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feat(Dynamics/Birkhoff): define Birkhoff sum and Birkhoff average (#6131
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/- | ||
Copyright (c) 2023 Yury Kudryashov. All rights reserved. | ||
Released under Apache 2.0 license as described in the file LICENSE. | ||
Authors: Yury Kudryashov | ||
-/ | ||
import Mathlib.Dynamics.BirkhoffSum.Basic | ||
import Mathlib.Algebra.Module.Basic | ||
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/-! | ||
# Birkhoff average | ||
In this file we define `birkhoffAverage f g n x` to be | ||
$$ | ||
\frac{1}{n}\sum_{k=0}^{n-1}g(f^{[k]}(x)), | ||
$$ | ||
where `f : α → α` is a self-map on some type `α`, | ||
`g : α → M` is a function from `α` to a module over a division semiring `R`, | ||
and `R` is used to formalize division by `n` as `(n : R)⁻¹ • _`. | ||
While we need an auxiliary division semiring `R` to define `birkhoffAverage`, | ||
the definition does not depend on the choice of `R`, | ||
see `birkhoffAverage_congr_ring`. | ||
-/ | ||
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section birkhoffAverage | ||
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variable (R : Type _) {α M : Type _} [DivisionSemiring R] [AddCommMonoid M] [Module R M] | ||
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/-- The average value of `g` on the first `n` points of the orbit of `x` under `f`, | ||
i.e. the Birkhoff sum `∑ k in Finset.range n, g (f^[k] x)` divided by `n`. | ||
This average appears in many ergodic theorems | ||
which say that `(birkhoffAverage R f g · x)` | ||
converges to the "space average" `⨍ x, g x ∂μ` as `n → ∞`. | ||
We use an auxiliary `[DivisionSemiring R]` to define division by `n`. | ||
However, the definition does not depend on the choice of `R`, | ||
see `birkhoffAverage_congr_ring`. -/ | ||
def birkhoffAverage (f : α → α) (g : α → M) (n : ℕ) (x : α) : M := (n : R)⁻¹ • birkhoffSum f g n x | ||
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theorem birkhoffAverage_zero (f : α → α) (g : α → M) (x : α) : | ||
birkhoffAverage R f g 0 x = 0 := by simp [birkhoffAverage] | ||
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@[simp] theorem birkhoffAverage_zero' (f : α → α) (g : α → M) : birkhoffAverage R f g 0 = 0 := | ||
funext <| birkhoffAverage_zero _ _ _ | ||
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theorem birkhoffAverage_one (f : α → α) (g : α → M) (x : α) : | ||
birkhoffAverage R f g 1 x = g x := by simp [birkhoffAverage] | ||
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@[simp] | ||
theorem birkhoffAverage_one' (f : α → α) (g : α → M) : birkhoffAverage R f g 1 = g := | ||
funext <| birkhoffAverage_one R f g | ||
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theorem map_birkhoffAverage (S : Type _) {F N : Type _} | ||
[DivisionSemiring S] [AddCommMonoid N] [Module S N] | ||
[AddMonoidHomClass F M N] (g' : F) (f : α → α) (g : α → M) (n : ℕ) (x : α) : | ||
g' (birkhoffAverage R f g n x) = birkhoffAverage S f (g' ∘ g) n x := by | ||
simp only [birkhoffAverage, map_inv_nat_cast_smul g' R S, map_birkhoffSum] | ||
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theorem birkhoffAverage_congr_ring (S : Type _) [DivisionSemiring S] [Module S M] | ||
(f : α → α) (g : α → M) (n : ℕ) (x : α) : | ||
birkhoffAverage R f g n x = birkhoffAverage S f g n x := | ||
map_birkhoffAverage R S (AddMonoidHom.id M) f g n x | ||
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theorem birkhoffAverage_congr_ring' (S : Type _) [DivisionSemiring S] [Module S M] : | ||
birkhoffAverage (α := α) (M := M) R = birkhoffAverage S := by | ||
ext; apply birkhoffAverage_congr_ring | ||
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theorem Function.IsFixedPt.birkhoffAverage_eq [CharZero R] {f : α → α} {x : α} (h : IsFixedPt f x) | ||
(g : α → M) {n : ℕ} (hn : n ≠ 0) : birkhoffAverage R f g n x = g x := by | ||
rw [birkhoffAverage, h.birkhoffSum_eq, nsmul_eq_smul_cast R, inv_smul_smul₀] | ||
rwa [Nat.cast_ne_zero] | ||
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end birkhoffAverage | ||
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/-- Birkhoff average is "almost invariant" under `f`: | ||
the difference between `birkhoffAverage R f g n (f x)` and `birkhoffAverage R f g n x` | ||
is equal to `(n : R)⁻¹ • (g (f^[n] x) - g x)`. -/ | ||
theorem birkhoffAverage_apply_sub_birkhoffAverage (R : Type _) [DivisionRing R] | ||
