Cube spaces and the multiple term return times theorem

0.16 MB·English

by  Pavel Zorin-Kranich

CUBE SPACES AND THE MULTIPLE TERM RETURN TIMES THEOREM
PAVELZORIN-KRANICH
ABSTRACT. We give a new proof of Rudolph’s multiple term return times theorem based on
Host-Kra structure theory. Our approach also yields a multiple term Wiener-Wintner-type
returntimestheoremfornilsequences.
2
1
0 1. INTRODUCTION
2
Inthisarticleweareconcernedwithuniversally goodweightsforpointwiseconvergence
t
c
of ergodic averages, i.e., sequences (a ) such that, for every measure-preserving system
O n
(Y,S) and every g ∈ L∞(Y), the averages

8
1
1 N
S] liNm N ang(Sny)
Xn=1
D
. converge for a.e. y ∈Y. Bourgain’s return timestheorem [BFKO89] asserts that, given any
h
t ergodicmeasure-preservingsystem(X,T),forevery f ∈ L∞(X)anda.e. x ∈X thesequence
a
m of weights a = f(Tnx) is universally good for pointwise convergence. The name “return
n
[ times theorem” comes from the case of a characteristic function f = 1 , A ⊂ X. Then
A
the theorem can be equivalently formulated by saying that, for a.e. x ∈ X, the pointwise
1
v ergodic theorem on any system Y holds along the sequence of return times of x to A.
2
This has been extended to averages involving multiple terms by Rudolph [Rud98]. In
0
2 order to formulate his result and for future convenience we now introduce some notation.
5 By a system we mean an ergodicregular measure-preserving system (X,µ,T) with a distin-
.
0 guishedcountablesubset D⊂ L∞(X)thatissufficientlylarge(wewillformulatetheprecise
1
conditionon D in Defintion 2.8).
2
1
: Definition 1.1. Let P be a statement about ergodic regular measure-preserving systems
v
i (X ,µ ,T), functions f ∈ L∞(X ) and points x ∈ X , i = 0,...,k. We say that P holds for
X i i i i i i i
universally almost every (u.a.e.) tuple x ,...,x if
r 0 k
a
(0) For every system (X ,µ ,T ,D ) there exists a set of full measure X˜ ⊂X such that
0 0 0 0 0 0
(1) foreverysystem(X ,µ ,T ,D )thereexistsameasurableset X˜ ⊂ X ×X suchthat
1 1 1 1 1 0 1
for every ~x ∈X˜ the set {x :(~x ,x )∈X˜ } has full measure in X and
0 0 1 0 1 1 1
.
.
.
(k) for every system (X ,µ ,T ,D ) there exists a measurable set X˜ ⊂ X ×···× X
k k k k k 0 k
such that for every ~x ∈ X˜ the set {x :(~x ,x )∈ X˜ } has full measure in X
k−1 k−1 k k−1 k k k
and
we have P(f ,..., f ,~x ) for every ~x ∈X˜ and every f ∈ D , i =0,...,k.
0 k k k k i i
Date:October19,2012.
2010MathematicsSubjectClassification. 28D05.
Keywordsandphrases. cubespace,returntimestheorem,Wiener-Wintner theorem.
1
2 PAVELZORIN-KRANICH
With this convention Rudolph’s multiple term return times theorem says that for every
k∈N the averages
1 N
(1.2) f (Tnx )··· f (Tnx )
N 0 0 0 k k k
Xn=1
converge for u.a.e. x ,...,x . We call this statement RTT(k). Birkhoff’s pointwise ergodic
0 k
theorem[Bir31]isessentiallyRTT(0)andBourgain’sreturntimestheoremisRTT(1). More
about the historyof these and relatedresults canbe foundin arecent survey byAssani and
Presser [AP12a].
It is known that the Host-Kra-Ziegler nilfactor Z (X ) is characteristic for RTT(k) in the
k 0
sense that if f ⊥Z (X ), thenthe averages (1.2) convergeto zerou.a.e. [AP12b, Theorem
0 k 0
4]. However, the proofof this fact hitherto depends on the convergence result RTT(k).
In this article we prove both results, RTT(k) and characteristicity, simultaneously by
induction on k using the Host-Kra structure theory. We also obtain the following Wiener-
Wintner return times theorem for nilsequences thereby generalizing [ALR95, Theorem 1].
Theorem 1.3 (Wiener-Wintner return times theorem for nilsequences). Let k,l ∈ N and
f ∈ L∞(X ), i = 0,...,k. Then for u.a.e. x ,...,x and every l-step nilsequence (a ) the
i i 0 k n n
averages
1 N k
a f (Tnx )
N n i i i
Xn=1 Yi=0
converge (to zero if in addition f ⊥Z (X ) or f ⊥Z (X ) for some i =1,...,k).
