INVARIANT MEASURES AND ASYMPTOTICS FOR SOME
SKEW PRODUCTS
J. AARONSON, H. NAKADA, O. SARIG, R. SOLOMYAK
Abstract:
For certain group extensions of uniquely ergodic transformations, we identify
all locally nite, ergodic, invariant measures. These are Maharam-type measures.
We also establish the asymptotic behaviour for these group extensions proving
logarithmic ergodic theorems, and bounded rational ergodicity. c 1999
x0 Introduction and General Framework
Let (X B) be a standard measurable space, and let : X ! X be an invertible
measurable map. Let G be a locally compact, Abelian, Polish (LCAP) topological
group and let : X ! G be measurable.
The skew product transformation : X G ! X G is dened by
(x y) := (x y + (x)):
A measure m : BB(G ) ! 0 1] is called locally nite if m(X K) < 1 8 K G compact.
Our program is to identify all -invariant locally nite measures and study their
asymptotic behaviour.
It is known (Fu], Pa]) that if is a uniquely ergodic homeomorphism of a
compact metric space (with invariant probability p), G is compact (with Haar
probability measure mG) and : X ! G is continuous, then ergodicity of with
respect to the product p mG is equivalent to the unique ergodicity of .
For non-compact G , it is well known that if is uniquely ergodic (with invariant
probability p), and is ergodic with respect to pmG , then there is no -invariant
probability on X G (see e.g. A1] chapter 8, or Sc2]).
It is natural to ask (as in Ve]) for -invariant locally nite measures. There is a
natural class of -invariant locally nite measures: the Maharam measures which
we proceed to describe.
Let (X B) and be as above and let h : X ! R+ be measurable. We call a
probability 2 P (X B) (h )-conformal if and dd = h -a.e..
Now let : X ! G be measurable, and let : G ! R be a continuous homomorphism. Let = be a (e )-conformal probability on (X B).
The associated Maharam measure is m : B B(G ) ! 0 1] dened by
dm (x y) := e;(y) d(x)dy (where dy denotes Haar measure on G ). The reason
for this terminology is that Maharam measures were rst considered for G = R in
Mah].
Date : 24/03/00.
1
2
J. AARONSON, H. NAKADA, O. SARIG, R. SOLOMYAK
A Maharam measure is easily seen to be -invariant, the dilation from the rst
coordinate being cancelled by the translation in the second.
The transformations considered here have the following properties:
Unique conformal probabilities:
For each continuous homomorphism : G ! R, there is a unique (e )conformal probability = on (X B)
Maharam measures are ergodic:
For each continuous homomorphism : G ! R, the Maharam measure m is
ergodic (for )
Ergodic measures are Maharam:
The only ergodic -invariant locally nite measures are Maharam measures.
Remarks
1) For G compact, the only continuous homomorphism : G ! R is 0, the
only Maharam measures are of form m mG, and the above properties for are
equivalent to its unique ergodicity.
2) As shown in Sc2], there are abundances of (e )-conformal innite measures, and of non-locally nite, -invariant, -nite measures.
We attempt our program in two cases. In x1, we treat the so called cylinder ow
R : T R ! T R dened by R(x y) := (x + y + (x)) where 2 Tn Q
and where (x) = ( + 1) 10 +1 ) ; (some > 0), the rest of the paper being
devoted to certain group extensions of adic transformations by symmetric cocycles
(see below).
Let S be a nite, ordered set, let A : S S ! f0 1g be an irreducible, aperiodic
matrix and let = A S Nbe the corresponding (topologically mixing) subshift
of nite type (SFT).
Let V be the adding machine on S N. The adic transformation on is the induced
transformation of V onn dened (in x2) for all except countably many points x 2 by (x) = V minfn1: V (x)2g (x).
For f :P ! G , we consider the symmetric cocycle f : ! G dened by
i
i
f (x) := 1
0 (f(T x) ; f(T (x))) where T : ! is the shift, the sum termii
i
nating as T (x) = T (x) 8 large i 1.
In x2 we show that the class of f -invariant, locally nite measures for f aperiodic having nite memory is collection of mixtures of the canonical Maharam
measures (theorems 2.1 and 2.2).
