathlib.Meta.NormNum.isNat_ofNat, Mathlib.Meta.NormNum.instAtLeastTwo, Nat.cast_one, Int, inv_one and 3 more
55 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem mandelbrotSet_eq : mandelbrotSet = {c | ∀ k, ‖(fun z ↦ z ^ 2 + c)^[k] 0‖ ≤ 2} := by
…
```
Mentions: Mandelbrot.mandelbrotSet, c
theorem Mandelbrot.MLC The MLC conjecture, stating that the mandelbrot set is locally connected sorry line 111 source
Source documentation
The MLC conjecture, stating that the mandelbrot set is locally connected.

depends on sorry what this means
Statement uses: LocallyConnectedSpace, Complex, Mandelbrot.mandelbrotSet, Membership.mem, Set
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
21 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem MLC : LocallyConnectedSpace mandelbrotSet := by
…
```
Mentions: Mandelbrot.mandelbrotSet
theorem Mandelbrot.MLC_general_exponent A stronger version of the MLC conjecture, stating that all multibrots are locally connected sorry line 118 source
Source documentation
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.

depends on sorry what this means
Statement uses: Nat, LocallyConnectedSpace, Complex, Mandelbrot.multibrotSet, Membership.mem, Set
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
21 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem MLC_general_exponent (n : ℕ) : LocallyConnectedSpace (multibrotSet n) := by
…
```
Mentions: Mandelbrot.multibrotSet
def Mandelbrot.IsAttractingCycle Documented declaration: IsAttractingCycle line 124 source
Source documentation
We say that z : ℂ is part of an attracting cycle of period n of f : ℂ → ℂ if it is an n-periodic point (i.e. f^[n] z = z), f^[n] is differentiable at z, ‖deriv f^[n] z‖ is strictly less than one, and n > 0.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Complex, Nat
Proof uses: And, LT.lt, OfNat.ofNat, Function.IsPeriodicPt, DifferentiableAt, Nat.iterate, Real, Norm.norm, deriv
22 instances, coercions and auxiliary constants left out.
```
def IsAttractingCycle (f : ℂ → ℂ) (n : ℕ) (z : ℂ) : Prop :=
…
```
theorem Mandelbrot.isAttractingCycle_z_squared_minus_one For example, 0 is part of an attracting 2-cycle of z ↦ z ^ 2 - 1 line 129 source
Source documentation
For example, 0 is part of an attracting 2-cycle of z ↦ z ^ 2 - 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Mandelbrot.IsAttractingCycle, Complex, HSub.hSub, HPow.hPow, Nat, OfNat.ofNat
Proof uses: And.intro, LT.lt, And, Function.IsPeriodicPt, DifferentiableAt, Nat.iterate, Real, Norm.norm, deriv, of_decide_eq_true, id, Eq, Bool, Decidable.decide, Bool.true, Eq.refl, of_eq_true, Eq.trans, True, congrFun' and 33 more
140 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
```
theorem isAttractingCycle_z_squared_minus_one : IsAttractingCycle (fun z ↦ z ^ 2 - 1) 2 0 :=
…
```
Mentions: Mandelbrot.IsAttractingCycle
theorem Mandelbrot.not_isAttractingCycle_z_squared_minus_two On the other hand, while 2 is part of a 1-cycle of z ↦ z ^ 2 - 2, that cycle is not attracting line 135 source
Source documentation
On the other hand, while 2 is part of a 1-cycle of z ↦ z ^ 2 - 2, that cycle is not attracting.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Not, Mandelbrot.IsAttractingCycle, Complex, HSub.hSub, HPow.hPow, Nat, OfNat.ofNat
Proof uses: of_eq_true, Eq.trans, Function.IsPeriodicPt, LT.lt, Real, HMul.hMul, True, And, congrArg, DifferentiableAt, Nat.iterate, Norm.norm, deriv, congr, Eq, eq_self, congrFun', Function.iterate_one, deriv_fun_sub, deriv_fun_pow and 21 more
125 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem not_isAttractingCycle_z_squared_minus_two : ¬ IsAttractingCycle (fun z ↦ z ^ 2 - 2) 1 2 := by
…
```
Mentions: Mandelbrot.IsAttractingCycle
theorem Mandelbrot.no_attractingCycle_period_zero No function has an attracting cycle of period 0 line 141 source
Source documentation
No function has an attracting cycle of period 0. This is important in that it means we don't need to require 0 < n in the conjectures below.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Complex, Not, Mandelbrot.IsAttractingCycle, OfNat.ofNat, Nat
Proof uses: of_eq_true, Eq.trans, False, True, congrArg, And, LT.lt, Function.IsPeriodicPt, DifferentiableAt, Nat.iterate, Real, Norm.norm, deriv, congr, congrFun', congrFun, id, deriv_id', NormOneClass.norm_one, and_false and 2 more
37 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem no_attractingCycle_period_zero (f : ℂ → ℂ) (z : ℂ) : ¬ IsAttractingCycle f 0 z := by
…
```
Mentions: Mandelbrot.IsAttractingCycle
theorem Mandelbrot.density_of_hyperbolicity The density of hyperbolicity conjecture, stating that the set of all parameters c for which fun z ↦ z ^ 2 + c has an attracting cycle is dense in the Mandelbrot set sorry line 147 source
Source documentation
The density of hyperbolicity conjecture, stating that the set of all parameters c for which fun z ↦ z ^ 2 + c has an attracting cycle is dense in the Mandelbrot set.

depends on sorry what this means
Statement uses: LE.le, Set, Complex, Mandelbrot.mandelbrotSet, closure, Set.ofPred, Exists, Nat, Mandelbrot.IsAttractingCycle, HAdd.hAdd, HPow.hPow, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
32 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem density_of_hyperbolicity :
    mandelbrotSet ⊆ closure {c | ∃ m z, IsAttractingCycle (fun z ↦ z ^ 2 + c) m z} := by
…
```
Mentions: Mandelbrot.mandelbrotSet, c, Mandelbrot.IsAttractingCycle
theorem Mandelbrot.density_of_hyperbolicity_general_exponent Documented declaration: density_of_hyperbolicity_general_exponent sorry line 155 source
Source documentation
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.

depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Set, Complex, Mandelbrot.multibrotSet, closure, Set.ofPred, Exists, Mandelbrot.IsAttractingCycle, HAdd.hAdd, HPow.hPow
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
34 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
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
…
```
Mentions: Mandelbrot.multibrotSet, c, Mandelbrot.IsAttractingCycle
theorem Mandelbrot.multibrotSet_frontier_measurable The boundary of any Multibrot set is measurable because it is closed, so it makes sense to ask about its area line 162 source
Source documentation
The boundary of any Multibrot set is measurable because it is closed, so it makes sense to ask about its area.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, MeasurableSet, Complex, frontier, Mandelbrot.multibrotSet
Proof uses: IsClosed.measurableSet, isClosed_frontier
18 instances, coercions and auxiliary constants left out.
```
theorem multibrotSet_frontier_measurable {n : ℕ} : MeasurableSet (frontier (multibrotSet n)) := isClosed_frontier.measurableSet
…
```
Mentions: Mandelbrot.multibrotSet
theorem Mandelbrot.volume_frontier_mandelbrotSet_eq_zero The boundary of the Mandelbrot set is conjectured to have zero area sorry line 166 source
Source documentation
The boundary of the Mandelbrot set is conjectured to have zero area.

depends on sorry what this means
Statement uses: Eq, ENNReal, MeasureTheory.Measure, Complex, Set, MeasureTheory.MeasureSpace.volume, frontier, Mandelbrot.mandelbrotSet, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, Nat
37 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem volume_frontier_mandelbrotSet_eq_zero : volume (frontier mandelbrotSet) = 0 := by
…
```
Mentions: Mandelbrot.mandelbrotSet
theorem Mandelbrot.volume_frontier_multibrotSet_eq_zero The boundary of any Multibrot set is conjectured to have zero area sorry line 173 source
Source documentation
The boundary of any Multibrot set is conjectured to have zero area. Note that we don't need to exclude the trivial cases n = 0 and n = 1 because the conjecture holds for them.

depends on sorry what this means
Statement uses: Nat, Eq, ENNReal, MeasureTheory.Measure, Complex, Set, MeasureTheory.MeasureSpace.volume, frontier, Mandelbrot.multibrotSet, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
37 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem volume_frontier_multibrotSet_eq_zero {n : ℕ} : volume (frontier (multibrotSet n)) = 0 := by
…
```
Mentions: Mandelbrot.multibrotSet