0 k+l 0 i k+l+1−i i
The general strategy is to consider not only the X ’s but also the Host-Kra cube spaces of
i
all orders simultaneously and to use the convergence criterion due to Bourgain, Fursten-
berg,KatznelsonandOrnstein(Proposition2.11)topassfromktok+1. Afterapreparatory
Section 2 we formulate our central convergence Theorem 3.2 on cube spaces, generalizing
RTT(k) and giving some information about characteristic factors. The remaining part of
Section3 is devoted to the proofof Theorem 3.2. Theorem 1.3 is then provedin Section4.
2. NOTATION AND TOOLS
Ergodic decomposition. For the purposes of this article we find it illuminating to think of
the ergodic decomposition in a particular way (that will be generalized in Section 3). Let
(X,µ,T) be a regular ergodic measure-preserving system, i.e. X is a compact metric space,
T : X → X is an invertible continuous map and µ is a T-invariant ergodic Borel probability
measure. Bythepointwiseergodictheorema.e. x ∈X isgenericforsome T-invariantBorel
probabilitymeasure m on X, i.e. 1 N f(Tnx)→ fdm for every f ∈C(X). It follows
x N n=1 x
easily that the function x 7→mx is mPeasurable and R
(2.1) µ= m dµ(x)
x
Z
Inparticular, forµ-a.e. x themeasure m isdefinedform -a.e. y. Tosee that m isergodic
y x x
for µ-a.e. x it suffices to verify that
2
(2.2) fdm − fdm dm (y)dµ(x)=0 for every f ∈C(X),
y x x
Z Z Z Z
(cid:12) (cid:12)
(cid:12) (cid:12)
since this says pre(cid:12)cisely that the ergod(cid:12)ic averages of f converge pointwise m -a.e. to an
x
m -essentially constant function for µ-a.e. x, and the latter full measure set can be chosen
x
independently from f since C(X) is separable. By definition of m ,m , the dominated
x y
convergence theorem and (2.1) we can rewrite the integral in (2.2) as
CUBE SPACES AND THE MULTIPLE TERM RETURN TIMES THEOREM 3
1 N 1 N 1 N
2lim ( Tnf)2(x)dµ(x)−2lim ( Tnf)(x) ( Tnf)(y)dm (y)dµ(x)
N Z N N Z N Z N x
Xn=1 Xn=1 Xn=1
1 N 1 N 1 M
=2lim ( Tnf)2(x)dµ(x)−2limlim ( Tnf)(x)( Tmf)(x)dµ(x),
N Z N N M Z N M
Xn=1 Xn=1 Xn=1
and this vanishes by the pointwise ergodic theorem and the dominated convergence theo-
rem.
Host-Kra structure theory. We recall the basic definitions and main results surrounding
the uniformity seminorms [HK05]. Let (X,µ,T) be a regular ergodic measure-preserving
system. The cube measures µ[l] on X[l] := X2l are defined inductively starting with µ[0] :=
µ. In the inductive step, given µ[l], fix an ergodic decomposition
µ[l] = m dµ[l](x)
x
Z
X[l]
as in (2.1). The space on which m is defined can be inferred from the subscript x. Define
x
(2.3) µ[l+1] := δ ⊗m dµ[l](x).
x x
Z
X[l]
To see that this coincides with the conventional definition one can use (2.1) and (2.2) to
write the above integral as
(2.4) µ[l+1] = δ ⊗m dm (y)dµ[l](x)
y y x
Z Z
= δ ⊗m dm (y)dµ[l](x)= m ⊗m dµ[l](x).
y x x x x
Z Z Z
The uniformity seminorms are
(2.5) kfk2l+1 := ⊗ fdµ[l+1] = E ⊗ f|I[l] 2dµ[l],
Ul+1 Z ε∈{0,1}l+1 Z ε∈{0,1}l
(cid:0) (cid:1)
where I[l] is the T[l]-invariant sub-σ-algebra on X[l]. For fε ∈ L∞(X), ε ∈ {0,1}l, we
will abbreviate f[l] := ⊗ fε. The uniformity seminorms satisfy the Cauchy-Schwarz-
ε∈{0,1}l
Gowers inequality [HK05, Lemma 3.9.(1)]
(2.6) f[l]dµ[l] ≤ kfεk .
Ul
Z
(cid:12) (cid:12) ε∈Y{0,1}l
(cid:12) (cid:12)
The Ul+1-seminorms determine(cid:12)factors Z ((cid:12)X) by the relation
l
f ⊥ L2(Z (X)) ⇐⇒ kfk =0 for f ∈ L∞(X).
l Ul+1
The factors Z (X) are inverse limits of rotations on l-step nilmanifolds (in fact, topological
l
inverse limits [HKM10, Theorem A.1]).
