File size: 5,574 Bytes
92416a2 | 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 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 | import Mathlib
namespace OAI
open Filter Set Uniformity
open scoped Topology ENNReal
namespace Problem340
lemma log_nat_div_tendsto_zero :
Tendsto (fun n : ℕ => Real.log (n : ℝ) / n) atTop (𝓝 0) := by
simpa [Function.comp_def] using (Real.tendsto_pow_log_div_mul_add_atTop 1 0 1 one_ne_zero).comp
(tendsto_natCast_atTop_atTop : Tendsto (fun n : ℕ => (n : ℝ)) atTop atTop)
lemma expGrowthSup_nat_pow (d : ℕ) :
ExpGrowth.expGrowthSup (fun n : ℕ => (n : ℝ≥0∞) ^ d) = 0 := by
have h : Tendsto (fun n : ℕ => (d : ℝ) * (Real.log (n : ℝ) / n))
atTop (𝓝 0) := by
simpa using log_nat_div_tendsto_zero.const_mul (d : ℝ)
have h' := EReal.tendsto_coe.mpr h
have heq : (fun n : ℕ => ENNReal.log ((n : ℝ≥0∞) ^ d) / (n : EReal))
=ᶠ[atTop] (fun n : ℕ => (((d : ℝ) * (Real.log (n : ℝ) / n) : ℝ) : EReal)) := by
filter_upwards [eventually_gt_atTop 0] with n hn
have hnR : (0 : ℝ) < n := by exact_mod_cast hn
rw [ENNReal.log_pow]
have hlog : ENNReal.log (n : ℝ≥0∞) = (Real.log (n : ℝ) : EReal) := by
rw [← ENNReal.ofReal_natCast, ENNReal.log_ofReal_of_pos hnR]
rw [hlog]
change (((d : ℝ) * Real.log (n : ℝ) / n : ℝ) : EReal) = _
congr 1
ring
exact (h'.congr' heq.symm).limsup_eq
lemma expGrowthSup_le_zero_of_polynomial_bound {u : ℕ → ℝ≥0∞} (C d : ℕ)
(h : ∀ᶠ n in atTop, u n ≤ (C : ℝ≥0∞) * (n + 1) ^ d) :
ExpGrowth.expGrowthSup u ≤ 0 := by
have hbound : ∀ᶠ n in atTop,
u n ≤ ((C : ℝ≥0∞) * 2 ^ d) * (n : ℝ≥0∞) ^ d := by
filter_upwards [h, eventually_ge_atTop 1] with n hn hn1
apply hn.trans
calc
(C : ℝ≥0∞) * (n + 1) ^ d ≤ (C : ℝ≥0∞) * (2 * (n : ℝ≥0∞)) ^ d := by
gcongr
exact_mod_cast (show n + 1 ≤ 2 * n by omega)
_ = ((C : ℝ≥0∞) * 2 ^ d) * (n : ℝ≥0∞) ^ d := by rw [mul_pow, mul_assoc]
exact (ExpGrowth.expGrowthSup_le_of_eventually_le (by finiteness) hbound).trans_eq
(expGrowthSup_nat_pow d)
lemma coverEntropy_eq_zero_of_polynomial_cover_bound {X : Type*} [UniformSpace X]
(T : X → X) {F : Set X} (hF : F.Nonempty)
(h : ∀ U ∈ 𝓤 X, ∃ C d : ℕ, ∀ n : ℕ,
Dynamics.coverMincard T F U n ≤ (C * (n + 1) ^ d : ℕ)) :
Dynamics.coverEntropy T F = 0 := by
apply le_antisymm _ (Dynamics.coverEntropy_nonneg T hF)
apply iSup₂_le
intro U hU
obtain ⟨C, d, hbound⟩ := h U hU
apply expGrowthSup_le_zero_of_polynomial_bound C d
exact Filter.Eventually.of_forall fun n => by
simpa using ENat.toENNReal_mono (hbound n)
lemma coverEntropy_eq_zero_of_polynomial_net_bound {X : Type*} [UniformSpace X]