In x3 and x4, we consider the asymptotic properties of f with respect to Maharam measures, where f : ! Rd is an aperiodic Holder continuous function.
For 2 Rd, consider the Maharam measure m : B( Rd) ! 0 1] dened by
dm (x y) = e;y d(x)dy where = is the (e(f ) )- conformal measure. In
x4, nwe show that f is boundedly rationally ergodic with return sequence a(n) (see A2], and/or x4) with respect to m0 . Bounded rational ergodicity is a
(log n) d2
strong form of rational ergodicity, and so this entails a kind of absolutely normalized
ergodic theorem:
Sn (f) Z fdm 8 f 2 L1 (m )
0
0
a(n)
X
where
fn f if 8 m` " 1 9 nk = m`k " 1 such that 8 pj = nkj " 1, we have
1 PN fp ! f a.e. as N ! 1 (see A1]).
N j =1 j
INVARIANT MEASURES AND ASYMPTOTICS FOR SOME SKEW PRODUCTS
3
For 6= 0, f is squashable with respect to m (see A1]) and there is no such
kind of ergodic theorem. Nevertheless, we show in x3 that the logarithmic ergodic
theorem holds:
P ;1 F k
log nk=0
f ;! h (T) m -a.e. as n ! 1 8F 2 L1 (m )
+
logn
htop (T) where is the equilibrium measure of f (see Bo]).
There is some relation between the results of x2 and results in P-S] remarked
at the end of x2. The program in x3 and x4 has been previously carried out in full
in A-W] for = f0 1gN f(x) = x1 . Bounded rational ergodicity of certain of the
cylinder ows was established in A-K].
Horocycle ows on Abelian covers of compact, hyperbolic surfaces can be considered as \smooth analogues" of the skew products considered here. Ergodic, Maharam measures for these horocycle ows were introduced, and their asymptotics
considered in B-L].
We conclude this introduction with a Basic Lemma, to be used in x1 and x2.
For a 2 G , dene Qa : G ! G by Qa (x y) := (x y + a), then Qa = Qa .
If m is an ergodic -invariant locally nite measure, then so is m Qa (a 2 G )
whence, as is well known, either m Qa ? m or m Qa = cm for some c 2 R+.
For m an ergodic -invariant locally nite measure, set
H = Hm := fa 2 G : m Qa mg:
0.1 Basic Lemma
(i) H is closed
(ii) If H = G , then m is a Maharam measure.
Proof
(i) By unicity of absolutely continuous invariant measures, 9 a multiplicative
homomorphism : H ! R+ such that
Z
Z
f Qadm = (a)
fdm 8 a 2 H f 2 L1 (m):
X G
X G
For f : X G ! R continuous with compact support, we have that f Qan ! f Qa
uniformly as an ! a in G . Suppose that an 2 H an ! a 2= H. This forces
(an) ! 0 since 8 > 0 9 f : X G ! R+ continuous with compact support
such that
Z
Z
fdm = 1
f Qa dm < X G
X G
whence
Z
Z
>
f Qadm f Qan dm = (an):
X G
X G
On the other hand
R 9 f : X G ! R continuous and everywhere positive, whence
f Qa > 0 and X G f Qa dm > 0 contradicting (an) ! 0 and showing that
a 2 H.
(ii) There is a measurable (hence continuous) homomorphism : G ! R such
that m Qa = e;(a) m. Dene the measure m : B(X G ) ! 0 1] by dm(x y) :=
e(y) dm(x y). It follows that m Qa = m. For A 2 B(X) B 2 B(G ) and a 2 G ,
we have
m(A (B + a)) = m Qa (A B) = m(A B)
4
J. AARONSON, H. NAKADA, O. SARIG, R. SOLOMYAK
whence by unicity of Haar measure on G 8 A 2 B(X) 9 (A) 2 R+ such that
m(A B) = (A)mG(B) (B 2 B(G )):
It follows that is a nite measure on X, and and that
dm(x y) = e;(y)d(x)dy:
The -invariance of m now implies that with dd = e (it being
necessary to cancel the dilation due to translation of the second coordinate by
dilation of the rst).
x1 Cylinder flows
Let T := R=Z
= 0 1) denote the additive circle (the multiplicative circle being
1
S := e2iT
C ) and let R(x) := x + mod 1. The natural distance function on
T is given by the norm kxk := minn2Z
jx + nj.