We also need the classical fact that the Kronecker factor is characteristic for L2 conver-
gence ofergodicaverages with arbitraryboundedscalar weights, see e.g.[HK09, Corollary
7.3] for a more general version.
Lemma2.7. Let(X,T)beanergodicmeasure-preservingsystemand f ∈ L2(X)beorthogonal
to Z (X). Then for any bounded sequence (a ) one has
1 n n
1 N
lim a Tnf =0 in L2(X).
N N n
Xn=1
4 PAVELZORIN-KRANICH
Conventions about cube measures. In the sequel we will have to consider systems for
whichacertainapproximatingprocedurecanbecarriedoutwithintheirdistinguishedsets.
Definition 2.8. A system is a regular ergodic measure-preserving system (X,µ,T) with a
distinguished set D⊂ L∞(X) that satisfies the followingconditions.
(1) (Cardinality) D is countable.
(2) (Density) D contains an L∞-dense subset of C(X).
(3) (Algebra) D is closed under pointwise multiplication and absolute value.
(4) (Decomposition)For every f ∈ D and l ∈N there exist decompositions
Dec(l) f = f + f + f , j ∈N,
⊥ Z,j err,j
such that f , f , f ∈ D, f ⊥ Z (X), f ∈ C(Z ), where Z is a nilfactor of
⊥ Z,j err,j ⊥ l Z,j j j
Z (X), kf k is uniformly bounded in j and kf k →0 as j →∞.
l err,j L∞(µ) err,j L1(µ)
Clearly,foranyregularergodicmeasure-preservingsystem(X,µ,T)anycountablesubset
of L∞(X) is contained in a set D that satisfies the above conditions.
Lemma 2.9. Let (X,µ,T,D) be a system. Then there exist measurable subsets Y ⊂ X[l] such
l
that for every l ∈N the following statements hold.
(1) µ[l](Y)=1 and for every y ∈Y we have m (Y)=1.
l l y l
(2) For every y ∈Y the measure m is ergodic and one has
l y
(2.10) m ⊗m = m d(m ⊗m )(x).
y y x y y
Z
Y[l+1]
(3) Y ⊂(X˜)[l], where X˜ ⊂X is the set of points that are generic for each f ∈ D w.r.t. µ.
l
(4) For every y ∈ Y, every k ∈ N and any functions f ∈ D, ε ∈ {0,1}l, such that
l ε
f ⊥Z (X) for some ε we have f[l]⊥Z (X[l],m ).
ε k+l k y
Proof. The fact that the last condition holds for full measure sets of y follows from the
l
Cauchy-Schwarz-Gowersinequality (2.6). The remaining conditionscan be achieved using
(2.4) and the pointwise ergodic theorem. (cid:3)
Bourgain-Furstenberg-Katznelson-Ornstein criterion. Ourmaintoolforprovingconver-
gence u.a.e. is the following criterion that reduces the search for a universal set of x ∈ X
(that a prioriinvolves uncountably many systems Y) to a problemabout X2.
Proposition 2.11([BFKO89,Proposition]). Let(X,T)beanergodicmeasure-preservingsys-
tem and f ∈ L∞(X)∩Z (X)⊥. Assume that x ∈X is generic for f and
1
1 N
f(Tnx)f(Tnξ)→0 for a.e. ξ∈X.
N
Xn=1
Then for every measure-preserving system (Y,S) and g ∈ L∞(Y) we have
1 N
f(Tnx)g(Sny)→0 for a.e. y ∈Y.
N
Xn=1
This has been generalized to averages along Følner sequences in countable amenable
groups satisfying the Tempelman condition [OW92, §3]. It seems plausible that a corre-
sponding statement for tempered Følner sequences (for which the pointwise ergodic theo-
rem is known to hold by [Lin01]) would also be true, but we concentrate on the standard
Cesàro averages.
CUBE SPACES AND THE MULTIPLE TERM RETURN TIMES THEOREM 5
Ameasure-theoretic lemma. Thenext lemmaisourmaintoolfordealingwithcubemea-
sures. Informally, it shows that a certain kind of universality for µ[1] ⊗ν[1] implies some
universality for (µ×ν)[1].
