Table Of ContentTHE RECURRENCE TIME FOR ERGODIC SYSTEMS OF
INFINITE MEASURES
6
0
0
STEFANOGALATOLO,DONGHANKIM,ANDKYEWONKOHPARK
2
n
a
J
7 Abstract. We investigate quantitative recurrence in systems having an in-
2 finite measure. We extend the Ornstein-Weiss theorem for a general class of
infinite systems estimating return time in decreasing sequences of cylinders.
] Then we restrict to a class of one dimensional maps with indifferent fixed
S points and calculate quantitative recurrence in sequences of balls, obtaining
D thatthisisrelatedtothebehaviorofthemapnearthefixedpoints.
.
h
1. Introduction
t
a
m ThewellknownPoincar´erecurrencetheoremstatesthatundersuitableassump-
tions a typical trajectory of the system comes back infinitely many times in any
[
neighborhood of its starting point. This result does not give an estimation about
1 the time necessaryforanobittocome backnearthe startingpoint. Thisnaturally
v
raises questions like: how much iterations of an orbit is necessary to come back
2
within a distance r from the starting point? The quantitative recurrence theory
7
6 investigates this kind of questions.
1 In the literature such indicators of quantitative recurrence have been defined in
0 severalways by measuring the rate of the firstreturn time of anorbit with respect
6
toadecreasingsequenceofneighborhoodsofthestartingpoint. Anaturalapproach
0
is to use the metric of the spaceX andconsider adecreasingsequence ofballs ([7],
/
h [4], [3], [25], [26], [18], [16]) in the above question. More precisely, let us define
t
a τr(x) to be the first return time of x into the ball B(x,r) centered in x and with
m radius r. It turns out that in many chaotic systems the asymptotic behaviour of
: τr(x), when r is small is related to the local dimension of the invariant measure at
v
x:
i
X
(1.1) τ (x)≃rdµ(x).
r
r
a This is proved for a.e. x in “fast” mixing systems ([25]). And in general systems
the asymptotic behavior of τ (x) only gives a lower bound for the dimension.
r
Other interestingsequences ofneighborhoodsofx canalso be defined, using the
symbolic dynamics generated by a partition ξ. Let ξ = ∨k−1T−iξ. We consider
k i=0
a decreasing sequence of cylinders ξ (x) containing x. Here we define R (x) to be
k k
the smallest n such that Tn(x) ∈ ξ (x). A partition ξ = {ξ0,...,ξℓ−1} naturally
k
associates to a symbolic name to a.e. x. The symbolic name (x )∞ of x is given
i i=1
by the rule
x =k ⇐⇒Tix∈ξk, k =0,...,ℓ−1.
i
HenceR (x)denotethefirsttimegreaterthankthatisnecessarytoseethek-name
k
ofxagaininthe orbitofx. Itis wellknownthatthe asymptoticbehaviorofR (x)
k
ThethirdauthorissupportedinpartbyKRFR05-2004-000-10864-0.
1
2 STEFANOGALATOLO,DONGHANKIM,ANDKYEWONKOHPARK
as k increases is related to the entropy of the system. If ξ is generating then for
a.e. x ([23],[24])
R (x)≃2hµ(T)k.
k
Other quantitative recurrence indicators consider the forward images of the whole
cylinder ξ and consider the first return for the cylinder, which is the minimum n
k
satisfying Tn(ξ )∩ξ 6=∅ ([26],[6]).
k k
In computer simulations or experimental situations, the quantitative recurrence
indicatorscanbeeasilyestimatedbylookingatthebehaviorofa“typical,random”
orbit and its first return time in a sequence of sets centered around the starting
point. Bytheaboverelationsrecurrenceandrelatedconceptswereusedtoestimate
the dimension of attractors (see [15] and references therein, or [8], [13], [10], [9],
[16]) or entropy (see e.g. [12]).
