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''Wiener–Wintner ergodic theorem''
 
''Wiener–Wintner ergodic theorem''
  
A strengthening of the pointwise ergodic theorem (cf. also [[Ergodic theory|Ergodic theory]]) announced in [[#References|[a21]]] and stating that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130110/w1301101.png" /> is a [[Dynamical system|dynamical system]], then given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130110/w1301102.png" /> one can find a set of full [[Measure|measure]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130110/w1301103.png" /> such that for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130110/w1301104.png" /> in this set the averages
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A strengthening of the pointwise ergodic theorem (cf. also [[Ergodic theory|Ergodic theory]]) announced in [[#References|[a21]]] and stating that if $( X , \mathcal{F} , \mu , T )$ is a [[Dynamical system|dynamical system]], then given $f \in L ^ { 1 } ( \mu )$ one can find a set of full [[Measure|measure]] $X _ { f }$ such that for $x$ in this set the averages
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130110/w1301105.png" /></td> </tr></table>
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\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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130110/w1301106.png" />. In other words, the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130110/w1301107.png" /> "works"  for an uncountable number of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130110/w1301108.png" />. This introduces into ergodic theory the study of general phenomena in which sampling is  "good"  for an uncountable number of systems. Since [[#References|[a21]]], several proofs of the  "Wiener–Wintner theorem"  have appeared (e.g., see [[#References|[a11]]] for a spectral path and [[#References|[a14]]] for a path using the notion of disjointness in [[#References|[a13]]]).
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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 [[#References|[a21]]], several proofs of the  "Wiener–Wintner theorem"  have appeared (e.g., see [[#References|[a11]]] for a spectral path and [[#References|[a14]]] for a path using the notion of disjointness in [[#References|[a13]]]).
  
 
==Uniform Wiener–Wintner theorem and Kronecker factor.==
 
==Uniform Wiener–Wintner theorem and Kronecker factor.==
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130110/w1301109.png" /> an ergodic dynamical system (cf. also [[Ergodicity|Ergodicity]]), the Kronecker factor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130110/w13011010.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130110/w13011011.png" /> is defined as the closed linear span in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130110/w13011012.png" /> of the eigenfunctions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130110/w13011013.png" />. The orthocomplement of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130110/w13011014.png" /> can be characterized by the Wiener–Wintner theorem. More precisely, a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130110/w13011015.png" /> is in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130110/w13011016.png" /> if and only if for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130110/w13011017.png" />-a.e. with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130110/w13011018.png" />,
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For $( X , \mathcal{F} , \mu , T )$ an ergodic dynamical system (cf. also [[Ergodicity|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 $x$,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130110/w13011019.png" /></td> </tr></table>
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\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 [[#References|[a9]]]. Other proofs of this result can be found in [[#References|[a1]]] and [[#References|[a15]]], for instance.
 
This theorem was announced by J. Bourgain [[#References|[a9]]]. Other proofs of this result can be found in [[#References|[a1]]] and [[#References|[a15]]], for instance.
  
A sequence of scalars <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130110/w13011020.png" /> is a good universal weight (for the pointwise ergodic theorem) if the averages
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A sequence of scalars $a _ { n }$ is a good universal weight (for the pointwise ergodic theorem) if the averages
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130110/w13011021.png" /></td> </tr></table>
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\begin{equation*} \frac { 1 } { N } \sum _ { n = 1 } ^ { N } a _ { n } g ( S ^ { n } y ) \end{equation*}
  
converge <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130110/w13011022.png" />-a.e. for all dynamical systems <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130110/w13011023.png" /> and all functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130110/w13011024.png" />. Bourgain's return-time theorem states that given a dynamical system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130110/w13011025.png" /> and a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130110/w13011026.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130110/w13011027.png" />, then for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130110/w13011028.png" />-a.e. with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130110/w13011029.png" />, the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130110/w13011030.png" /> is a good universal weight (see [[#References|[a8]]]). By applying this result to the irrational rotations on the one-dimensional torus given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130110/w13011031.png" /> and to the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130110/w13011032.png" />, one easily obtains the Wiener–Wintner theorem. Another proof of his result can be found in [[#References|[a10]]] and [[#References|[a19]]]. Previous partial results can be found in [[#References|[a11]]].
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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 [[#References|[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 [[#References|[a10]]] and [[#References|[a19]]]. Previous partial results can be found in [[#References|[a11]]].
  