Recall that, forergodicmeasure-preservingsystems(X,µ),(Y,ν), theprojectionontothe
invariantfactorofX×Y hastheformφ(x, y)=ψ(π (x),π (y)),whereπ areprojections
1 1 1
ontotheKroneckerfactorsandψisthequotientmapofZ (X)×Z (Y)bytheorbitclosure
1 1
of the identity. To see this, recall that by Lemma 2.7 the function f ⊗ g, f ∈ L∞(X),
g ∈ L∞(Y), is orthogonal to the invariant factor of X × Y whenever f ⊥ Z (X) or g ⊥
1
Z (Y). Thus the invariant sub-σ-algebra on X ×Y is contained in Z (X)×Z (Y), i.e. it is
1 1 1
(isomorphic to) the invariant sub-σ-algebra of a product of two compact group rotations
(cf. e.g. [Rud95, Theorem 1.9]). In particular, for an ergodic system Y the invariant factor
of Y ×Y is isomorphicto Z (Y).
1
Lemma 2.12. Let (X,µ),(Y,ν) be ergodic measure-preserving systems and fix measure disin-
tegrations
µ= µ dκ, ν = µ dλ.
κ λ
Z Z
κ∈Z (X) λ∈Z (Y)
1 1
This induces an ergodic decomposition
ν ⊗ν = (ν ⊗ν) dλ, (ν ⊗ν) = ν ⊗ν dλ′.
λ λ λ′ λ′λ−1
Z Z
λ∈Z (Y) λ′∈Z (Y)
1 1
Let x ∈X and Λ⊂Z (Y) be a full measure set. Assume that for µ-a.e. ξ and every λ∈Λ, for
1
(ν⊗ν) -a.e. (η,η′) one some statement P(x,ξ,η,η′) holds. Then P(x,ξ, y,η) also holds for
λ
ν-a.e. y and m˜ -a.e. (ξ,η), where
x,y
m˜ = µ ⊗ν d(κ,λ),
x,y κ λ
Z
κ∈Z (X),λ∈Z (Y):ψ(π (x),π (y))=ψ(κ,λ)
1 1 1 1
thehomomorphismψisasaboveandtheintegralistakenoveranaffinesubgroupwithrespect
to its Haar measure.
Proof. Recallthatkerψhasfullprojectionsonbothcoordinates. Therefore,forevery x there
is a full measure set of ξ such that the set Λ has full measure in {λ:ψ(π (x)π (ξ)−1,λ)=
1 1
id} (note that this is a closedaffine subgroup of Z (Y) that thereforehas a Haar measure).
1
In particular, for a full measure set of ξ (that depends on Y) the hypothesis holds for
a.e. λ with ψ(π (x)π (ξ)−1,λ)=id, i.e. we have P(x,·) for a set of full measure w.r.t. the
1 1
measure
δ ⊗ (ν ⊗ν) dλdµ(ξ)
ξ λ
Z Z
ξ∈X λ∈Z (Y):ψ(π (x)π (ξ)−1,λ)=id
1 1 1
= µ ⊗(ν ⊗ν) dλdκ
κ λ
Z Z
κ∈Z (X) λ∈Z (Y):ψ(π (x)κ−1,λ)=id
1 1 1
= µ ⊗ ν ⊗ν dλ′dλdκ
κ λ′ λ′λ−1
Z Z Z
κ∈Z (X) λ∈Z (Y):ψ(π (x)κ−1,λ)=id λ′∈Z (Y)
1 1 1 1
= µ ⊗ δ ⊗ν dν(y)dλdκ
κ y π(y)λ−1
Z Z Z
κ∈Z (X) λ∈Z (Y):ψ(π (x)κ−1,λ)=id y∈Y
1 1 1
= µ ⊗δ ⊗ν dλdκdν(y)
κ y π(y)λ−1
Z Z Z
y∈Y κ∈Z (X) λ∈Z (Y):ψ(π (x)κ−1,λ)=id
1 1 1
6 PAVELZORIN-KRANICH
= µ ⊗δ ⊗ν d(κ,λ)dν(y)
κ y λ
Z Z
y∈Y κ∈Z (X),λ∈Z (Y):ψ(π (x),π (y))=ψ(κ,λ)
1 1 1 1
= δ ⊗m˜ dν(y).
y x,y
Z
y∈Y
This gives P(x,ξ, y,η) for ν-a.e. y and m˜ -a.e. pair (ξ,η) as required. (cid:3)
x,y
The next lemma provides us with means for using the measure m˜ in a higher step
x,y
setting.
Lemma 2.13. Let (Z,g),(Z′,g′) be ergodic nilrotations and ψ : Z (Z)×Z (Z′) → H the
1 1
factor map modulo the orbit closure of (π (g),π (g′)). Then for every λ ∈ Z (Z) and a.e.
1 1 1
λ′ ∈Z (Z′) the rotation by (g,g′) on the nilmanifold
1
N ={(z,z′)∈ Z ×Z′ :ψ(π (z),π (z′))=ψ(λ,λ′)}
λ,λ′ 1 1
is uniquely ergodic.
Proof. By [Lei05, 2.17-2.20] it suffices to prove ergodicityto obtain unique ergodicity.
