| import OAI.Dynamics.StandardMap.ForwardChart |
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| namespace OAI |
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| open MeasureTheory Set |
| open scoped ENNReal BigOperators |
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| open Set Filter Metric |
| open scoped Topology |
| namespace StandardMapEntropy |
| structure BridgeScalarControl (k : ℝ) : Prop where |
| nonneg : 0 ≤ k |
| prefix_geom : growthBase k^(-(49/25:ℝ)) ≤ 1/2 |
| prefix_small : 3*growthBase k^(-(47/50:ℝ)) ≤ 1/4 |
| graph_geom : growthBase k^(-(3/5:ℝ)) ≤ 1/2 |
| graph_small : (384*Real.pi)*growthBase k^(-(7/10:ℝ)) ≤ 1/2 |
| slope_small : 24/growthBase k^((4/5:ℝ)) ≤ 1/2 |
| lemma BridgeScalarControl.prefix_dirichlet {k : ℝ} (h : BridgeScalarControl k) |
| (B n : ℕ) (hn : 1 ≤ n) (hB : B=n ∨ B=n+1) (hBn : 2 ≤ B) |
| (z : ℝ × ℝ) (hz : prefixEligible k B (n+1) z) : |
| tSolution (orbitCoefficient k z.1 z.2) (n+1) ≠ 0 ∧ ∀ p, 1 ≤ p → p ≤ n → |
| |dirichletSolution (orbitCoefficient k z.1 z.2) (n+1) p| ≤ |
| 12*growthBase k^(-(9/10:ℝ)*(p:ℝ)) := by |
| have hJ : n+1=B ∨ n+1=B+1 := by rcases hB with rfl|rfl <;> omega |
| have hh := eligible_prefix_dirichlet (orbitCoefficient k z.1 z.2) (growthBase k) B (n+1) |
| hBn hJ (by have := growthBase_ge_four k h.nonneg; linarith) h.prefix_geom h.prefix_small |
| (fun i _ _ => potential_bound k _ h.nonneg) hz.1 hz.2 |
| exact ⟨hh.1,fun p hp hpn => hh.2 p hp (by omega)⟩ |
| lemma eventually_bridgeScalarControl : ∀ᶠ k : ℝ in atTop, BridgeScalarControl k := by |
| have he (a C ε : ℝ) (ha : 0 < a) (hε : 0 < ε) : |
| ∀ᶠ k : ℝ in atTop, C*growthBase k^(-a) ≤ ε := by |
| have ht := ((tendsto_rpow_neg_atTop ha).comp tendsto_growthBase).const_mul C |
| have hh := ht.eventually (gt_mem_nhds (show C*0 < ε by simpa using hε)) |
| exact hh.mono fun k hk => hk.le |
| filter_upwards [eventually_ge_atTop (0:ℝ),he (49/25) 1 (1/2) (by norm_num) (by norm_num), |
| he (47/50) 3 (1/4) (by norm_num) (by norm_num),he (3/5) 1 (1/2) (by norm_num) (by norm_num), |
| he (7/10) (384*Real.pi) (1/2) (by norm_num) (by norm_num),he (4/5) 24 (1/2) (by norm_num) (by norm_num)] |
| with k hk h1 h2 h3 h4 h5 |
| refine ⟨hk,by simpa only [one_mul] using h1,h2,by simpa only [one_mul] using h3,h4,?_⟩ |
| simpa only [Real.rpow_neg (by have := growthBase_ge_four k hk; linarith : 0 ≤ growthBase k),div_eq_mul_inv] using h5 |
| end StandardMapEntropy |
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| end OAI |
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