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f205175 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 | import Mathlib
namespace OAI
section
section
noncomputable section
open Set Function
open scoped Classical
namespace CriticalPercolation.Boundary
def balancedPermutation {V E : Type*} [Fintype E] (tail head : E → V)
(hb : ∀ v, Fintype.card {e // head e=v}=Fintype.card {e // tail e=v}) : Equiv.Perm E :=
Equiv.ofFiberEquiv (fun v => Fintype.equivOfCardEq (hb v))
lemma balancedPermutation_tail {V E : Type*} [Fintype E] (tail head : E → V)
(hb : ∀ v, Fintype.card {e // head e=v}=Fintype.card {e // tail e=v}) (e : E) :
tail (balancedPermutation tail head hb e)=head e :=
Equiv.ofFiberEquiv_map _ e
theorem exists_closed_trail {V E : Type*} [Fintype E] (tail head : E → V)
(hb : ∀ v, Fintype.card {e // head e=v}=Fintype.card {e // tail e=v}) (e₀ : E) :
∃ n : ℕ, 0<n ∧ ∃ e : ℕ → E, e 0=e₀ ∧ e n=e₀ ∧
(∀ i, head (e i)=tail (e (i+1))) ∧ Set.InjOn e (Set.Iio n) := by
let p := balancedPermutation tail head hb
let n := Function.minimalPeriod p e₀
have hn : 0<n := Function.minimalPeriod_pos_of_mem_periodicPts (p.injective.mem_periodicPts e₀)
refine ⟨n,hn,(fun i => p^[i] e₀),rfl,?_,?_,Function.iterate_injOn_Iio_minimalPeriod⟩
· exact Function.isPeriodicPt_minimalPeriod p e₀
· intro i
dsimp only
rw [Function.iterate_succ_apply']
exact (balancedPermutation_tail tail head hb _).symm
end CriticalPercolation.Boundary
end
end
end
end OAI
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