Since N only depends on ψ(λ,λ′) and kerψ has full projection on Z (Z) it suffices
λ,λ′ 1
to verify the conclusion for a full measure set of (λ,λ′). For this end it suffices to check
that for any f ∈ C(Z), f′ ∈ C(Z′) the limit of the ergodic averages of f ⊗ f′ is essentially
constant on N . We decompose f = f + f with f ⊥ Z (Z) and f ∈ L∞(Z (Z)), and
λ,λ′ ⊥ Z ⊥ 1 Z 1
analogously for f′. For f ⊗ f′ the limitisessentially constant on N forany (λ,λ′) since
Z Z λ,λ′
the rotation is ergodic on (π ×π )(N ).
1 1 λ,λ′
On the other hand, the limit of the ergodic averages of tensor products involving f
⊥
vanishes on Z ×Z′ a.e. by Lemma 2.7, hence also a.e. on a.e. fiber N . (cid:3)
λ,λ′
3. RETURN TIMES THEOREM ON CUBE SPACES
In order to concisely state our central result, Theorem 3.2, we need a cube version of
Definition 1.1. Recall that we write f[l] =⊗ f , where f ∈ L∞(X ).
i ε∈{0,1}l i,ε i,ε i
Definition 3.1. Let P be a statement about ergodic regular measure-preserving systems
(X ,µ ,T), functions f[l] and points x ∈ X[l], i = 0,...,k. We say that P holds for [l]-
i i i i i i
universally almost every ([l]-u.a.e.) tuple x ,...,x if
0 k
(0) For every system (X ,µ ,T ,D ) there exists a measurable set X˜[l] ⊂ X[l] such that
0 0 0 0 0 0
for every y ∈Y we have m (X˜[l])=1 and
0 0,l y0 0
(1) foreverysystem (X ,µ ,T ,D ) thereexists a measurable set X˜[l] ⊂ X[l]×X[l] such
1 1 1 1 1 0 1
that for every ~x ∈X˜[l] and every y ∈Y we have m {x :(~x ,x )∈X˜ }=1 and
0 0 1 1,l y1 1 0 1 1
.
.
.
(k) foreverysystem (X ,µ ,T ,D ) there exists a measurable set X˜[l] ⊂ X[l]×···×X[l]
k k k k k 0 k
such that for every ~x ∈ X˜[l] and every y ∈ Y we have m {x : (~x ,x ) ∈
k−1 k−1 k k,l yk k k−1 k
X˜ }=1 and
k
we have P(f[l],..., f[l],~x ) for every ~x ∈X˜ and any f ∈ D , 0≤i ≤k, ε∈{0,1}l.
0 k k k k i,ε i
With this definition “u.a.e.” correspondsto “[0]-u.a.e.”.
Our main theorem below states that certain nilfactors are characteristic for return time
averages on cube spaces.
Theorem 3.2. For any k,l ∈ N the ergodic averages of ⊗k f[l] converge [l]-u.a.e. If in
i=0 i
addition
CF(k,l) ∃ε∈{0,1}l s.t. f ⊥Z (X ) or f ⊥Z (X ) for some 1≤ i ≤ k
0,ε k+l 0 i,ε k+l+1−i i
CUBE SPACES AND THE MULTIPLE TERM RETURN TIMES THEOREM 7
then the limit vanishes [l]-u.a.e.
The case l = 0 of Theorem 3.2 is RTT(k) with additional information CF(k,0) about
characteristic factors.
The proof is by induction on k and the inductive loop of the proof spans this whole
section. Thebasecase k=0followsbydefinitionofY andthepointwiseergodictheorem.
0,l
Thus we assume Theorem 3.2 for some fixed k ∈ N and prove all intermediate results
for this value of k assuming that they hold for lower values of k. The base cases of the
intermediate results will be proved as we state them.
In order to pass from k to k+1 in Theorem 3.2 we write
(3.3) X[l+1]×···×X[l+1] =(X[l]×···×X[l])2 =:X2.
0 k 0 k
Our aim is to apply Proposition 2.11 with this X and Y = X[l] . The remaining part of this
k+1
section is devoted to verification of the hypotheses of that proposition. This involves the
followingsteps. FirstweuseRTT(k)toconstructacertainuniversal measuredisintegration
with built-in genericity properties on a product of ergodic systems (Theorem 3.4). We use
characteristic factors for RTT(k) to represent measures in this disintegration in a different
way. Finally, we verify a certain instance of RTT(k+1) (Lemma 3.12).
Universal disintegration of product measures. The return times theorem can be seen
as a statement about measure disintegration, cf. [ALR95, Theorem 4] for the special case
k=1.
