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Quasi-discrete spectrum and the Wiener-Wintner ergodic theorem for polynomials. (Spectre quasi-discret et théorème ergodique de Wiener-Wintner pour les polynômes.) (French) Zbl 0803.47013
Let $(\Omega,\mu)$ be a separable probability space, $T: \Omega\to \Omega$ be measure-preserving. Fix $f\in L\sp 1$. The ergodic theorem of Wiener-Wintner asserts that for $\mu$-a.e. $\omega$, $${1\over n} \sum\sp{n-1}\sb{k=0} \exp(ik\alpha)\cdot f(T\sp k \omega)\quad\text{converges for all }\alpha\in \bbfR.$$ The author uses the Van der Corput inequality to show the following generalization: For $\mu$-a.e. $\omega$, $${1\over n} \sum\sp{n-1}\sb{k=0} \phi(P(k))\cdot f(T\sp k \omega)\quad\text{converges}$$ for all real polynomials $P$ and all continuous periodic functions $\phi$ on $\bbfR$. If, in addition, $T$ is weakly mixing and $\int f d\mu= 0$, then for $\mu$-a.e. $\omega$, for all $M\in \bbfN$ and all continuous periodic $\phi$, $$\sup\sb P \left\vert{1\over n} \sum\sp{n-1}\sb{k=0} \phi(P(k))\cdot f(T\sp k \omega)\right\vert\to 0,$$ where the supremum is taken over all real polynomials of degree bounded by $M$. The proof of the Wiener-Wintner theorem is based on an equivalence which the author generalizes in the following way: Let $E\sb 0$ be the set of eigenvalues of $T$ (as an operator on $L\sp 2$) and, for $m\ge 1$, set $E\sb m:= \{f\in L\sp 2: \vert f\vert\equiv 1,\ f\circ T\cdot \bar f\in E\sb{m-1}\}$. Suppose the linear span of $\bigcup\sb m E\sb m$ is dense in $L\sp 2$, $f\in L\sp 1$ and $M\ge 1$. Then $\int fgd\mu= 0$ holds for all $g\in E\sb M$ if and only if for $\mu$-a.e. $\omega$, for all real polynomials of degree less than or equal to $M$ and every continuous periodic $\phi$, $${1\over n} \sum\sp{n-1}\sb{k=0} \phi(P(k))\cdot f(T\sp k \omega)\to 0.$${}.

47A35Ergodic theory of linear operators
28D05Measure-preserving transformations