## Tag Archives: pushforward

### Ergodic decomposition

Let $X$ be a compact metric space and $f:X\to X$ be a homeomorphism. Let $\mathcal{M}(f)$ be the set of $f$-invariant measures and $\mathcal{E}(f)$ be the set of $f$-invariant ergodic measures.

Let $\mu$ be an invariant measure. A distribution $\tau$ on $\mathcal{E}(f)$ is said to be the ergodic decomposition of $\mu$ if for each continuous function $\phi:X\to\mathbb{R}$, the following holds:

$\int_{\mathcal{E}(f)}(\int_X \phi d\nu) d\tau(\nu)=\int_X \phi d\mu$.

The following approach is attributed to R. Mane.
Let $G_\nu$ be the set of generic points of $\nu$ and $G=\bigcup_{\nu\in \mathcal{E}(f)}G_\nu$. Then $G$ is a Borel subset of $X$ and of full probability. Consider the map $\beta:G\to \mathcal{E}(f),x\mapsto\nu_x=\lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}\delta_{f^kx}$.

Proposition: The map $\beta:G\to \mathcal{E}(f)$ is Borel.

So the pushforwad $\tau=\beta_*(\mu)$ is a Borel distribution on $\mathcal{E}(f)$. Moreover for each continuous map $\Phi:\mathcal{M}(X)\to \mathbb{R}$, we have
$\int_{\mathcal{E}(f)}\Phi(\nu)d\tau(\nu)=\int_G\Phi(\beta(x))d\mu(x)$.

In particular letting $\Phi(\nu)=\int_X \phi d\nu$, we have $\Phi(\beta(x))=\int\phi d\nu_x=\phi^*(x)$ and

$\int_{\mathcal{E}(f)}(\int_X \phi d\nu)d\tau(\nu)=\int_G \phi^*(x)d\mu(x)=\int_X\phi(x)d\mu(x)$, where $\phi^*$ is the Birkhoff average and the last equality follows from Brikhoff ergodic theorem. So $\tau=\beta_*(\mu)$ is the ergodic decomposition of $\mu$.