| import OAI.Dynamics.StandardMap.ScaledChart |
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| namespace OAI |
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| open MeasureTheory Set |
| open scoped ENNReal BigOperators |
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| open MeasureTheory Set Filter Metric |
| open scoped ENNReal Topology |
| namespace StandardMapEntropy |
| lemma continuous_liftProjection : Continuous liftProjection := |
| (((AddCircle.continuous_mk' (1:ℝ)).comp continuous_fst).prodMk |
| ((AddCircle.continuous_mk' (1:ℝ)).comp continuous_snd)) |
| lemma interval_small_move (y y' : ℝ) (hy : y ∈ Icc (0:ℝ) 1) (hmove : |y'-y| ≤ 1) : |
| y' ∈ Ioc (-2:ℝ) 2 := by |
| have hh := abs_le.mp hmove |
| constructor <;> linarith [hy.1,hy.2,hh.1,hh.2] |
| lemma actual_backward_slice (k : ℝ) (n B : ℕ) (hn : 1 ≤ n) (hB : B=n ∨ B=n+1) (hk : 0 ≤ k) |
| (hq : growthBase k^(-(3/5:ℝ)) ≤ 1/2) |
| (hsmall : (384*Real.pi)*growthBase k^(-(7/10:ℝ)) ≤ 1/2) |
| (ha : 24/growthBase k^((4/5:ℝ)) ≤ 1/2) |
| (D : Set ℝ) (hD : MeasurableSet D) (X X' : ℝ → ℝ) (l u : ℝ) |
| (hlen : u-l < 1/4) (hlu : Icc l u ⊆ Icc (0:ℝ) 1) |
| (hDunit : D ⊆ Icc (0:ℝ) 1) |
| (hXI : ∀ y ∈ D, X y ∈ Icc l u) |
| (hX : ∀ y ∈ D, HasStrictDerivAt X (X' y) y) |
| (hXs : ∀ y ∈ D, |X' y| ≤ 24/growthBase k^((4/5:ℝ))) |
| (hXL : ∀ y ∈ D, ∀ z ∈ D, |X y-X z| ≤ (24/growthBase k^((4/5:ℝ)))*|y-z|) |
| (ht : ∀ y ∈ D, tSolution (orbitCoefficient k (X y) y) (n+1) ≠ 0) |
| (hU : ∀ y ∈ D, ∀ p, 1 ≤ p → p ≤ n → |
| |dirichletSolution (orbitCoefficient k (X y) y) (n+1) p| ≤ |
| 12*growthBase k^(-(9/10:ℝ)*(p:ℝ))) |
| (W : Torus → ℝ) (hW : Continuous W) (hW0 : ∀ z, 0 ≤ W z) (δ : ℝ) (hδ : 0 ≤ δ) |
| (htransport : ∀ z w : ℝ × ℝ, |
| dist (liftIter k (-(B:ℤ)) z) (liftIter k (-(B:ℤ)) w) ≤ |
| 8/growthBase k^((4/5:ℝ)*(B-1:ℕ)) → |
| |W (liftProjection z)-W (liftProjection w)| ≤ δ) : |
| (ENNReal.ofReal (Real.exp (-1152*Real.pi)*(1-(24/growthBase k^((4/5:ℝ)))^2))*volume (Icc l u))* |
| (∫⁻ y in D, ENNReal.ofReal (W (liftProjection (X y,y)))) ≤ |
| 16*(∫⁻ z, ENNReal.ofReal (W z) ∂area)+ |
| (ENNReal.ofReal (Real.exp (-1152*Real.pi)*(1-(24/growthBase k^((4/5:ℝ)))^2))*volume (Icc l u))* |
| ENNReal.ofReal δ*volume D := by |
| classical |
| have hp : 0 < growthBase k := by have := growthBase_ge_four k hk; linarith |
| have ha0 : 0 ≤ 24/growthBase k^((4/5:ℝ)) := by positivity |
| have htrans : (24/growthBase k^((4/5:ℝ)))^2 < 1 := by nlinarith |
| obtain ⟨Θ,J,hΘ,hbase,hlevel,hinj,hJ,hdet,hclose⟩ := actual_backward_carry_full k n hn hk hq hsmall D X X' l u hlen hXI hX hXs hXL htrans ht hU |
| let w : ℝ → ℝ≥0∞ := fun y => ENNReal.ofReal (W (liftProjection (X y,y))) |
| have hwc : ContinuousOn w D := by |
| apply ENNReal.continuous_ofReal.comp_continuousOn |
| apply hW.comp_continuousOn |
| apply continuous_liftProjection.comp_continuousOn |
| have hxc : ContinuousOn X D := fun y hy => (hX y hy).hasDerivAt.continuousAt.continuousWithinAt |
| exact hxc.prodMk continuousOn_id |
| have hwm : Measurable (D.indicator w) := by |
| exact hwc.measurable_piecewise continuousOn_const hD |
| have hbound : (fun z : ℝ × ℝ => (z.1,Θ z)) '' (Icc l u ×ˢ D) ⊆ |
| Ioc (-2:ℝ) 2 ×ˢ Ioc (-2:ℝ) 2 := by |
| rintro z ⟨v,hv,rfl⟩ |
| have hvu := hlu hv.1 |
| have hxu := hlu (hXI v.2 hv.2) |
| have hd : |v.1-X v.2| ≤ 1 := by apply abs_le.mpr; constructor <;> linarith [hvu.1,hvu.2,hxu.1,hxu.2] |
| have hc := (hclose v.2 hv.2 v.1 hv.1).1 |
| refine ⟨⟨by linarith [hvu.1],by linarith [hvu.2]⟩,interval_small_move v.2 (Θ v) (hDunit hv.2) ?_⟩ |
| exact hc.trans (by nlinarith [abs_nonneg (v.1-X v.2)]) |
| have htpt (z : ℝ × ℝ) (hz : z ∈ Icc l u ×ˢ D) : |
| D.indicator w z.2 ≤ ENNReal.ofReal (W (liftProjection (z.1,Θ z)))+ENNReal.ofReal δ := by |
| rw [indicator_of_mem hz.2] |
| have he := graph_backward_endpoint_close k hk (z.1,Θ z) (X z.2,z.2) n B hn hB |
| (hlevel z.2 hz.2 z.1 hz.1) (hclose z.2 hz.2 z.1 hz.1).2 |
| have hh := htransport (X z.2,z.2) (z.1,Θ z) (by simpa only [dist_comm] using he) |
| calc |
| _ ≤ ENNReal.ofReal (W (liftProjection (z.1,Θ z))+δ) := ENNReal.ofReal_le_ofReal (by have := (abs_le.mp hh).2; linarith) |
| _ = _ := ENNReal.ofReal_add (hW0 _) hδ |
| have hh := sweep_slice_bound (Icc l u) D (fun z : ℝ × ℝ => (z.1,Θ z)) J |
| (D.indicator w) (fun z => ENNReal.ofReal (W z)) _ (ENNReal.ofReal δ) |
| measurableSet_Icc hD hwm (ENNReal.continuous_ofReal.comp hW).measurable |
| hJ hinj (fun z hz => (hdet z hz).1) hbound htpt |
| simpa only [setLIntegral_indicator hD,inter_self,w] using hh |
| end StandardMapEntropy |
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| end OAI |
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