For > 0, let G R be the closed subgroup generated by 1 and . Note
that G = Zif 2 Q and G = R if 2= Q. Consider for > 0, the function
: T ! G dened by
= () := 10 ) ; 1 1) = ( + 1)10 ) ; +1
+1
+1
and the skew products (or cylinder ows) R( ) : T G ! T G dened by
R( ) (x y) = (x + y + () (x)) for 2= Q > 0.
The goal here is to identify all the locally nite, -nite, R( ) -invariant meaP ;1 () Rk .
sures. Write (n) := nk=0
We recall some information about the continued fraction expansion
= 1=a1 + 1=a2 + 1=a3 + : : :
of 2 0 1) n Q. This can be found in Kh].
The positive integers an are called the partial quotients of .
Dene pn qn 2 Z+ gcd (pn qn) = 1 by
pn := 1=a + 1=a + 1=a + + 1=a 1
2
3
n
qn
then
q0 = 1 q1 = a1 qn+1 = an+1qn + qn;1
p0 = 0 p1 = 1 pn+1 = an+1 pn + pn;1
and
p2n < < p2n+1
q2n
q2n+1
pn ; pn+1 = (;1)n+1 :
qn qn+1 qnqn+1
The rationals pqnn are called the convergents of , and the numbers qn are called
(principal) denominators of .
W
Recall the Denjoy-Koksma inequality,
that kFqn k1R TF for any function
W
F : T! R of bounded variation ( TF < 1) such that TF(t)dt = 0. In particular,
j
(qn)j 2( + 1):
INVARIANT MEASURES AND ASYMPTOTICS FOR SOME SKEW PRODUCTS
5
1.1 Proposition
8 2= Q > 0 and > 0 9 a unique (( ) R)-conformal probability measure
= 2 P (T).
Proposition 1.1 follows from a more general \folklore theorem" (pointed out to the
authors by J-P. Conze and K. Schmidt):
Theorem
Let 2= Q and suppose that h : T ! R has bounded variation and
R
h(x)dx
=
0, then there is a unique (eh R)-conformal 2 P (T).
T
Moreover is nonatomic.
Proof
We rst prove existence.
S Rn ;. As
Let ; be the (countable) set of discontinuities of h and let ;1 := n2Z
shown in Ke]:
9 X a compact metric space, T : X ! X a homeomorphism, : X ! 0 1) H :
X ! R continuous such that
(i) T = R , (ii) 8 x 2= ;1 j;1fxgj = 1 H(;1 x) = h(x).
It follows fromW the Denjoy-Koksma inequality that
(iii) jHqn (x)j Th 8 x 2 X n ;1;1 and hence (by continuity) 8 x 2 X.
By theorem 4.1 in Sc2], 9 2 P (X) and c 2 R such that T and
d T = eH +c . Since
d
1 = (T qn X) =
Z
X
eHqn +cqn d ecqn
as n ! 1, we must have c = 0.
We claim that is nonatomic. Otherwise 9 x 2 X with (fxg) > 0 whence
9 2 P (X) with = Pn2ZanT n xPwhere an > 0. By ddT = eH an =
ceHn (x) for some c > 0 entailing (X) c n2Z
eHqn (x) = 1 and contradicting
2 P (X).
Now dene 2 P (T) by = ;1 . It follows that is nonatomic, whence
(;1) = 0 and R and ddR = eh -a.e..
Existence and nonatomicity are now established and we turn to the proof of
unicity.
We prove that if R and ddR = eh -a.e., then R is -ergodic. This
suces since nonunicity implies existence of with R and ddR = eh a.e., and R not -ergodic.
As above, is non-atomic, and by minimality of R, (J) > 0 8 intervals J. Thus
if : 0 1) ! 0 1) is dened by (x) := ((0 x)) then is an orientation preserving
homeomorphism of T, and ;1 = Lebesgue. It follows that S = R ;1
is absolutely continuous with S 0 = eh and by theorem 2b in dM-vS] S is ergodic
(Lebesgue). It follows that R is ergodic ().
