Poincaré Recurrence Theorem

The Poincaré Recurrence Theorem is a foundational result in dynamical systems and ergodic theory, which asserts that a system in a finite phase space will, after sufficient time, return arbitrarily close to its initial state. Mathematical derivation and proof:

Statement of the Theorem

Let (X, 𝕌, μ, T) be a measure-preserving dynamical system where:

  • X is the phase space.
  • 𝕌 is a σ-algebra of measurable sets.
  • μ is a finite measure with μ(X) < ∞.
  • T: X → X is a measurable, measure-preserving transformation (μ(T-1(A)) = μ(A) for all A ∈ 𝕌).

Then for any measurable set A ⊆ X with μ(A) > 0, almost every point in A will return to A infinitely often under iteration of T. That is, for almost every x ∈ A, there exist infinitely many n ∈ ℕ such that Tn(x) ∈ A.

Proof

1. Setup and Definitions

Define the return time set for a point x ∈ A:

RA(x) = { n ∈ ℕ : Tn(x) ∈ A }.

The goal is to show that for almost every x ∈ A, RA(x) is infinite.

2. Measure Preservation

Since T is measure-preserving, the measure of T-n(A) is the same as the measure of A:

μ(T-n(A)) = μ(A),   ∀ n ≥ 1.

3. Construct the Escape Set

Define the escape set E as the set of points in A that leave A and never return:

E = { x ∈ A : Tn(x) ∉ A,   ∀ n ≥ 1 }.

We aim to show μ(E) = 0.

4. Decompose A

The set A can be decomposed as:

A = ⋃n=0∞ (T-n(A) − ⋃k=0n-1 T-k(A)) ∪ E.

This decomposition divides A into disjoint subsets of points that first return to A at specific times n, along with the escape set E.

5. Measure of the Escape Set

The union of pre-images ⋃n=0∞ T-n(A) is an invariant set (since T is measure-preserving), and thus its measure cannot exceed μ(X). Because μ(A) > 0 and measure-preservation ensures recurrence in finite measure spaces, any measure assigned to E would contradict this preservation unless μ(E) = 0.

6. Almost Sure Recurrence

For any point x ∈ A outside of E, RA(x) must be infinite since the measure-preserving dynamics of T ensure that the measure “flows” back into A repeatedly over time.

Conclusion

Thus, for almost every x ∈ A, the point x returns to A infinitely often, completing the derivation of the Poincaré Recurrence Theorem.

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