Exercise 16, § 7

Let $X$ be a locally compact space, $\mu$ a positive measure of total mass 1 on $X$ and $f: X\rightarrow X$ a mapping such that $f(\mu) =\mu$.

(a) The mapping $\varphi \mapsto \varphi \circ f$ induces a unitary endomorphism $u_f$ of $L^2(X, \mu)$.

(b) The real number 1 is an eigenvalue of $u_f$ of multiplicity 1 if and only if, for every measurable subset A of X such that $f^{-1}(A) = A$, one has $\mu(A)\in \{0,1\}$. One then says that $f$ is $\mu$-ergodic.

(c) Let $E\subset L^2(X, \mu)$ be the proper subspace of $u_f$ for the eigenvalue 1. The sequence $(v_N)_{N\in\mathbf{N}}$ defined by

converges to the orthogonal projection onto E in the space $\mathscr{L}(L^2(X, \mu))$ endowed with the topology of simple convergence (“von-Neumann ergodic theorem”).(Note that the orthogonal of E is the closure of the image of $u_f-1_E$; verify then that $v_N(\varphi )\rightarrow 0$ when $\varphi \in E^{\circ}.$)

(d) One defines an endomorphism $\widetilde{u}_f$ of $L^1(X, \mu)$ as in question a), and one sets

For every $\varphi \in L^1(X, \mu)$, the sequence $(\widetilde{v}_N(f))_{N\in\mathbf{N}}$ converges in $L^1(X, \mu)$ towards an element $\widetilde{\varphi}$ of the proper subspace of $\widetilde{u}_f$ for the eigenvalue 1. (First consider the case where $\varphi$ is bounded, and apply the preceding question.)

(e) Let I be a finite set and $X = I^{\mathbf{Z}}$ endowed with the product measure of the measures $\mu_n$ on I such that $\mu_n(i) = 1/$Card(I) for every $i\in I$. Put $f((i_n)) = (i_{n+1})$ for every $(i_n)\in X$. Then $f(\mu) =\mu$ and $f$ is $\mu$-ergodic (“Bernoulli shift”).

(f) Let $X =\mathbf{R}/\mathbf{Z}$ and $\mu$ the normalised Haar measure on X. Let $a\in X$ and put $f(x) =x+a$. Then $f(\mu) =\mu$ and $f$ is $\mu$-ergodic if and only if $a$ is the class of an irrational number.