Wiener-Wintner theorem

Wiener–Wintner ergodic theorem

A strengthening of the pointwise ergodic theorem (cf. also Ergodic theory) announced in [a21] and stating that if $( X , \mathcal{F} , \mu , T )$ is a dynamical system, then given $f \in L ^ { 1 } ( \mu )$ one can find a set of full measure $X _ { f }$ such that for $x$ in this set the averages

\begin{equation*} \frac { 1 } { N } \sum _ { n = 1 } ^ { N } f ( T ^ { n } x ) e ^ { 2 \pi i \varepsilon } \end{equation*}

converge for all real numbers $\varepsilon$. In other words, the set $X _ { f }$ "works" for an uncountable number of $\varepsilon$. This introduces into ergodic theory the study of general phenomena in which sampling is "good" for an uncountable number of systems. Since [a21], several proofs of the "Wiener–Wintner theorem" have appeared (e.g., see [a11] for a spectral path and [a14] for a path using the notion of disjointness in [a13]).

Uniform Wiener–Wintner theorem and Kronecker factor.[edit]

For $( X , \mathcal{F} , \mu , T )$ an ergodic dynamical system (cf. also Ergodicity), the Kronecker factor $\mathcal{K}$ of $T$ is defined as the closed linear span in $L ^ { 2 } ( \mu )$ of the eigenfunctions of $T$. The orthocomplement of $\mathcal{K}$ can be characterized by the Wiener–Wintner theorem. More precisely, a function $f$ is in $\mathcal{K} ^ { \perp }$ if and only if for $\mu$-a.e. with respect to $x$,

\begin{equation*} \operatorname { lim } _ { N \rightarrow \infty } \operatorname { sup } _ { \varepsilon } \left| \frac { 1 } { N } \sum _ { n = 1 } ^ { N } f ( T ^ { n } x ) e ^ { 2 \pi i n \varepsilon } \right| = 0. \end{equation*}

This theorem was announced by J. Bourgain [a9]. Other proofs of this result can be found in [a1] and [a15], for instance.

A sequence of scalars $a _ { n }$ is a good universal weight (for the pointwise ergodic theorem) if the averages

\begin{equation*} \frac { 1 } { N } \sum _ { n = 1 } ^ { N } a _ { n } g ( S ^ { n } y ) \end{equation*}

converge $\nu$-a.e. for all dynamical systems $( Y , \mathcal{B} , \nu , S )$ and all functions $g \in L ^ { 1 } ( \mu )$. Bourgain's return-time theorem states that given a dynamical system $( X , \mathcal{F} , \mu , T )$ and a function $f$ in $L^{\infty}$, then for $\mu$-a.e. with respect to $x$, the sequence $f ( T ^ { n } x )$ is a good universal weight (see [a8]). By applying this result to the irrational rotations on the one-dimensional torus given by $S _ { \alpha } ( y ) = y + \alpha$ and to the function $g ( y ) = e ^ { 2 \pi i y }$, one easily obtains the Wiener–Wintner theorem. Another proof of his result can be found in [a10] and [a19]. Previous partial results can be found in [a11].

Wiener–Wintner return-time theorem and the Conze–Lesigne algebra.[edit]

A natural generalization of the return-time theorem is its Wiener–Wintner version, in which averages of the sequence $f ( T ^ { n } x ) g ( S ^ { n } y ) e ^ { 2 \pi i n \varepsilon }$ are considered. Such a generalization was obtained in [a7] and one of the tools used to prove it was the Conze–Lesigne algebra. This algebra of functions was discovered by J.P. Conze and E. Lesigne [a12] in their study of the $L^1$ norm convergence of the averages

\begin{equation} \tag{a1} \frac { 1 } { N } \sum _ { n = 1 } ^ { N } \prod _ { i = 1 } ^ { H } f _ { i } \circ T ^ { i n } \end{equation}

for $H = 3$. These averages were introduced by H. Furstenberg. (The functions $f_i$ are in $L ^ { \infty } ( \mu )$. The $L^1$-norm convergence of (a1) for $H \geq 4$ is still an open problem (as of 2001).) It is shown in [a7] that the orthocomplement of the Conze–Lesigne factor characterizes those functions for which outside a single null set of $x$ independent of $g$ or $S$ one has $\nu$-a.e.

\begin{equation*} \operatorname { lim } _ { N \rightarrow \infty } \operatorname { sup } _ { \varepsilon } \left\| \frac { 1 } { N } \sum _ { n = 1 } ^ { N } f ( T ^ { n } x ) g ( S ^ { n } y ) e ^ { 2 \pi i n \varepsilon } \right\| = 0. \end{equation*}

Several results related to the ones above can be found in [a2], [a3], [a4], [a16], [a18], [a20], [a17], and [a22]. In [a5] it was shown that many dynamical systems have a Wiener–Wintner property, based on the speed of convergence in the uniform Wiener–Wintner theorem; this allows one to derive the results in [a8] and [a9] for such systems in a much simpler way.

References[edit]