The aim of this work is to investigate the quantitative recurrence in dynamical
systemspreservinganinfinite measure. Suchsystemsareusedfor modelsofstatis-
tically anomalous phenomena such as intermittency and anomalous diffusion (e.g.
[22][14])andthey do haveinterestingstatisticalbehavior. Many classicaltheorems
of finite measure preserving systems from ergodic theory can be extended to the
infinite measure preserving case. In particular we cite the Hopf ratio ergodic theo-
rem which will be used later. Let T be conservative and ergodic (see. e.g. [2]) and
f,g ∈L1 such that fdµ, gdµ6=0 , then
R R f(Tk(x)) fdµ
k∈[0,n−1]
→
g(Tk(x)) gdµ
Pk∈[0,n−1] R
as n→∞ for a.e. x. P R
The notion of entropy can be extended to the infinite case (see e.g. [28], [21]).
Let T be a conservative, ergodic measure preserving transformation on a σ-finite
space (X,A,µ). Then the entropy of T can be defined as
h (T)=µ(Y)h (T )
µ µY Y
where Y ∈ A with 0 < µ(Y) < ∞ and T is the first return map of Y (T (x) =
Y Y
TRY(x) where
R (x)=min{n≥1:Tn(x)∈Y}
Y
when x∈Y) and µ is the induced measure (µ (E)= µ(E∩Y)) which is invariant
Y Y µ(Y)
and ergodic under T . Since this notion does not depend on the choice of Y, we
Y
can well define the entropy of infinite measure preserving systems as above.
To give examples of the particular behavior of infinite measure preserving maps
let us consider a particularly representative class of maps on the unit interval: the
Manneville-Pomeaumaps defined by:
x+2z−1xz if 0≤x<1/2,
(1.2) T(x)=
(2x−1 if 1/2≤x<1.
these maps have an indifferent “slowly repulsive” fixed point at the origin. When
z ∈ [2,∞) this forces the natural invariant measure for this map to be infinite
and absolutely continuous with respect to Lesbegue. It is not hard to see that
ξ = {[0,1/2),[1/2,1)} is a generating partition and the entropy h (T) is positive
µ
andfinite. MoreoverwehaveaversionoftheShannon-McMillan-BreimanTheorem
THE RECURRENCE TIME FOR ERGODIC SYSTEMS OF INFINITE MEASURES 3
([17]): for every f ∈L1(µ), with f 6=0
−logR(µ(ξn(x))) hµ(T)
→
S (f,x) fdµ
n
as n → ∞ for a.e. x. Here ξn(x) is the elemeRnt of the iterated partition ξn =
∨n−1T−iξ containing x and S (f,x) is the partial sums of f along the orbit of x:
i=1 n
S (f,x)= f(Tk(x)).
n
k∈[X0,n−1]
This gives among other applications a result showing how the behavior of these
infinitesystemsis“weaklychaotic”([28],[5])givingsymbolicorbitswhichhavean
algorithmicinformationcontentgrowingasasublinearpowerlawastimeincreases,
while for a system having finite measure and positive entropythe informationcon-
tent increases linearly with time.
InthisworkwegeneralizeOrnstein-Weisstheoremtoawideclassofconservative,
ergodic, infinite measure preserving systems. More precisely we prove that :
Rn(x)≃2αhµ(fTd)µSn(f,x)
R
almost everywhere, where α is a constant depending on the system under consid-
eration and f and S are as above. This is proved in a weak form (Theorem 2.3)
n
with a “limsup” statement for generalsystems. The statement is also provedin a
stronger form (Theorem 3.2) for a system satisfying a strong mixing condition.
Insection4weapplythepreviousresulttoaclassofsmoothmapsontheinterval
with a neutral fixed point containing the Manneville-Pomeau family (1.2) and we
calculate for this family the quantitative recurrence indicators into balls, showing
thattheyarerelatedtothebehaviorofthemapneartotheneutralfixedpointand
not on the local dimension (recall (1.1)). More precisely, we see that if a map is of
the form as in (1.2) then
τ (x)≃rz−1
r
as r→0 (see theorem 4.1 for the general statement).