Remark The (( ) R)-conformal = 2 P (T) can also be obtained using
the methods of Her] (as in N1] and N2]):
Dene the continuous f = f
: R ! R by
f
(x) =
(
x x 2 0 a( )))
; (x ; a( )) + a( )
x 2 a( ) 1)
6
J. AARONSON, H. NAKADA, O. SARIG, R. SOLOMYAK
where a( ) := +1;;11 (this value of a is forced by the slopes, and continuity of
f
).
By the theory of rotation numbers, 9 0 < b < 1, and an orientation preserving
homeomorphism : T ! T with (0) = 0 (1) = 1 such that ;1 f = R
where f := f
+ b.
It can be shown that if := m , then ddR = .
Invariant measures for the cylinder flow R(1)
Recall that q 2 N is called a Legendre denominator for if 9 p 2 N such that
j ; pq j < 21q2 . This is because of Legendre's theorem that a Legendre denominator
for is a principal denominator for .
It is well known that there are innitely many odd Legendre denominators for
any 2= Q.
1.2 Sublemma
Suppose that q is an odd Legendre denominator for , then j
(1)
q j 1.
Proof in case j ; qp j < 21q2 .
Firstly f kpq mod 1 : 0 k q ; 1g =: f0 = a1 < a2 < < aq 1g with
ai := kqi p and f kpq + 12 mod 1 : 0 k q ; 1g =: f0 = b1 < b2 < < bq 1g
satisfy a1 < b1 < a2 < b2 < < aq < bq 1 with bi ; ai = ai+1 ; bi = 21q .
Now let ki `i (0 i q ; 1) be such that ai = kqi p mod 1 and bi = `qi p mod 1.
Set ai := ki mod 1 and bi = `i mod 1.
We claim that a1 < b1 < a2 < b2 < < aq < bq 1. The reason for this is
that jk ; kpq j < 21q (0 k q ; 1) whence in case > pq ,
ai < ai < ai + 21q = bi < bi < bi + 21q = ai+1 < : : :
and in case < pq ,
ai+1 > ai+1 > ai+1 ; 21q = bi > bi > bi ; 21q = ai > : : ::
Now (1)
q is a step function with points of discontinuity 1 ; a1 > 1 ; b1 > 1 ; a2 >
1 ; b2 > > 1 ; aq > 1 ; bq 0, and jumps of +2 at 1 ; ai (1 i q) and
;2 at 1 ; ai (1 i q). The values of (1)
q are
R of form fv v + 2g for some
v 2 Z. The only v 2 Zpermitted by the condition T
(1)
q (t)dt = 0 is v = ;1. Thus
(1)
j
q j 1.
This subsection is based on the following lemma which is obtained from sublemmas 1.1 and 1.2:
1.3 Lemma
9 nk ! 1 such that j
(1)
qnk j 1 8 k 1.
Remark
Sublemma 1.2 can be strengthened: j
(1)
q j 1 whenever q is an odd principal
denominator for . This is shown in N1].
For > 0 2 Tn Q dene the R(1) -invariant, Maharam measure m
on
B(T Z) by
m
(A fng) := ;n 1
(A):
INVARIANT MEASURES AND ASYMPTOTICS FOR SOME SKEW PRODUCTS
7
1.4 Theorem
1) 8 2= Q and > 0 (T Z B(T Z) m
R(1) ) is a conservative, ergodic
measure preserving transformation.
2) If m is a locally nite measure on T Zsuch that (T Z B(T Z) m R(1) )
is ergodic and measure preserving, then 9 c > 0 such that m = cm
.
Proof
The ergodicity of (T Z B(T Z) m
R(1) ) was established in N1] (see
C-K] and also A-K] for the Lebesgue case = 1) and is standard using Sc1] and
lemma 1.3:
9 nk ! 1 (odd Legendre convergents) such that j
(1)
nk j 1 and
n
k
1
(R AA) ! 0 8 A 2 B(T):
We prove (2). Let m be a R(1) -ergodic locally nite measure on T Z. We
claim that m = cm
for some c > 0. By the Basic Lemma and proposition 1.1,
it suces to prove that H := fn 2 Z: m Qn mg = Z.
