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Arch. Math. 97 (2011), 187–197
c 2011 Springer Basel AG
0003-889X/11/020187-11
published online July 16, 2011
DOI 10.1007/s00013-011-0285-7
Archiv der Mathematik
Approximately transitive dynamical systems and simple
spectrum
E. H. el Abdalaoui and M. Lemanczyk
Abstract. For some countable discrete torsion Abelian groups we give
examples of their finite measure-preserving actions which have simple
spectrum and no approximate transitivity property.
Mathematics Subject Classification (2010). Primary 37A25;
Secondary 37A30.
Keywords. Ergodic theory, Dynamical system, AT property,
Funny rank one, Haar spectrum.
1. Introduction. The property of approximate transitivity (AT) has been
introduced to ergodic theory by Connes and Woods [ 5 ] in 1985 in connec-
tion with their measure-theoretic characterization of ITPFI hyperfinite factors
in the theory of von Neumann algebras. Since then, approximately transi-
tive dynamical systems have been objects of study in several papers, e.g.,
[ 1 , 9 , 12 , 13 , 15 , 16 , 22 ]. All systems with the AT property have zero entropy [ 5 , 8 ]
and it is only recently [ 1 , 9 ] that explicit examples of zero entropy systems
without the AT property have been discovered.
In view of the form of its definition (see below) the AT property seems
to be related to spectral properties, more precisely to the spectral multiplic-
ity of the measure-preserving system. Indeed, it was in the 1990’s when
J.-P. Thouvenot, using some generic type arguments, observed that the AT
property implies the existence of a cyclic vector in the L 1 -space of the under-
lying dynamical system. Moreover, a modification of the definition of the AT
property (in which we replace L 1 norms by L 2 norms [ 16 ]) implies simplicity
of the L 2 spectrum. However, in general it is still an interesting open prob-
lem whether the original AT property implies simplicity of the L 2 spectrum,
Research supported by the EU Program Transfer of Knowledge “Operator Theory Methods
for Differential Equations” TODEQ and the Polish MNiSzW grant N N201 384834.
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E. H. el Abdalaoui and M. Lemanczyk
188
Arch. Math.
i.e., simplicity of the spectrum of the classical Koopman representation of the
dynamical system. Moreover, as noted in [ 9 ], the other implication was not
known. The aim of this note is to answer this latter question—we give exam-
ples of systems which are not AT but have simple L 2 spectrum.
One further motivation for this note is the problem of a measure-theoretic
characterization of the simplicity of the L 2 spectrum. This problem has a long
history in ergodic theory, see for example [ 20 , 21 ] and the references therein,
and our examples also give the answer to a further open question (stated
explicitly in case of
-actions in [ 11 ]aswellasin[ 9 ]): namely we give a neg-
ative answer to the question of whether the class of funny rank one systems
[ 10 , 19 ] coincides with the class of systems with simple L 2 spectrum; indeed a
funny rank one system enjoys the AT property (see below).
We would like however to emphasize that in the note we do not con-
sider
Z
-actions which are the most popular objects of study in ergodic theory,
and therefore the problems we mentioned above remain open for the class of
Z
Z
-actions (and actions of many other groups). The systems which are consid-
ered below are actions of countable discrete Abelian groups with torsion. More
precisely G = n =0 Z
where ( p n ) n≥ 0 is an increasing sequence of prime
numbers. We will then deal with the so-called Morse extensions given by some
special
/p n Z
-valued cocycles over a discrete spectrum action of G studied by
M. Guenais in [ 14 ] (in that paper she proved that the resulting G -actions have
Haar component of multiplicity one in the L 2 spectrum). All such systems
have simple spectrum. We will show that for each sufficiently fast choice of
parameters in Guenais’ construction we can apply the criterion for being a
non-AT system formulated in [ 1 ] (this criterion is an elaborated version of the
method already used in [ 9 ]). More precisely, we will prove the following.
Theorem 1.1. Assume that G = n =0 Z
Z
/ 2
Z
/p n Z
with ( p n ) n≥ 0
an increasing
5 2( n +1) ,n
sequence of prime numbers and p n ≥
0 . Then there exists a simple
L 2 spectrum action of G without the AT property.
≥
Following Connes and Woods [ 5 ] we now recall the definition of the AT
property.
Let G be a countable discrete Abelian group. Assume that this group acts
as measure preserving maps: g
→
T g
∈
Aut( X,
B
,μ ), where ( X,
B
,μ )isa
standard probability Borel space. The action ( X,
,μ,T ) with T =( T g ) g∈G
(or simply T ) is called AT if for an arbitrary family of nonnegative functions
f 1 ,...,f l ∈
B
L 1 +
( X,
B
,μ ) ,l
≥
2 , and any ε> 0, there exist a positive integer s ,
elements g 1 ,...,g s
∈
G , real numbers λ j,k > 0 ,j =1 ,...,l,k =1 ,...,s and
L 1 +
f
∈
( X,
B
,μ ) such that
1
s
f j −
λ j,k f
◦
T g k
<ε,
1
≤
j
≤
l.