 
==Wiener–Wintner return-time theorem and the Conze–Lesigne algebra.==
 
==Wiener–Wintner return-time theorem and the Conze–Lesigne algebra.==
A natural generalization of the return-time theorem is its Wiener–Wintner version, in which averages of the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130110/w13011033.png" /> are considered. Such a generalization was obtained in [[#References|[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 [[#References|[a12]]] in their study of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130110/w13011034.png" /> norm convergence of the averages
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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 [[#References|[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 [[#References|[a12]]] in their study of the $L^1$ norm convergence of the averages
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130110/w13011035.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
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\begin{equation} \tag{a1} \frac { 1 } { N } \sum _ { n = 1 } ^ { N } \prod _ { i = 1 } ^ { H } f _ { i } \circ T ^ { i n } \end{equation}
  
for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130110/w13011036.png" />. These averages were introduced by H. Furstenberg. (The functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130110/w13011037.png" /> are in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130110/w13011038.png" />. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130110/w13011039.png" />-norm convergence of (a1) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130110/w13011040.png" /> is still an open problem (as of 2001).) It is shown in [[#References|[a7]]] that the orthocomplement of the Conze–Lesigne factor characterizes those functions for which outside a single null set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130110/w13011041.png" /> independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130110/w13011042.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130110/w13011043.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130110/w13011044.png" />-a.e.
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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 [[#References|[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.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130110/w13011045.png" /></td> </tr></table>
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\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 [[#References|[a2]]], [[#References|[a3]]], [[#References|[a4]]], [[#References|[a16]]], [[#References|[a18]]], [[#References|[a20]]], [[#References|[a17]]], and [[#References|[a22]]]. In [[#References|[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 [[#References|[a8]]] and [[#References|[a9]]] for such systems in a much simpler way.
 