Theorem 3.4. Let(X ,µ ,T,D ), i =0,...,k, be systems. Then[l]-u.a.e. x ,...,x is generic
i i i i 0 k
for some measure m on X[l]×···×X[l] and every function ⊗k f[l], f ∈ D .
x0,...,xk 0 k i=0 i i,ε i
Moreover, for [l]-u.a.e. x ,...,x and every y ∈Y one has
0 k−1 k l,k
(3.5) m ⊗m = m dm (x ).
x ,...,x y x ,...,x y k
0 k−1 k Z 0 k k
Proof. By Theorem 3.2 with l =0 we obtain convergence of the averages
1 N k
f[l](Tnx )
N i i i
Xn=1Yi=0
for [l]-u.a.e. x ,...,x and any f ∈ D . For continuous functions f ∈ D we define
0 k i,ε i i,ε i
m (⊗k f[l]) as the limit of these averages. By the Stone-Weierstraß theorem these
x0,...,xk i=0 i
tensorproductsspanadensesubspaceC(X[l]×···×X[l]),sobydensitytheabove(bounded)
0 k
linear form admits a unique continuous extension.
In order to obtain (3.5) it suffices to verify that the integrals of functions of the form
⊗k f[l], f ∈ D ,withrespecttobothmeasurescoincide. Bygenericityandthedominated
i=0 i i,ε i
convergence theorem we have for [l]-u.a.e. x ,...,x that
0 k−1
⊗ f[l]⊗ f[l]dm dm (x )
Z Z i<k i k x0,...,xk yk k
1 N
= lim f[l](Tnx )· f[l](Tnx )dm (x )
Z N N i i i k k k yk k
Xn=1Yi<k
1 N
=lim f[l](Tnx )· f[l](Tnx )dm (x )
N N i i i Z k k k yk k
Xn=1Yi<k
= ⊗ f[l]dm f[l]dm
Z i<k i x0,...,xk−1Z k yk
8 PAVELZORIN-KRANICH
as required. (cid:3)
Properties of the universal disintegration. We will now represent the measure m
x ,...,x
0 k
for [l]-u.a.e. x ,...,x in the form m˜ in the notation of Lemma 2.12. At this step we
0 k x,y
have to use the information about characterisitc factors. We begin with a preliminary ob-
servation.
Lemma3.6. IfsomepropertyPholds[l]-u.a.e.then,for[l]-u.a.e. x ,...,x ,Pholdsm -
0 k x ,...,x
0 k
a.e.
Proof. For k=0thisfollowsfrom(3.5). Assume that the conclusionisknownfor k−1and
show it for k.
By the induction hypothesis, for [l]-u.a.e. x ,...,x , m -a.e., for every y ∈ Y ,
0 k−1 x ,...,x k k,l
P holds m -a.e. in x . The conclusion followsfrom (3.5). 0 k−1 (cid:3)
y k
k
Lemma 3.7. For [l]-u.a.e. x ,...,x we have
0 k
m =m˜ ,
x ,...,x x,y
0 k
whereweusethenotationofLemma2.12with(X,µ)=(X[l]×···×X[l] ,m ),(Y,ν)=
0 k−1 x0,...,xk−1
(X[l],m ), x =(x ,...,x ) and y = x .
k xk 0 k−1 k
Proof. Toverifythatthemeasurescoincideitsufficestocheckthattheintegralsoffunctions
of the form ⊗k f[l], f ∈ D coincide. For this end consider the splittings f = f +
i=0 i i,ε i i,ε i,ε,⊥
f + f , j ∈N, given by Dec(k+l+1−i).
i,ε,Z,j i,ε,err,j
Projections of tensor products that involve f on one of the Kronecker factors vanish
i,ε,⊥
a.e. for [l]-u.a.e. x, y by Corollary 3.13 for k−1 that is part of the induction hypothesis
for this section. Since kerψ has full projections on both coordinates the corresponding
integrals w.r.t. m˜ also vanish. The integrals w.r.t. m vanish for [l]-u.a.e. x, y by
x,y x,y
Theorem 3.2.
For the main terms we have
(3.8) ⊗k f[l] dm˜
Z i=0 i,Z,j x,y
= E(⊗k−1f[l] |Z (X))(κ)E(f [l] |Z (Y))(λ)d(κ,λ).