Indeed, write mk (A) := m(A fkg) and suppose that H 6= Z, then m :=
m;1 + m1 ? m0 . 9 U T open, such that m0 (U) = 1 and m(U) < 51 , whence
9 I T, an open interval such that m(I) < m05(I) :
Given 0 < p < 1 and an open interval L = (a ; r a + r), denote by Lp the
subinterval (a ; pr a + pr). Note that if x 2 Lp and jyj < (1;p2)jLj then x + y 2 L.
9 0 < p(1;<p)j1Ij such that
m0 (Ip ) > m02(I ) . By lemma 1.3, 9 k 1 such that
kqnk k < 2 and j
(1)
qnk j 1.
It follows that
q nk
R
(1) (Ip f0g) I f;1 1g
whence
m0 (I ) < m (I ) = m(I f0g)
0 p
p
2
qnk
= m(R(1) (Ip f0g)) m(I f;1 1g)
= m(I) < m05(I ) :
The contradiction shows the impossibility of H 6= R, and thus proves 2).
Invariant measures for the cylinder flow R( )
For > 0 2 Tn Q dene the locally nite measure m
on B(T R) by
dm
(x y) := ;y d
(x)dy:
Evidently m
R( ) = m
.
Fix 2 Tn Q. For t 2 R, consider the set
L(t) = L (t) := fa 2 0 1] : 9 nk ! 1 qnk t mod 1 ! ag
(where fqn : n 1g are the denominators of ).
It is shown in Ku-Ni] that L(t) = 0 1] for Lebesgue-a.e. t 2 R, and it is shown
in Kr-Li] that for 2= Q with bounded partial quotients and t 2 R, L(t) is nite
i t 2 Q + Q.
1.5 Lemma mod 1 ! a, then 8 x 2 T,
If a 2 L( +1 ) and qnk +1
(qn)k (x) ! ( + 1)fN ; a : N = ;1 0 1 2g:
8
J. AARONSON, H. NAKADA, O. SARIG, R. SOLOMYAK
Proof
; N ; aj < , then qn = ( +1)(N +
Let > 0 N 2 Zand suppose that jqn +1
a ), whence
(qn) = ( + 1)(10 ) )qn ; qn = ( + 1)(L ; a )
+1
where L := (10 ) )qn ; N 2 Z.
+1
Recalling that j
(qn) j 2( +1) we see that ;2+a ; L 2+a+. It follows
that for a > 0 and suciently small > 0: L = ;1 0 1 2.
1.6 Theorem
) is innite, then
Suppose that 2= Q > 0 are such that L( +1
1) For each > 0, (T R B(T R)m
R( ) ) is a conservative, ergodic
measure preserving transformation.
2) If m is a locally nite measure on T R such that (T R B(T R)m R( ) )
is ergodic and measure preserving, then 9 c > 0 such that m = cm
.
Proof
The ergodicity of (T R B(T R) m
R( ) ) was established in N2] and
in St] for = 1 (Lebesgue measure).
We prove (2). Let m be a R( ) -ergodic locally nite measure on T R. We
claim that m = cm
for some c > 0. By the Basic Lemma and proposition
1.1, it suces to prove that H := fa 2 R : m Qa mg = R.
Suppose otherwise, then H 6= R and 9 q 0 such that H = qZ. Since H is
) with ( + 1)(N ; a) 2= H 8 N = ;1 0 1 2.
discrete, it follows that 9 a 2 L( +1
Fix such an a and set EP:= f( + 1)(N ; a) : N = ;1 0 1 2g then E R n H
and E is nite. Set m := j 2E m Qj , then m ? m and 9 K T R compact
such that m(K) > 0 m(K) = 0. 9 U S Y open and precompact, such that
K U and m(U) < m5(Kn ) where n is the Besicovitch covering constant for R2
(n 16, see W-Z]).
For each z = (x y) 2 K 9 an open rectangle R(z) with diameter less than
1 minfjj ; j 0 j : j j 0 2 E j 6= j 0 g such that z 2 R(z) U. 9 a nite set ; K
2
S
P
such that K V := z2; R(z) and z2; 1R(z) n. Evidently m(V ) < m5(Kn ) .