(1.1)
k =1
Recall now that T has the funny rank one property [ 10 , 19 ] if for every
A
∈B
and ε> 0 we can find F
∈B
,g 1 ,...,g N
∈
G such that the family
R
, called a funny Rokhlin tower , consists of sets which
are disjoint and for some J
=
{
T g 1 F,...,T g N F
}
⊂{
1 ,...,N
}
Vol. 97 (2011)
Approximately transitivity
189
μ A
T g i F <ε.
(1.2)
i
∈
J
Alternatively, T is of funny rank one if and only if there exists a sequence
(
R n ) n≥ 1
of funny towers with bases F n such that each set A
∈B
can be
ε -approximated [as in ( 1.2 )] by the union of some levels for each
R n whenever
n
n ε . From this it easily follows that each system which is of funny rank one
is AT. Indeed, the set of functions which are constant on levels of
≥
R n for some
1 (and zero outside R n ) are dense in L 1 , and the set of those which are
additionally positive is dense in L 1 +
n
≥
L 1 +
; it follows that given f 1 ,...,f l ∈
( X, μ )
and ε> 0 it is enough to take f =
[in ( 1.1 )] for n
≥
1 large enough.
1 F n
,μ ) induces a (continuous) unitary repre-
sentation, called a Koopman representation ,of G in the space L 2 ( X,
The action T of G on ( X,
B
B
,μ )
L 2 ( X,
given by U T g f = f
G . We recall that a
Koopman representation is said to have simple spectrum if for some f
◦
T g ,f
∈
B
,μ )and g
∈
∈
L 2 ( X,
L 2 ( X, μ )we
define its spectral measure σ f (or in a more precise notation, σ U T ,f )tobea
finite Borel measure on the dual
,μ ) ,L 2 ( X,
B
B
,μ )=span
{
f
◦
T g :
g
∈
G
}
.Given f
∈
G of G determined by
σ f ( g ):=
G
χ ( g ) dσ f ( χ )=
X
σ U T ,f ( g )=
f
◦
T g ·
fdμ, for all g
∈
G.
See [ 20 , 21 ] for more information on the spectral theory of G -actions.
2. A criterion for a system to be non-AT. Let G be a countable discrete
Abelian group which we assume to act on a probability standard Borel space
( X,
B
,μ ) as measure-preserving maps: g
→
T g . Assume that
P
=
{
P 0 ,P 1 }
is
a partition of X (with P 0 ∈B
). Through its
P
-names every point x
∈
X can
be now coded: x
→
π ( x )=( x g ) g∈G where
x g = 0if T g ( x )
∈
P 0
1 if not.
Let Λ be a finite subset of G .Bya funny word on the alphabet
{
0 , 1
}
based
on Λ we mean a sequence ( W g ) g∈ Λ with W g ∈{
0 , 1
}
,g
∈
Λ. For any two funny
words W, W based on the same set Λ
⊂
G their Hamming distance is given
by
1
d Λ ( W, W )=
= W g }|
| |{
g
∈
Λ: W g
.
|
Λ
As noted in [ 1 ] we have the following extension of Dooley–Quas’ [ 9 ] neces-
sary condition for a system T =( T g ) g∈G to have the AT property.
Proposition 2.1. [ 9 ] Let ( X,
,μ,T ) be an AT dynamical system. Then for any
ε> 0 there exist a finite set Λ
B
⊂
G andafunnyword W based on Λ such that
|
Λ
|
μ (
{
x
∈
X : d Λ ( π ( x )
| Λ ,W ) <ε
}
) > 1
−
ε.
The contrapositive of Proposition 2.1 gives a criterion for a system to be
non-AT. Some further work has been done in [ 1 ] to formulate a condition
stronger than the negation of the assertion in Proposition 2.1 and which may
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E. H. el Abdalaoui and M. Lemanczyk
190
Arch. Math.
be applied to many systems (the criterion obtained this way is an elaborated
version of the method already used in [ 9 ]). We now present this criterion (Prop-
osition 2.2 below) in its generality needed for this note.