Several results related to the ones above can be found in [[#References|[a2]]], [[#References|[a3]]], [[#References|[a4]]], [[#References|[a16]]], [[#References|[a18]]], [[#References|[a20]]], [[#References|[a17]]], and [[#References|[a22]]]. In [[#References|[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 [[#References|[a8]]] and [[#References|[a9]]] for such systems in a much simpler way.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I. Assani,  "A Wiener–Wintner property for the helical transform"  ''Ergod. Th. Dynam. Syst.'' , '''12'''  (1992)  pp. 185–194</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  I. Assani,  "A weighted pointwise ergodic theorem"  ''Ann. IHP'' , '''34'''  (1998)  pp. 139–150</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  I. Assani,  "Uniform Wiener–Wintner theorems for weakly mixing dynamical systems"  ''Preprint unpublished''  (1992)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  I. Assani,  "Strong laws for weighted sums of independent identically distributed random variables"  ''Duke Math. J.'' , '''88''' :  2  (1997)  pp. 217–246</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  I. Assani,  "Wiener–Wintner dynamical systems"  ''Preprint''  (1998)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  I. Assani,  "Multiple return times theorems for weakly mixing systems"  ''Ann. IHP'' , '''36''' :  2  (2000)  pp. 153–165</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  I. Assani,  E. Lesigne,  D. Rudolph,  "Wiener–Wintner return times ergodic theorem"  ''Israel J. Math.'' , '''92'''  (1995)  pp. 375–395</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  J. Bourgain,  "Return times sequences of dynamical systems"  ''Preprint IHES''  (1988)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  J. Bourgain,  "Double recurrence and almost sure convergence"  ''J. Reine Angew. Math.'' , '''404'''  (1990)  pp. 140–161</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  J. Bourgain,  H. Furstenberg,  Y. Katznelson,  D. Ornstein,  "Appendix to: J. Bourgain: Pointwise ergodic theorems for arithmetic sets"  ''IHES'' , '''69'''  (1989)  pp. 5–45</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  A. Bellow,  V. Losert,  "The weighted pointwise ergodic theorem and the individual ergodic theorem along subsequences"  ''Trans. Amer. Math. Soc.'' , '''288'''  (1995)  pp. 307–345</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  J.P. Conze,  E. Lesigne,  "Théorèmes ergodiques pour des mesures diagonales"  ''Bull. Soc. Math. France'' , '''112'''  (1984)  pp. 143–175</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  H. Furstenberg,  "Disjointness in ergodic theory"  ''Math. Systems Th.'' , '''1'''  (1967)  pp. 1–49</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  E. Lesigne,  "Théorèmes ergodiques pour une translation sur une nilvariete"  ''Ergod. Th. Dynam. Syst.'' , '''9'''  (1989)  pp. 115–126</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top">  E. Lesigne,  "Spectre quasi-discret et thèoréme ergodique de Wiener–Wintner pour les polynômes"  ''Ergod. Th. Dynam. Syst.'' , '''13'''  (1993)  pp. 767–784</TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top">  E. Lesigne,  "Un théorème de disjonction de systèmes dynamiques et une généralisation du théorème ergodique de Wiener–Wintner"  ''Ergod. Th. Dynam. Syst.'' , '''10'''  (1990)  pp. 513–521</TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top">  D. Ornstein,  B. Weiss,  "Subsequence ergodic theorems for amenable groups"  ''Israel J. Math.'' , '''79'''  (1992)  pp. 113–127</TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top">  E.A. Robinson,  "On uniform convergence in the Wiener Wintner theorem"  ''J. London Math. Soc.'' , '''49'''  (1994)  pp. 493–501</TD></TR><TR><TD valign="top">[a19]</TD> <TD valign="top">  D. Rudolph,  "A joinings proof of Bourgain's return times theorem"  ''Ergod. Th. Dynam. Syst.'' , '''14'''  (1994)  pp. 197–203</TD></TR><TR><TD valign="top">[a20]</TD> <TD valign="top">  D. Rudolph,  "Fully generic sequences and a multiple-term return times theorem"  ''Invent. Math.'' , '''131''' :  1  (1998)  pp. 199–228</TD></TR><TR><TD valign="top">[a21]</TD> <TD valign="top">  N. Wiener,  A. Wintner,  "Harmonic analysis and ergodic theory"  ''Amer. J. Math.'' , '''63'''  (1941)  pp. 415–426</TD></TR><TR><TD valign="top">[a22]</TD> <TD valign="top">  P. Walters,  "Topological Wiener–Wintner ergodic theorem and a random <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130110/w13011046.png" /> ergodic theorem"  ''Ergod. Th. Dynam. Syst.'' , '''16'''  (1996)  pp. 179–206</TD></TR></table>
