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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 | import OAI.Dynamics.StandardMap.ScaledChart
namespace OAI
open MeasureTheory Set
open scoped ENNReal BigOperators
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
end OAI
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