In the lastsectionwe showanexample ofa system havinga particularbehavior
of recurrence. In this system the asymptotic behavior of R (x) oscillates so that
n
lim logRn(x) does not exists.
n→∞ Sn
Acknowledgments. The authors wish to thank J-R Chazottes for fruitful
discussions,KoreaInstitute for AdvancedStudy andScuola Normale Superiorefor
logistic help and for the nice environment where the work has been done.
2. General case
Let us consider a measure preserving system (X,T,A,µ). Let S (A,x) be the
n
number of Tix∈A for 0≤i≤n−1, i.e.,
n−1
S (A,x)=S (1 ,x)= 1 (Ti(x)).
n n A A
i=0
X
Let ξ be a measurable partition of X and let
ξ =ξ∨T−1ξ∨···∨T−n+1ξ.
n
Given a partition ξ we denote the atom containing x by ξ(x). Define R (x) by
n
R (x)=min{j ≥1|ξ (x)=ξ (Tjx)}
n n n
4 STEFANOGALATOLO,DONGHANKIM,ANDKYEWONKOHPARK
considering a fixed set A∈A we also define R¯ (x) by
n
R¯ (x)=min{S (A,x)≥1|ξ (x)=ξ (Tjx)}.
n j n n
Note that R¯ (x)=S (A,x).
n Rn(x)
Lemma 2.1. Let T be a conservative, ergodic measure preserving transformation
(c.e.m.p.t.) on the σ-finite space (X,A,µ) with h (T) < ∞, and let ξ be a finite
µ
generatingpartition(modµ). AssumethatthereisanatomAinξ with0<µ(A)<
∞. For almost every x∈A
logR¯ (x) h (T)
n µ
lim = .
n→∞ Sn(x) µ(A)
Proof. Let ξ be the induced partition on A, where
A
ξ =∪ {V ∩{R =k}:V ∈A∩ξ }.
A k≥1 A k
Sinceageneratingpartitioninducesageneratingpartitionfortheinducedmap,the
induced map T on A is isomorphic to the shift map on (ξ )N. Let x˜ = (x˜ )∞ ∈
A A i i=1
(ξ )N, x˜ ∈ξ be the corresponding point of x in (ξ )N. Then
A i A A
R (x˜)=min{j ≥1|x˜ ...x˜ =x˜ ...x˜ }
n 1 n j+1 j+n
=min{j ≥1|(ξ ) (x)=(ξ ) (Tjx)}.
A n A n A
By the Ornstein-Weiss theorem for almost every x we have
logR (x˜)
n
lim =h (T ,ξ )=h (T ).
n→∞ n µA A Y µA A
Let m(x,n) be a positive integer defined by
m=R (x)+R (T x)+···+R (Tn−1x).
A A A A A
Then we have (ξA)n(x)=ξm(x). Since TAj =TSj(x)(x), we have
R (x˜)=min{j ≥1|(ξ ) (x)=(ξ ) (Tjx)}
n A n A n A
=min{S (x)≥1|ξ (x)=ξ (Tjx)}=R¯ (x).
j m m m
If S (x)=n then m(x,n−1)<k ≤m(x,n). Thus we have
k
R (x˜)=R¯ (x)≤R¯ (x)≤R¯ (x)=R (x˜)
n−1 m(n−1) k m(n) n
and
logR (x˜) logR¯ (x) logR (x˜)
n−1 k n
≤ ≤ .
n S (x) n
k
Hence for almost every x∈A we have
logR¯ (x) h (T)
k µ
lim = .
k→∞ Sk(x) µ(A)
(cid:3)
LetusrecallthefollowingfactalsoknownastheAaronsonergodictheorem([2]).