(T : X → X) {F : Set X} (hF : F.Nonempty)
(h : ∀ U ∈ 𝓤 X, ∃ C d : ℕ, ∀ n : ℕ,
Dynamics.netMaxcard T F U n ≤ (C * (n + 1) ^ d : ℕ)) :
Dynamics.coverEntropy T F = 0 := by
apply le_antisymm _ (Dynamics.coverEntropy_nonneg T hF)
rw [Dynamics.coverEntropy_eq_iSup_netEntropyEntourage]
apply iSup₂_le
intro U hU
obtain ⟨C, d, hbound⟩ := h U hU
apply expGrowthSup_le_zero_of_polynomial_bound C d
exact Filter.Eventually.of_forall fun n => by
simpa using ENat.toENNReal_mono (hbound n)
lemma netMaxcard_le_card_of_orbit_code {X ι : Type*} [Fintype ι]
(T : X → X) (F : Set X) (U : SetRel X X) [U.IsRefl] (n : ℕ)
(code : X → ι)
(hcode : ∀ x ∈ F, ∀ y ∈ F, code x = code y →
(x, y) ∈ Dynamics.dynEntourage T U n) :
Dynamics.netMaxcard T F U n ≤ (Fintype.card ι : ℕ) := by
classical
apply iSup₂_le
intro s hs
have hinj : Set.InjOn code (s : Set X) := by
intro x hx y hy hxy
exact Set.PairwiseDisjoint.elim_set hs.2 hx hy y
(hcode x (hs.1 hx) y (hs.1 hy) hxy)
(SetRel.rfl (Dynamics.dynEntourage T U n))
have hcard : s.card ≤ Fintype.card ι := by
simpa using Finset.card_le_card_of_injOn code (fun x _ => Finset.mem_univ (code x)) hinj
exact_mod_cast hcard
lemma coverEntropy_eq_zero_of_eventually_polynomial_net_bound {X : Type*} [UniformSpace X]
(T : X → X) {F : Set X} (hF : F.Nonempty)
(h : ∀ U ∈ 𝓤 X, ∃ C d : ℕ, ∀ᶠ n in atTop,
Dynamics.netMaxcard T F U n ≤ (C * (n + 1) ^ d : ℕ)) :
Dynamics.coverEntropy T F = 0 := by
apply le_antisymm _ (Dynamics.coverEntropy_nonneg T hF)
rw [Dynamics.coverEntropy_eq_iSup_netEntropyEntourage]
apply iSup₂_le
intro U hU
obtain ⟨C, d, hbound⟩ := h U hU
apply expGrowthSup_le_zero_of_polynomial_bound C d
filter_upwards [hbound] with n hn
simpa using ENat.toENNReal_mono hn
lemma coverEntropy_eq_zero_of_polynomial_orbit_codes {X : Type*} [PseudoMetricSpace X]
(T : X → X) {F : Set X} (hF : F.Nonempty)
(h : ∀ ε : ℝ, 0 < ε → ∃ C d : ℕ, ∀ᶠ n in atTop,
∃ code : X → Fin (C * (n + 1) ^ d),
∀ x ∈ F, ∀ y ∈ F, code x = code y →
∀ j < n, dist (T^[j] x) (T^[j] y) < ε) :
Dynamics.coverEntropy T F = 0 := by
apply coverEntropy_eq_zero_of_eventually_polynomial_net_bound T hF
intro U hU
obtain ⟨ε, hε, hεU⟩ := Metric.mem_uniformity_dist.mp hU
obtain ⟨C, d, hcode⟩ := h ε hε
refine ⟨C, d, ?_⟩
filter_upwards [hcode] with n hn
obtain ⟨code, hshadow⟩ := hn
have : SetRel.IsRefl U := isRefl_of_mem_uniformity hU
simpa using netMaxcard_le_card_of_orbit_code T F U n code (by
intro x hx y hy hxy
apply Dynamics.mem_dynEntourage.mpr
intro j hj
exact hεU (hshadow x hx y hy hxy j hj))
end Problem340
end OAI
|