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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 | import OAI.Dynamics.EntropyCounterexample.PeriodicEntropy
namespace OAI
noncomputable section
namespace Problem340
namespace ClockAtInfinity
abbrev Clock := AddCircle (100 : ℝ)
def step (t : Clock) : Clock := t + ((1 : ℝ) : Clock)
@[continuity, fun_prop]
theorem continuous_step : Continuous step :=
continuous_id.add continuous_const
theorem step_period : step^[100] = id := by
have h : (100 : ℕ) • ((1 : ℝ) : Clock) = 0 := by
rw [← AddCircle.coe_nsmul]
norm_num [nsmul_eq_mul]
funext t
change (fun x : Clock => x + ((1 : ℝ) : Clock))^[100] t = t
rw [add_right_iterate]
simp only [h, add_zero]
theorem coverEntropy_step : Dynamics.coverEntropy step (Set.univ : Set Clock) = 0 := by
let : Fact (0 < (100 : ℝ)) := ⟨by norm_num⟩
exact PeriodicEntropy.coverEntropy_eq_zero_of_period (by norm_num) step_period
theorem coverEntropy_clockImage {X : Type*} [UniformSpace X]
{T : X → X} {φ : Clock → X} (hφ : Continuous φ)
(hsem : Function.Semiconj φ step T) :
Dynamics.coverEntropy T (Set.range φ) = 0 := by
let : Fact (0 < (100 : ℝ)) := ⟨by norm_num⟩
apply le_antisymm _ (Dynamics.coverEntropy_nonneg T (Set.range_nonempty φ))
have h := Dynamics.coverEntropy_image_le_of_uniformContinuous hsem
(CompactSpace.uniformContinuous_of_continuous hφ) Set.univ
simpa only [Set.image_univ, coverEntropy_step] using h
end ClockAtInfinity
end Problem340
end
end OAI
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