Theorem 2.2. Suppose that T be a c.e.m.p.t. of the σ-finite, measure space
(X,B,m), and let
a(n)
a(n)↑∞, ↓0 as n↑∞,
n
THE RECURRENCE TIME FOR ERGODIC SYSTEMS OF INFINITE MEASURES 5
then (1) If ∃A∈B, 0<m(A)<∞ such that
a(R )dm<∞,
A
ZA
then
S (f)
n →∞ a.e.∀f ∈L1(m)
+
a(n)
(2) Otherwise,
S (f)
liminf n =0 a.e.∀f ∈L1(m)
+
n→∞ a(n)
Theorem 2.3. Let T be a c.e.m.p.t. on the σ-finite space (X,A,µ) with h (T)<
µ
∞, and let ξ ⊂A be a finite generating partition (mod µ). Assume that there is an
atom A in ξ with 0<µ(A)<∞. Then for any f ∈L1(µ) with fdµ6=0,
logR (x) h (T)
n µ R
limsup = a.e.,
n→∞ Sn(f)(x) α fdµ
almost everywhere, where
R
α=sup{β : (R )βdm<∞}.
A
ZA
Moreover, if α=0, then the limsup goes to infinity.
Proof. For any C if n is large, then by Theorem 2.2 (1)
S (x)≥Cnα−ǫ,
n
so for large k
R¯ (x)=S (x)≥C(R (x))α−ǫ.
k Rk(x) k
Taking the logarithm we have
logR¯ (x)≥(α−ǫ)logR (x)+logC.
k k
which implies
logR (x) 1 logC
k
≤ 1− .
logR¯ (x) α−ǫ logR¯ (x)
k (cid:18) k (cid:19)
Hence we conclude
logR (x) 1
k
(2.1) limsup ≤ .
logR¯ (x) α
k→∞ k
For the reversed inequality, suppose that there is η >1 such that
logR (x) 1
k
limsup ≤ .
logR¯ (x) η3α
k→∞ k
Thenforallk largeenoughwehave(R (x))η2α <R¯ (x). SinceR¯ (x)=S (x),
k k k Rk(x)
if S =R¯ and n=R for k large enough, we have
n k k
(2.2) S >nη2α.
n
But by Theorem 2.2 and the definition of α we have infinitely many n’s such that
S (x)<nηα.
n
Let n be such an n and m = S (x) < n ηα. Then, since S is increasing, if
0 n0 0 n
m<S <mη, then ℓ>n and
ℓ 0
S <mη <n η2α <ℓη2α,
ℓ 0
6 STEFANOGALATOLO,DONGHANKIM,ANDKYEWONKOHPARK
whichimplies thatS 6=R¯ foranyk by(2.2). Therefore,thereareinfinitely many
ℓ k
m’s such that
(2.3) R¯ ∈/ (m,mη) for any k.
k
By Lemma 2.1 for any ǫ>0, if k is large, then
logR¯ (x) h
k µ
− <ǫ.
S (x) µ(A)
(cid:12) k (cid:12)
(cid:12) (cid:12)
Since S increases by 1 at mo(cid:12)st, if k is large, we(cid:12)have
k (cid:12) (cid:12)
(2.4) R¯k+1 <2Sk+1(h¯+ǫ) ≤2(Sk+1)(h¯+ǫ) =2h¯+ǫ(2Sk(h¯−ǫ))hh¯¯−+ǫǫ <2h¯+ǫ(R¯k)hh¯¯−+ǫǫ,
where ¯h = h /µ(A). Therefore, for large k we have R¯ < R¯η which contradicts
µ k+1 k
to (2.3). Hence we have
logR (x) 1
k
limsup ≥
logR¯ (x) α
k→∞ k
and combined with (2.1) we have
logR (x) 1
k
limsup = .
logR¯ (x) α
k→∞ k
By applying Lemma 2.1 we obtain
logR (x) h (T)
n µ
limsup = .