Z i=0 i,Z,j 1 k,Z,j 1
κ∈Z (X),λ∈Z (Y):ψ(π (x),π (y))=ψ(κ,λ)
1 1 1 1
Since the underlying nilmanifold of a nilrotation is a bundle of nilmanifolds over its Kro-
necker factor, the conditional expectation above is just integration in the fibers, and by
uniqueness of the Haar measure the whole integral equals
⊗k−1f[l] (κ)f[l] (λ)d(κ,λ),
Z i=0 i,Z,j k,Z,j
κ∈Z ,λ∈Z′:ψ(π (x),π (y))=ψ(π (κ),π (λ))
j j 1 1 1 1
where Z is the orbit closure of x in k−1Z[l] and Z′ is the orbit closure of y in Z[l]. By
j i=0 i,j j k,j
Lemma 2.13, the above fibers of Z ×QZ′ are uniquely ergodic for every x and a.e. y, and
j j
the integral then equals
1 N
lim ⊗k−1f[l] (Tnx)f[l] (Sny)= ⊗k f[l] dm .
N N i=0 i,Z,j k,Z,j Z i=0 i,Z,j x,y
Xn=1
Itremainstotreattheerrorterms,i.e. thecase f = f forsomei′,ε′. ByLemma2.9(3),
i′,ε′ i′,ε′,err,j
for [l]-u.a.e. x, y we have
⊗k f[l]dm ®kf k →0 as j →∞.
Z i=0 i x,y i′,ε′ L1(µi)
CUBE SPACES AND THE MULTIPLE TERM RETURN TIMES THEOREM 9
Similarly, we have |⊗k−1 f[l]|dm ® kf k if i′ < k and |f[l]|dm ® kf k
i=0 i x i′,ε′ L1(µi) k yk k,ε L1(µi)
if i′ = k for [l]-u.aR.e. x, y. This implies that either E(⊗k−1f[l]|RZ (X)) or E(f[l]|Z (Y))
i=0 i 1 k 1
converges to zero in probabilityfor [l]-u.a.e. x, y, so
⊗k f[l]dm˜ →0 as j →∞
Z i=0 i x,y
for [l]-u.a.e. x ,...,x since kerψ has full projectionson coordinates. (cid:3)
0 k
Corollary 3.9. For [l]-u.a.e. x ,...,x the measure m =m˜ is ergodic.
0 k x ,...,x x,y
0 k
Note that even for a non-ergodic invariant measure on a regular system there may exist
generic points, so the mere fact that ~x is generic for m does not suffice.
~x
Proof. In order to see that m is ergodic it suffices to verify that for any continuous
x ,...,x
0 k
functions f ∈C(X ) we have
i,ε i
1 N
(3.10) lim ⊗k f[l](Tnξ~)= ⊗k f[l]dm for m -a.e. ξ~.
N N i=0 i Z i=0 i x0,...,xk x0,...,xk
Xn=1
Recallthatfor[l]-u.a.e. x ,...,x thelimitontheleft-handsideof(3.10)existsform -
0 k x ,...,x
0 k
a.e. ξ~ by Lemma 3.6 and equals ⊗k f[l]dm . Splitting the f ’s as before it suffices to
i=0 i ξ~ i,ε
verify (3.10) for the main terms, aRnd this followsdirectly from (3.8). (cid:3)
Thesufficientspecialcaseofconvergencetozero. ThelasthypothesisofProposition2.11
isa certainspecial case of its conclusion. Recall that we already have u.a.e. convergence to
zero on X2 (as defined in (3.3)), but not yet in the required sense. This is now corrected
using Lemma 2.12.
Lemma 3.11 (Change of order in the cube construction). Let l ∈ N and P be a statement
about points of k X[l+1]. Assume that for [l +1]-u.a.e. x ,...,x we have P(x ,...,x ).
i=0 i 0 k 0 k
Then for [l]-u.a.e. x ,...,x , for m -a.e. x′, we have P(x ,...,x ,x′).
Q 0 k x ,...,x 0 k
0 k
Strictlyspeaking, the coordinatesof x′ in (x ,...,x ,x′) should be attached to x ,...,x
0 k 0 k
but we do not want to introduce additional notation at this point.
Proof. The base case k=0 followsdirectly from (2.10).
Assume now that k >0. By the inductive hypothesis of this section the conclusion holds
for k−1, sofor[l]-u.a.e. x ,...,x , form -a.e. x′,forevery y ∈Y andm -a.e.
0 k−1 x ,...,x k k,l+1 y
0 k−1 k
x , we have P(x ,...,x ,x′,x ).
k 0 k−1 k
Using (2.10) we can rewrite the emphasized part of the statement as “for m -a.e.
x ,...,x
0 k−1
x′, for every ˜y ∈ Y , for every ergodic component µ of (m )2 from a fixed full measure
k k,l e ˜y
set, for µ -a.e. x ” The conclusion followsby Lemma 2.12 andkLemma 3.7. (cid:3)
e k
Lemma 3.12. Let l,l′ ∈N and assume CF(k,l +l′). Then for [l]-u.a.e. ~x =(x ,...,x ), for
0 0 k
m -a.e. ~x ,...,form -a.e. ~x theergodicaveragesofthefunction⊗k f[l+l′] converge
~x0 1 ~x0,...,~xl′−1 l′ i=0 i
to zero at (~x ,...,~x ).