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<table><tr><td valign="top">[a1]</td> <td valign="top">  I. Assani,  "A Wiener–Wintner property for the helical transform"  ''Ergod. Th. Dynam. Syst.'' , '''12'''  (1992)  pp. 185–194</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  I. Assani,  "A weighted pointwise ergodic theorem"  ''Ann. IHP'' , '''34'''  (1998)  pp. 139–150</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  I. Assani,  "Uniform Wiener–Wintner theorems for weakly mixing dynamical systems"  ''Preprint unpublished''  (1992)</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  I. Assani,  "Strong laws for weighted sums of independent identically distributed random variables"  ''Duke Math. J.'' , '''88''' :  2  (1997)  pp. 217–246</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  I. Assani,  "Wiener–Wintner dynamical systems"  ''Preprint''  (1998)</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  I. Assani,  "Multiple return times theorems for weakly mixing systems"  ''Ann. IHP'' , '''36''' :  2  (2000)  pp. 153–165</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  I. Assani,  E. Lesigne,  D. Rudolph,  "Wiener–Wintner return times ergodic theorem"  ''Israel J. Math.'' , '''92'''  (1995)  pp. 375–395</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  J. Bourgain,  "Return times sequences of dynamical systems"  ''Preprint IHES''  (1988)</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  J. Bourgain,  "Double recurrence and almost sure convergence"  ''J. Reine Angew. Math.'' , '''404'''  (1990)  pp. 140–161</td></tr><tr><td valign="top">[a10]</td> <td valign="top">  J. Bourgain,  H. Furstenberg,  Y. Katznelson,  D. Ornstein,  "Appendix to: J. Bourgain: Pointwise ergodic theorems for arithmetic sets"  ''IHES'' , '''69'''  (1989)  pp. 5–45</td></tr><tr><td valign="top">[a11]</td> <td valign="top">  A. Bellow,  V. Losert,  "The weighted pointwise ergodic theorem and the individual ergodic theorem along subsequences"  ''Trans. Amer. Math. Soc.'' , '''288'''  (1995)  pp. 307–345</td></tr><tr><td valign="top">[a12]</td> <td valign="top">  J.P. Conze,  E. Lesigne,  "Théorèmes ergodiques pour des mesures diagonales"  ''Bull. Soc. Math. France'' , '''112'''  (1984)  pp. 143–175</td></tr><tr><td valign="top">[a13]</td> <td valign="top">  H. Furstenberg,  "Disjointness in ergodic theory"  ''Math. Systems Th.'' , '''1'''  (1967)  pp. 1–49</td></tr><tr><td valign="top">[a14]</td> <td valign="top">  E. Lesigne,  "Théorèmes ergodiques pour une translation sur une nilvariete"  ''Ergod. Th. Dynam. Syst.'' , '''9'''  (1989)  pp. 115–126</td></tr><tr><td valign="top">[a15]</td> <td valign="top">  E. Lesigne,  "Spectre quasi-discret et thèoréme ergodique de Wiener–Wintner pour les polynômes"  ''Ergod. Th. Dynam. Syst.'' , '''13'''  (1993)  pp. 767–784</td></tr><tr><td valign="top">[a16]</td> <td valign="top">  E. Lesigne,  "Un théorème de disjonction de systèmes dynamiques et une généralisation du théorème ergodique de Wiener–Wintner"  ''Ergod. Th. Dynam. Syst.'' , '''10'''  (1990)  pp. 513–521</td></tr><tr><td valign="top">[a17]</td> <td valign="top">  D. Ornstein,  B. Weiss,  "Subsequence ergodic theorems for amenable groups"  ''Israel J. Math.'' , '''79'''  (1992)  pp. 113–127</td></tr><tr><td valign="top">[a18]</td> <td valign="top">  E.A. Robinson,  "On uniform convergence in the Wiener Wintner theorem"  ''J. London Math. Soc.'' , '''49'''  (1994)  pp. 493–501</td></tr><tr><td valign="top">[a19]</td> <td valign="top">  D. Rudolph,  "A joinings proof of Bourgain's return times theorem"  ''Ergod. Th. Dynam. Syst.'' , '''14'''  (1994)  pp. 197–203</td></tr><tr><td valign="top">[a20]</td> <td valign="top">  D. Rudolph,  "Fully generic sequences and a multiple-term return times theorem"  ''Invent. Math.'' , '''131''' :  1  (1998)  pp. 199–228</td></tr><tr><td valign="top">[a21]</td> <td valign="top">  N. Wiener,  A. Wintner,  "Harmonic analysis and ergodic theory"  ''Amer. J. Math.'' , '''63'''  (1941)  pp. 415–426</td></tr><tr><td valign="top">[a22]</td> <td valign="top">  P. Walters,  "Topological Wiener–Wintner ergodic theorem and a random $L^{2}$ ergodic theorem"  ''Ergod. Th. Dynam. Syst.'' , '''16'''  (1996)  pp. 179–206</td></tr></table>