n→∞ Sn(x) αµ(A)
The resultfollowsreplacing1 by Hopf’s ergodictheoremwithanyf ∈L1 with
A
0< fdµ<∞. (cid:3)
R
3. Mixing case
A set A is called a Darling-Kac set, if there is a sequence {a } such that
n
n
1
lim Tˆk1 =µ(A)
A
n→∞an
k=1
X
almost uniformly on A. A function f is slowly varying at ∞ if
f(xy)
→1 as x→∞,∀y >0.
f(x)
Suppose that T has a Darling-Kacset, whose return time process is uniformly -
or strongly mixing from below (see [1] e.g.) and
(3.1) a (T)∼nαs(n),
n
where s(n) is a slowly varying function. The Darling-Kac Theorem states that
S (x)
n
→Y , in distribution,
α
a (T)
n
where Y has the normalized Mittag-Leffler distribution of order α. By Theo-
α
rem 2.2, α in (3.1) is the same α in Theorem 2.3.
By Aaronson’s work we have the following theorem:
THE RECURRENCE TIME FOR ERGODIC SYSTEMS OF INFINITE MEASURES 7
Fact 3.1 ([1]). Let L (n)=log(logn) and K = Γ(1+α) .
2 α αα(1−α)1−α
n
1
limsup f ◦Tk =K fdµ
n→∞ nαs(n/L2(n))L2(n)1−α k=1 αZ
X
a.e. for every f ∈L1.
+
Thus, for any ǫ>0 we have for almost every x
S (x)
n
(3.2) limsup =0.
nα+ǫ
n
Theorem 3.2. Let T be a c.e.m.p.t. on the σ-finite space (X,A,µ) with h (T)<
µ
∞, and let ξ ⊂ A be a finite generating partition (mod µ). Assume that there is
an atom A in ξ with 0<µ(A)<∞. Suppose that T has a Darling-Kac set, whose
return time process is uniformly - or strongly mixing from below and
a (T)∼nαs(n),
n
where s(n) is a slowly varying function. Then for any f ∈L1(µ) with fµ6=0,
logR (x) h (T) R
n µ
lim = a.e.
n→∞ Sn(f)(x) α fdµ
Proof. By Theorem 2.3 we have only to showRthat
logR (x) h (T)
n µ
liminf ≥ .
n→∞ Sn(f)(x) α fdµ
For any C, if n is large, then by (3.2) R
S (x)≤Cnα+ǫ.
n
Hence for large k
R¯ (x)=S (x)≤C(R (x))α+ǫ.
k Rk(x) k
Taking logarithm
logR¯ (x)≤(α+ǫ)logR (x)+logC.
k k
which implies
logR (x) 1 logC
k
≥ 1− .
logR¯ (x) α+ǫ logR¯ (x)
k (cid:18) k (cid:19)
Hence we have
logR (x) 1
k
(3.3) liminf ≥ .
k→∞ logR¯k(x) α
By applying Lemma 2.1 we obtain
logR (x) h (T)
n µ
liminf ≥ .
n→∞ Sn(x) αµ(A)
The result follows replacing 1 by Hopf’s ergodic theorem with any f ∈ L1 with
A
0< fdµ<∞. (cid:3)
R
8 STEFANOGALATOLO,DONGHANKIM,ANDKYEWONKOHPARK
4. Manneville Pomeau like maps
In this section we consider a particular class of maps. We apply the results of
the previoussectionto calculaterecurrencetime into balls. The mapswearegoing
toconsiderareageneralizationofthe socalledManneville-Pomeaumaps(Eq. 1.2)
and are included in the class of maps studied by [27].
We say that a map T : [0,1] → [0,1] is a Manneville-Pomeau map (MP map)
with exponent z if it satisfies the following conditions:
(1) there is c ∈ (0,1) such that, if I = [0,c] and I = (c,1], then T and
0 1 (0,c)
T extend to C1 diffeomorphisms, T(I ) = [0,1], T(I ) = (0,1] and
(c,1) 0 1 (cid:12)
(cid:12)
T(0)=0;
(cid:12)
(2) th(cid:12)ere is λ > 1 such that T′ ≥ λ on I , whereas T′ > 1 on (0,c] and
1
T′(0)=1;
(3) the map T has the following behaviour when x→0+
T(x)=x+rxz +o(xz)
for some constant r >0 and z >1.