[AddCommGroup M] [Module R M] (f : α → α) (g : α → M) (n : ℕ) (x : α) : | ||
birkhoffAverage R f g n (f x) - birkhoffAverage R f g n x = | ||
(n : R)⁻¹ • (g (f^[n] x) - g x) := by | ||
simp only [birkhoffAverage, birkhoffSum_apply_sub_birkhoffSum, ← smul_sub] |
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/- | ||
Copyright (c) 2023 Yury Kudryashov. All rights reserved. | ||
Released under Apache 2.0 license as described in the file LICENSE. | ||
Authors: Yury Kudryashov | ||
-/ | ||
import Mathlib.Algebra.BigOperators.Basic | ||
import Mathlib.Dynamics.FixedPoints.Basic | ||
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/-! | ||
# Birkhoff sums | ||
In this file we define `birkhoffSum f g n x` to be the sum `∑ k in Finset.range n, g (f^[k] x)`. | ||
This sum (more precisely, the corresponding average `n⁻¹ • birkhoffSum f g n x`) | ||
appears in various ergodic theorems | ||
saying that these averages converge to the "space average" `⨍ x, g x ∂μ` in some sense. | ||
See also `birkhoffAverage` defined in `Dynamics/BirkhoffSum/Average`. | ||
-/ | ||
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open Finset Function | ||
open scoped BigOperators | ||
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section AddCommMonoid | ||
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variable {α M : Type _} [AddCommMonoid M] | ||
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/-- The sum of values of `g` on the first `n` points of the orbit of `x` under `f`. -/ | ||
def birkhoffSum (f : α → α) (g : α → M) (n : ℕ) (x : α) : M := ∑ k in range n, g (f^[k] x) | ||
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theorem birkhoffSum_zero (f : α → α) (g : α → M) (x : α) : birkhoffSum f g 0 x = 0 := | ||
sum_range_zero _ | ||
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@[simp] | ||
theorem birkhoffSum_zero' (f : α → α) (g : α → M) : birkhoffSum f g 0 = 0 := | ||
funext <| birkhoffSum_zero _ _ | ||
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theorem birkhoffSum_one (f : α → α) (g : α → M) (x : α) : birkhoffSum f g 1 x = g x := | ||
sum_range_one _ | ||
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@[simp] | ||
theorem birkhoffSum_one' (f : α → α) (g : α → M) : birkhoffSum f g 1 = g := | ||
funext <| birkhoffSum_one f g | ||
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theorem birkhoffSum_succ (f : α → α) (g : α → M) (n : ℕ) (x : α) : | ||
birkhoffSum f g (n + 1) x = birkhoffSum f g n x + g (f^[n] x) := | ||
sum_range_succ _ _ | ||
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theorem birkhoffSum_succ' (f : α → α) (g : α → M) (n : ℕ) (x : α) : | ||
birkhoffSum f g (n + 1) x = g x + birkhoffSum f g n (f x) := | ||
(sum_range_succ' _ _).trans (add_comm _ _) | ||
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theorem birkhoffSum_add (f : α → α) (g : α → M) (m n : ℕ) (x : α) : | ||
birkhoffSum f g (m + n) x = birkhoffSum f g m x + birkhoffSum f g n (f^[m] x) := by | ||
simp_rw [birkhoffSum, sum_range_add, add_comm m, iterate_add_apply] | ||
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theorem Function.IsFixedPt.birkhoffSum_eq {f : α → α} {x : α} (h : IsFixedPt f x) (g : α → M) | ||
(n : ℕ) : birkhoffSum f g n x = n • g x := by | ||
simp [birkhoffSum, (h.iterate _).eq] | ||
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theorem map_birkhoffSum {F N : Type _} [AddCommMonoid N] [AddMonoidHomClass F M N] | ||
(g' : F) (f : α → α) (g : α → M) (n : ℕ) (x : α) : | ||
g' (birkhoffSum f g n x) = birkhoffSum f (g' ∘ g) n x := | ||
map_sum g' _ _ | ||
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end AddCommMonoid | ||
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section AddCommGroup | ||
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variable {α G : Type _} [AddCommGroup G] | ||
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/-- Birkhoff sum is "almost invariant" under `f`: | ||
the difference between `birkhoffSum f g n (f x)` and `birkhoffSum f g n x` | ||
is equal to `g (f^[n] x) - g x`. -/ | ||
theorem birkhoffSum_apply_sub_birkhoffSum (f : α → α) (g : α → G) (n : ℕ) (x : α) : | ||
birkhoffSum f g n (f x) - birkhoffSum f g n x = g (f^[n] x) - g x := by | ||
rw [← sub_eq_iff_eq_add.2 (birkhoffSum_succ f g n x), | ||
← sub_eq_iff_eq_add.2 (birkhoffSum_succ' f g n x), | ||
← sub_add, ← sub_add, sub_add_comm] | ||
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end AddCommGroup |