Poincarré return theorem

Poincarré return theorem §

Theorem Let $g$ be a continuous, bijective transformation that preserves volumes defined in Euclidean space to itself. Suppose that there exists a finite region $D$ such that $gD=D$. Then for every neighourhood $U$ of every point of $D$, there exists a point $x\in U$ that eventually returns in $U$, that is

$$g^{n}x\in U\text{ for some }n>0$$

Proof Consider the images of the neighborhood $U$:

$$U,gU,\dots ,g^{n}U$$

since $g$ preserves volumes, they have the same (non-zero) volume. If they didn’t intersetc, the volume of $D$ would be infinite, so there exists some $k,l\ge 0$, $k>l$ such that

$$g^{k}U\bigcap g^{l}U\ne \varnothing$$

since $g$ is bijective, we can define its inverse. Applying the inverse of $g^l$ we get

$$g^{k-l}U\bigcap U\ne \varnothing$$

since this set is non-empty, we have just showd that there exists an element $y\in g^{k-l}U\bigcap U$ that is inside the starting neighborhood $U$ and also in $g^{k-l}U$. One could pick $y=g^{k-l}x$. $\square$