For example the map defined in equation 1.2 is in this family. In [27] it is
provedthatwhenz ≥2thesemapshaveaninfinite,absolutelycontinuousinvariant
measure µ with positive density and the entropy can be calculated as h (T) =
µ
log(T′)dµ.
[0,1]
ThesemapshaveDKsetswherethefirstreturnmapismixing(see[1]page103)
R
andhencethey satisfythe assumptionsofthe abovesection. Ifz >2(see e.g. [29])
we have a behavior of the return time sequence given by
a (T)=n1/(z−1)s(n),
n
wheres(n)isaslowlyvaryingfunction. ThereforesettingS (x)= 1 (Ti(x)),
n i≤n I1
we have by Theorem 3.2
P
logR (x) h (T)
n µ
lim = (z−1).
n→∞ Sn(x) µ(I1)
Theorem 4.1. Let (X,T,ξ) satisfy (1)-(3) and µ be the absolutely continuous
invariant measure then
logτ (x)
r
lim =z−1
r→0 −logr
for almost all points x (we recall that τ (x) is the first return time of x in the ball
r
B(x,r)).
In the proof of the above theorem we will use the following technical lemma on
the behavior of real sequences.
Lemma 4.2. Let r be a decreasing sequence suchthat r →0 and there is a c>0
n n
suchthat r >cr eventually as n increases. Let τ :R→R be decreasing. Then
n+1 n r
limn→∞ −lolgoτgrrnn =l=⇒limr→0 −lolgoτgrr =l.
Proof. If r ≥ r ≥ r ≥ cr then τ ≥ τ ≥ τ , moreover logr ≥ logr ≥
n n+1 n rn+1 r rn n
logr ≥logcr ≥logcr... hence for n big enough
n+1 n
logτ logτ logτ
rn+1 ≥ r ≥ rn ,
−logr −logr−logc −logr −2logc
n+1 n
which taking the limits gives the statement.
THE RECURRENCE TIME FOR ERGODIC SYSTEMS OF INFINITE MEASURES 9
(cid:3)
Proof of Theorem 4.1. By theorem 3.2, for almost each x,
logR (x) h (T)
n µ
lim = (z−1).
n→∞ Sn(x) µ(I1)
Moreover,bytheinfinitemeasureversionoftheSMBtheorem(seee.g. [27]or[29])
−log(µ(ξ (x))) h (T)
n µ
→ .
S (x) µ(I )
n 1
Dividing the two equations we have
logR (x)
n
lim =z−1.
n→∞−log(µ(ξn(x)))
Now, since the measure is absolutely continuous with positive density (which is
also bounded near each point x6=0) and since each element of the partition ξ (x)
n
is an interval, the diameter of such an element is equal to its measure up to a
multiplicative constant (which depends on x) and then
logR (x)
n
lim =z−1
n→∞−log(diam(ξn(x)))
for almost all x. Since obviously ξ (x)⊂B(x,diam(ξ (x))), we have that
n n
τ (x)≤R (x)
diam(ξn(x)) n
and then limsup τdiam(ξn(x))(x) ≤z−1. Now, we have that diam(ξ (x)) is
n→∞ −log(diam(ξn(x))) n
a decreasing sequence, moreover by the SMB theorem for almost each x it holds
diam(ξn(x)) ∼ 2−Const.∗Sn. Then diam(ξn(x)) → 0, moreover, since Sn increases
at most by 1 at each step then when n is big there is a constant c > 0 such that
diam(ξ (x))≥c∗diam(ξ (x)).Nowfrom4.2wehavelimsup logτr(x) ≤z−1.