Latest revision as of 17:03, 1 July 2020

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.

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.

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

[a1] I. Assani, "A Wiener–Wintner property for the helical transform" Ergod. Th. Dynam. Syst. , 12 (1992) pp. 185–194
[a2] I. Assani, "A weighted pointwise ergodic theorem" Ann. IHP , 34 (1998) pp. 139–150
[a3] I. Assani, "Uniform Wiener–Wintner theorems for weakly mixing dynamical systems" Preprint unpublished (1992)
[a4] I. Assani, "Strong laws for weighted sums of independent identically distributed random variables" Duke Math. J. , 88 : 2 (1997) pp. 217–246
[a5] I. Assani, "Wiener–Wintner dynamical systems" Preprint (1998)
[a6] I. Assani, "Multiple return times theorems for weakly mixing systems" Ann. IHP , 36 : 2 (2000) pp. 153–165
[a7] I. Assani, E. Lesigne, D. Rudolph, "Wiener–Wintner return times ergodic theorem" Israel J. Math. , 92 (1995) pp. 375–395
[a8] J. Bourgain, "Return times sequences of dynamical systems" Preprint IHES (1988)
[a9] J. Bourgain, "Double recurrence and almost sure convergence" J. Reine Angew. Math. , 404 (1990) pp. 140–161
[a10] J. Bourgain, H. Furstenberg, Y. Katznelson, D. Ornstein, "Appendix to: J. Bourgain: Pointwise ergodic theorems for arithmetic sets" IHES , 69 (1989) pp. 5–45
[a11] A. Bellow, V. Losert, "The weighted pointwise ergodic theorem and the individual ergodic theorem along subsequences" Trans. Amer. Math. Soc. , 288 (1995) pp. 307–345
[a12] J.P. Conze, E. Lesigne, "Théorèmes ergodiques pour des mesures diagonales" Bull. Soc. Math. France , 112 (1984) pp. 143–175
[a13] H. Furstenberg, "Disjointness in ergodic theory" Math. Systems Th. , 1 (1967) pp. 1–49
[a14] E. Lesigne, "Théorèmes ergodiques pour une translation sur une nilvariete" Ergod. Th. Dynam. Syst. , 9 (1989) pp. 115–126
[a15] E. Lesigne, "Spectre quasi-discret et thèoréme ergodique de Wiener–Wintner pour les polynômes" Ergod. Th. Dynam. Syst. , 13 (1993) pp. 767–784
[a16] E. Lesigne, "Un théorème de disjonction de systèmes dynamiques et une généralisation du théorème ergodique de Wiener–Wintner" Ergod. Th. Dynam. Syst. , 10 (1990) pp. 513–521
[a17] D. Ornstein, B. Weiss, "Subsequence ergodic theorems for amenable groups" Israel J. Math. , 79 (1992) pp. 113–127
[a18] E.A. Robinson, "On uniform convergence in the Wiener Wintner theorem" J. London Math. Soc. , 49 (1994) pp. 493–501
[a19] D. Rudolph, "A joinings proof of Bourgain's return times theorem" Ergod. Th. Dynam. Syst. , 14 (1994) pp. 197–203
[a20] D. Rudolph, "Fully generic sequences and a multiple-term return times theorem" Invent. Math. , 131 : 1 (1998) pp. 199–228
[a21] N. Wiener, A. Wintner, "Harmonic analysis and ergodic theory" Amer. J. Math. , 63 (1941) pp. 415–426
[a22] P. Walters, "Topological Wiener–Wintner ergodic theorem and a random $L^{2}$ ergodic theorem" Ergod. Th. Dynam. Syst. , 16 (1996) pp. 179–206
How to Cite This Entry:
Wiener-Wintner theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wiener-Wintner_theorem&oldid=50492
This article was adapted from an original article by I. Assani (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article