0 l′
Again, the tensor product ⊗k f[l+l′] should be arranged in a different order, but in our
i=0 i
opinion the above notation makes our goal more clear: it is not the function but the order
in which we build the product space that changes.
Proof. We use induction on l′. The case l′ = 0 is precisely Theorem 3.2. Assume that the
conclusionis known for l+1 and l′−1. The claim for l and l′ followsby Lemma3.11. (cid:3)
Corollary 3.13. Let l,l′ ∈N and assume CF(k,l +l′). Then for [l]-u.a.e. x ,...,x we have
0 k
f[l]⊗···⊗ f[l] ⊥Z (m ).
0 k l′ x0,...,xk
10 PAVELZORIN-KRANICH
Proof. ThisfollowsfromLemma3.12by Lemma3.6, the definitionof cube measures (2.3),
the characterization of uniformityseminorms (2.5) and the ergodic theorem. (cid:3)
Proof of Theorem 3.2 for k+1. Assume first CF(k,l). Then Lemma 3.12 with l′ = 1 states
that for [l]-u.a.e. x =(x ,...,x ), for m -a.e. x′, for any f ∈ D we have
0 k x ,...,x i,ε i
0 k
1 N
lim ⊗k f[l]((⊗k T[l])nx)·⊗k f[l]((⊗k T[l])nx′)=0.
N N i=0 i i=0 i i=0 i i=0 i
Xn=1
For [l]-u.a.e. x ,...,x we obtain genericity w.r.t. m by Theorem 3.4, ergodicity of
0 k x ,...,x
0 k
m byCorollary3.9andorthogonalityof⊗k f[l] totheKroneckerfactorofm by
x0,...,xk i=0 i x0,...,xk
Corollary3.13,soProposition2.11withX =(X[l]×···×X[l],m )andY =(X[l] ,m )
0 k x0,...,xk k+1 yk+1
impliesthe claimed convergence to zero [l]-u.a.e.
This takes care of the terms f in the splittings f = f + f + f given
i,ε,⊥ i,ε i,ε,⊥ i,ε,Z,j i,ε,err,j
by Dec(k + l + 1 − i) (resp. k + l for i = 0). By an approximation argument like in
the proof of Lemma 3.7 it suffices to consider the main terms, so we may assume that
k f[l]((T[l])nx) is a nilsequence. The claimed convergence a.e. in x then follows
i=0 i i k+1
from the Wiener-Wintner theorem for nilsequences [HK09, Theorem 2.22]. The limit is
Q
zero a.e. in x provided that f ⊥ Z (X ) for some ε by definition of Y and
k+1 k+1,ε l+1 k+1 k+1,l
Lemma 2.7. (cid:3)
4. WIENER-WINTNER RETURN TIMES THEOREM FOR NILSEQUENCES
The first step in the proof is the identification of characteristic factors in the spirit of
[ALR95, §4]. We refer to [EZK12] for the notation.
Lemma 4.1. Let f ∈ L∞(X ), i =0,...,k, and assume CF(k,l). Let further G/Γ be anilman-
i i
ifold, G a filtration on G of length l, and let a Mal’cev basis adapted to G be fixed. Then for
• •
u.a.e. x ,...,x we have
0 k
1 N k
(4.2) lim sup kFk−1 F(g(n)Γ) f (Tnx ) =0,
N→∞ W˜k,2l(G/Γ) N i i i
g∈P(Z,G ),F∈W˜k,2l(G/Γ) (cid:12) Xn=1 Yi=0 (cid:12)
• (cid:12) (cid:12)
where˜k= l (d −d ) l . (cid:12) (cid:12)
r=1 r r+1 r−1
P (cid:0) (cid:1)
Proof. By Corollary 3.13 we have ⊗k f ⊥ Z (m ) for u.a.e. ~x ∈ X × ··· × X and by
i=0 i l ~x 0 k
Theorem 3.4 u.a.e. ~x is fully generic for ⊗ f w.r.t. m . The claim follows by the uniform
i i ~x
Wiener-Wintner theorem [EZK12, Theorem 4.1]. (cid:3)
Theorem 1.3 now follows from equidistribution results on nilmanifolds.
Proof of Theorem 1.3 for the given k. ByLemma4.1itsufficestoconsider f ∈ L∞(Z (X )).
i l+k+1−i i
By the pointwise ergodic theorem we can assume that each f is a smooth function on a
i
nilsystemfactorof X . The conclusionnowfollowsfromequidistributionresultsonnilman-
i
ifolds [Lei05, Theorem B]. (cid:3)
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