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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 43 44 45 46 47 48 49 50 51 52 53 54 55 | import Mathlib
import OAI.Probability.CriticalPercolation.LinearGroups.PowerBoundedUnipotent
namespace OAI
section
noncomputable section
open Module
namespace CriticalPercolation.LinearGrowth
variable {V : Type*} [NormedAddCommGroup V] [NormedSpace ℂ V] [FiniteDimensional ℂ V]
def continuousEndAlgEquiv : End ℂ V ≃ₐ[ℂ] (V →L[ℂ] V) :=
AlgEquiv.ofLinearEquiv LinearMap.toContinuousLinearMap rfl (fun _ _ => rfl)
omit [FiniteDimensional ℂ V] in
lemma scalar_power_of_projective_bound (T : V →L[ℂ] V)
(δ : ℝ) (hδ : 0<δ)
(hb : ∀ n : ℕ, ∀ x : V, δ*‖T^n‖*‖x‖≤‖(T^n) x‖)
{v : V} (hv : v≠0) {a : ℂ} (ha : a≠0) (he : T v=a • v)
(m : ℕ) (hni : IsNilpotent (T^m-a^m • (1:V →L[ℂ] V))) :
T^m=a^m • (1:V →L[ℂ] V) := by
let U := (a^m)⁻¹ • T^m
have hU : IsNilpotent (U-1) := by
have hh := hni.smul ((a^m)⁻¹)
simpa only [U,smul_sub,smul_smul,inv_mul_cancel₀ (pow_ne_zero m ha),one_smul] using hh
have hepow (n : ℕ) : (T^n) v=a^n • v := by
induction n with
| zero => simp
| succ n ih =>
rw [pow_succ']
change T ((T^n) v)=_
rw [ih,map_smul,he,smul_smul,pow_succ]
have hnorm (n : ℕ) : δ*‖T^n‖≤‖a‖^n := by
have hh := hb n v
rw [hepow,norm_smul,norm_pow a n] at hh
exact (mul_le_mul_iff_left₀ (norm_pos_iff.mpr hv)).mp hh
have hUb (n : ℕ) : ‖U^n‖≤1/δ := by
dsimp [U]
rw [smul_pow,norm_smul,norm_pow ((a^m)⁻¹) n,norm_inv,norm_pow a m,inv_pow,←pow_mul,←pow_mul]
have hh := hnorm (m*n)
have hpos : 0<‖a‖^(m*n) := pow_pos (norm_pos_iff.mpr ha) _
rw [inv_mul_eq_div]
apply (div_le_div_iff₀ hpos hδ).mpr
nlinarith
have hU1 : U=1 := eq_one_of_power_bounded_unipotent U hU (1/δ) hUb
have hh := congrArg (fun x : V →L[ℂ] V => a^m • x) hU1
simpa only [U,smul_smul,mul_inv_cancel₀ (pow_ne_zero m ha),one_smul] using hh
end CriticalPercolation.LinearGrowth
end
end
end OAI
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