A probability Borel measure ρ defined on G is called a strong Blum–Hanson
measure (SBH measure shortly) if the following holds
χ ( θ )
2
√ k
θ∈ Θ
1
lim sup
k→ + ∞
sup
Θ ⊂G,| Θ | = k, ( η θ ) θ∈ Θ ∈{ 1 ,− 1 } k
η θ ·
dρ ( χ ) < 1+ ε 0 ,
G
for some ε 0 in [0 , 1). Clearly the Haar measure m G of G is an SBH measure;
more generally, every absolutely continuous probability measure with density
d
L 1 ( G, m G ) satisfying
[0 , 1), is an SBH
measure. It is shown in [ 1 ] that each SBH measure is a Rajchman measure
(i.e., its Fourier transform vanishes at infinity [ 4 ]).
∈
d
∞
< 1+ ε 0
for some ε 0 ∈
Proposition 2.2. Assume that a dynamical system ( X,
B
,μ,T ) is ergodic and
that there exists a partition
P
=
{
P 0 ,P 1 }
with the following properties:
(i) There exists S
∈
Aut( X,
B
,μ ) commuting with all elements T g ,g
∈
G ,
such that SP 0 = P 1 .
(ii) The spectral measure σ 1 P 0 − 1 P 1 is an SBH measure.
Then the system T =( T g ) g∈G is not AT.
Proof. (The proof is similar to the proof of Proposition 3.4 from [ 1 ]; we pro-
vide it for sake of completeness.) Let W be a funny word based on a subset
Θ=
{
g 1 ,g 2 ,...,g k }⊂
G and define Λ W on X by
k
Λ W ( x )= 1
k
A j ( x ) ,
x
∈
X
j =1
where A j
is defined as
( x )= 1
if W g j
= x g j
A j
−
1if t
(recall that ( x g )= π ( x )isthe
-name of x ). We claim that the distribution
Λ ∗ W of Λ W is symmetric. Indeed, we have π ( x )=1
P
−
π ( Sx ) and therefore
Λ W ( x )=
−
Λ W ( Sx )
and since S is measure-preserving the symmetry of Λ ∗ W follows.
Observe that
1) W g j (
A j ( x )=(
−
1 P 0 − 1 P 1 )( T g j x )
(2.1)
and that
Λ W ( x )=1
−
2 d Θ ( W, π ( x )
| Θ ) .
(2.2)
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Vol. 97 (2011)
Approximately transitivity
191
In view of ( 2.2 ), the symmetry of Λ ∗ W
and the Tchebychev inequality, for
0 <ε< 1 / 2, we obtain that
μ (
{
x
∈
X : d Θ ( W, π ( x )
| Θ ) <ε
}
)= μ (
{
x
∈
X :Λ W ( x ) > 1
−
2 ε
}
)
1
2 μ (
=
{
x
∈
X :
|
Λ W ( x )
|
> 1
−
2 ε
}
)
1
Λ W || 2
≤
2 ε ) 2 ||
.
(2.3)
2(1
−
But, in view of ( 2.1 ) and the Spectral Theorem
2
=
k
1
k
Λ W 2
A j
dμ
j =1
X
k
1
k 2
1) W g i (
=
(
−
1 P 0 − 1 P 1 )( T g i x )
i,j =1
X
1) W g j (
·
(
−
1 P 0 − 1 P 1 )( T g j x ) dμ ( x )
k
1
k 2
1) W g i + W g j σ 1 P 0 − 1 P 1
=
(
−
( g i −
g j )
i,j =1
1) W g i χ ( g i )
2
k
1
k
1
√ k
=
(
−
dσ 1 P 0 − 1 P 1
( χ ) .
i
=1
G
It follows that for k large enough we have
1
k (1 + ε 0 ) .
Λ W 2 <
(2.4)
Combining ( 2.3 ) and ( 2.4 ) we obtain that
1+ ε 0
2(1
kμ (
{
x
∈
X : d Θ ( W, π ( x )
| Θ ) <ε
}
)
≤
2 ε ) 2
−
1+ ε 0
2(1 − 2 ε ) 2
and since
ε for ε> 0 small enough and W was arbitrary, this
contradicts Proposition 2.1 .
< 1
−
3. Countable discrete Abelian group action with simple spectrum and without
the AT property. In this section we shall present the proof of Theorem 1.1 .
3.1. Cocycles for discrete spectrum actions of countable discrete Abelian
groups. For completeness, we now briefly present an extension of the classical
spectral theory of compact group extensions of
-actions to group actions.
Let G be a countable discrete Abelian group acting on a standard probabil-
ity Borel space ( X,
Z
,μ ).
Assume that Γ is a compact metric Abelian group (written multiplicatively)
with Haar measure m Γ .A cocycle associated to T =( T g ) g∈G with values in Γ
is a measurable function ϕ : X
B
,μ ) as measure-preserving maps: g
→
T g ∈
Aut( X,
B
Γ which satisfies
ϕ ( x, g + g )= ϕ ( x, g ) ϕ ( T g x, g )
×
G
→
(3.1)
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