📂Dynamics

Poincaré Recurrence Theorem Proof

Theorem

In the Euclidean space where a multidimensional map $g : \mathbb{R}^{n} \to \mathbb{R}^{n}$ is defined to be injective and continuous, and $D \subset \mathbb{R}^{n}$ is a compact invariant set, let’s say $g(D) = D$. For any $\overline{x} \in D$, given any neighborhood $U$, there exists $x \in U$ such that for some $n \in$, $g^{n} (x) \in U$ is satisfied.

Explanation

The statement is simple: if $D$ is a compact invariant set, then grabbing any $U$ within it means there’s a moment when it can’t immediately escape $U$ but will eventually return to it. This can be applied repeatedly, first, second, third time… so no matter how far $g$ takes $U$ through $U$, it inevitably returns to $U$.

Proof ¹

Consider repeatedly taking $g$ on $U$ as follows: $$U , g (U), g^{2} (U) , \cdots , g^{n} (U) , \cdots$$ Since $g$ was injective from the premise, they conserve the volume exactly. If they don’t overlap, then $D$ containing them would have infinite volume. However, as $D$ is compact from the premise, for some $k >l$ $$g^{k} (U) \cap g^{l} (U) \ne \emptyset$$ Taking the pre-image gives $$g^{k-l} \left( U \right) \cap U \ne \emptyset$$ Thus, for $n := k - l$, there exists $x$ that satisfies both $x \in U$ and $g^{n} (x) \in U$.

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