| import Mathlib |
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| namespace OAI |
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| universe uι |
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| section |
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| open scoped BigOperators ENNReal |
| open MeasureTheory |
| namespace EntropyRateDimension |
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| structure System (ι : Type uι) [Fintype ι] where |
| ratio : ι → ℝ |
| offset : ι → ℝ |
| weight : ι → ℝ |
| ratio_pos : ∀ i, 0 < |ratio i| |
| ratio_lt_one : ∀ i, |ratio i| < 1 |
| weight_pos : ∀ i, 0 < weight i |
| weight_sum : ∑ i, weight i = 1 |
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| namespace System |
| variable {ι : Type uι} [Fintype ι] |
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| def affine (S : System ι) (i : ι) (x : ℝ) : ℝ := S.ratio i * x + S.offset i |
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| def wordAffine (S : System ι) : List ι → ℝ × ℝ |
| | [] => (1, 0) |
| | i :: w => (S.ratio i * (S.wordAffine w).1, |
| S.ratio i * (S.wordAffine w).2 + S.offset i) |
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| def completeMap (S : System ι) {n : ℕ} (w : Fin n → ι) : ℝ × ℝ := |
| S.wordAffine (List.ofFn w) |
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| noncomputable def wordWeight (S : System ι) {n : ℕ} (w : Fin n → ι) : ℝ := |
| ∏ j, S.weight (w j) |
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| noncomputable def mapSupport (S : System ι) (n : ℕ) : Finset (ℝ × ℝ) := by |
| classical |
| exact Finset.univ.image (S.completeMap (n := n)) |
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| noncomputable def mapMass (S : System ι) (n : ℕ) (g : ℝ × ℝ) : ℝ := by |
| classical |
| exact ∑ w ∈ (Finset.univ : Finset (Fin n → ι)).filter (S.completeMap · = g), |
| S.wordWeight w |
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| noncomputable def walkEntropy (S : System ι) (n : ℕ) : ℝ := |
| (∑ g ∈ S.mapSupport n, Real.negMulLog (S.mapMass n g)) / Real.log 2 |
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| noncomputable def entropyRate (S : System ι) : ℝ := |
| sInf (Set.range fun n : ℕ => S.walkEntropy (n + 1) / (n + 1 : ℕ)) |
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| noncomputable def lyapunov (S : System ι) : ℝ := |
| -(∑ i, S.weight i * (Real.log |S.ratio i| / Real.log 2)) |
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| def SelfSimilar (S : System ι) (μ : Measure ℝ) : Prop := |
| μ = ∑ i, ENNReal.ofReal (S.weight i) • μ.map (S.affine i) |
| end System |
| noncomputable def lowerHausdorffDimension (μ : Measure ℝ) : ℝ≥0∞ := |
| ⨅ (E : Set ℝ) (_ : MeasurableSet E) (_ : 0 < μ E), dimH E |
| end EntropyRateDimension |
| end |
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| end OAI |
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