n+1 n r→0 −logr
For the opposite inequality we consider the set
1
J ={x:d(x,∂ξ (x))≥ diam(ξ (x))}
n n n2 n
which is a set made of the union of ‘central’ parts of each interval ξ (x). We have
n
that λ(J ) ≥ 1− 2 (λ denotes the Lesbegue measure) and then by the Borel-
n n2
Cantelli principle there is a full measure set J such that x ∈ J implies that there
is n s.t. x∈J ,∀k ≥n. Now if x∈J then τ (x)≥R (x), hence
k n n12diam(ξn(x)) n
log(τ (x))
liminf n12diam(ξn(x)) ≥z−1.
n→∞ −log(diam(ξn(x)))
By Theorem 2.2 we have liminf Sn = ∞ almost everywhere and then, by the
log(n)
SMB theorem log(n)=o(−logdiam(ξ (x))) almost everywhere. Then we have
n
log(τ (x)) log(τ (x))
lim n12diam(ξn(x)) = lim n12diam(ξn(x)) ≥z−1,
n→∞ −logdiam(ξn(x)) n→∞−log(n12diam(ξn(x)))
then, applying Lemma 4.2 as before we have the statement. (cid:3)
10 STEFANOGALATOLO,DONGHANKIM,ANDKYEWONKOHPARK
5. A counterexample
In this section we construct an example of a system having a “pathological”
behavior, for which limn→∞ lologgSnn does not exist, and then also limn→∞ logRSnn(x)
does not exist (it oscillates, having a different limsup and liminf behavior). The
system is constructed in an abstract way. We are not aware of any smooth, or
“concrete” systems presenting such a behavior.
Recall from Section 2 that if liminf logSn = α, then limsup logRn = hµ(T).
n logn n Sn αµ(A)
To show that lim logRn does not exist when logSn does not converges, we show
n Sn logn
the following lemma with the same assumptions in Section 2:
Lemma 5.1. If limsup logSn =β a.e., then we have
n→∞ logn
logR h (T)
n µ
liminf = a.e.
n→∞ Sn βµ(A)
Proof. If limsup logSn = β, then for any ǫ > 0 lim Sn = 0 a.e., which implies
n logn n nβ+ǫ
that
logR (x) 1
k
liminf ≥ .
k→∞ logR¯k(x) β
The proof is similar to Theorem 2.3. for the upper bound, we suppose that there
is λ<1 such that
logR (x) 1
k
liminf ≥ .
k→∞ logR¯k(x) λ3β
Hence (R (x))λ2β > R¯ (x) for all k large enough. Since R¯ (x) = S (x), if
k k k Rk(x)
S = R¯ and n = R for large k, we have S < nλβ. By the assumption we have
n k k n
infinitely many n’s such that S >nλβ. Hence, there are infinitely many m’s such
n
that
(5.1) R¯ ∈/ (mλ,m) for any k.
k
By(2.4)ifk is large,we haveR¯λ <R¯ whichcontradictsto (5.1). Therefore,we
k+1 k
have
logR (x) 1
k
liminf = .
k→∞ logR¯k(x) β
Combined with Lemma 2.1, we have
logR h (T)
n µ
liminf = a.e.
n→∞ Sn βµ(A)
(cid:3)
We will sketch the construction of an example (X,F,µ,T) which is ergodic and
measure preserving with the following properties.
1. There exists a subset X of finite measure where T| is Bernoulli (indepen-
0 X0
dent process) with respect to the partition {P ,P }.
0 1
2. The partition P = {P ,P ,P } of X generates the σ-algebra where X =
0 1 2 0
P ∪P andXc =P . Thatis,ifxandy aredifferentpointsofX,thenthereexists
0 1 0 2
a time k such that Tkx and Tky belong to different atoms of P.
3. Let 0<d< 1. For a.e. x∈X, liminf Sn(x) <1 and
2 n1/2
S (x)
n
limsup >1,
n1/2+d