We claim that (at least) one of the rectangles R = R(z) (z 2 ;) has the property
that m(R) < m(5R) , else
X
X
m(V ) n1 m(R(z)) 51n m(R(z)) 51n m(K):
z 2;
z2;
It follows from the restriction onS the diameter of R that fQj R : j 2 E g is a
disjoint collection, whence, if S := j 2E Qj R, then
m(S) = m(R) < m(5R) :
Write R = I J where I (0 1) and J R are open intervals. Given 0 < p < 1
and an open interval L = (a ; r a+r), denote by Lp the subinterval (a ; pr a+pr).
Note that if x 2 Lp and jyj < (1;2p)jLj then x + y 2 L.
9 0 < p < 1 such that m(Ip Jp ) > m(2R) . By lemma 1.5, 9 k 1 and A Ip
such that
kqnk k < (1;2p)jIj m(A Jp) > m(3R)
INVARIANT MEASURES AND ASYMPTOTICS FOR SOME SKEW PRODUCTS
and
It follows that
whence
9
min
j
() (x) ; jj < (1;2p)jJ j 8 x 2 A:
j 2E qnk
nk
Rq
( ) (A Jp ) S
< m(A Jp ) = m(qnk R( ) (A Jp )) m(S) < m(5R) :
The contradiction shows the impossibility of H 6= R.
m(R)
3
x2 Locally finite invariant measures for tail relations of skew products
Let S be a nite set of s 2 elements and let = A S Nbe a mixing
topological Markov shift and let T : ! be the shift. A cylinder set in is a
set of form
a1 : : : an] = fx 2 : xk = ak 8 1 k ng
where a1 : : : an 2 S. An admissible word (of length n) is an element (e1 : : : en ) 2
S n (or word) satisfying Aj j+1 = 1 8 1 j n;1. Note that a cylinder a1 : : : an]
is nonempty i its corresponding word (a1 : : : an) is admissible. We denote the
collection of admissible words of length n, or paths of length n ; 1 (the number of
steps) by Wn.
Consider T 's tail relation
T = T(T ) := f(x y) 2 2 : 9 n 0 T n x = T n yg:
Consider the reverse lexicographic order on T(T)-equivalence classes:
x y i xn0 < yn0 xn = yn for any n > n0:
It is easy to see that for any xed x y there are nitely many z such that
x z y so the type of ordering in each equivalence class is either Z or Z+
or Z;: Let 1 , ;1 be the set of maximal and minimal elements, respectively.
Introduce functions Pmax Pmin : S ! S
Note that
Pmax(a) = maxfi such that Aia = 1 g
Pmin(a) = minfi such that Aia = 1:g
x 2 1 =) xn;1 = Pmax(xn) for all n:
It follows that there are at most s maximal points, (similarly, at most s minimal
points) and all of them are periodic.
The adic transformation : n 1 ! n ;1 assigns to each x the smallest
y strictly greater than x: Specically, given x 2 n 1 9 ` 1 such that
xj = Pmax(xj +1) 8 1 j ` ; 1 and x` < Pmax(x`+1 )
(x) = (y1 y2 : : :) where the yk 's are dened reverse-inductively:
8
>
< xk k ` + 1
yk = > minfi 2 S : i > x` Aix`+1 = 1g k = `
: Pmin(yk+1 ) 1 k ` ; 1:
10
J. AARONSON, H. NAKADA, O. SARIG, R. SOLOMYAK
S
S
It is convenient to restrict to 0 := n j 0 j ;1 n j 0 j 1 :
Remarks
1) It is possible to visualize as the space of innite paths in the directed graph
; with vertex set S N and edges connecting (b n) to (c n + 1) i Abc = 1.
2) If ! = S Nis the full shift, and V is the adding machine, then is the
induced transformation V0 in the sense that (x) = V F (x) (x) (x 2 0) where
F (x) := minfn 1 : V n (x) 2 0 g.
3) Adic transformations were introduced in V1] (see also V2] and V3]) in the
more general setting of non-stationary Markov chains.
Let G be a locally compact, Abelian, Polish topological group. For f : !
consider the skew product transformation Tf : G ! G dened by
Tf (x y) := (Tx y + f(x)).
Now Tf 's tail relation is
T(Tf ) := f((x y) (x0 y0 )) 2 ( G )2 : 9 n 0 Tfn(x y) = Tfn (x0 y0 )g
= f((x y) (x0 y0 )) 2 ( G )2 : (x x0) 2 T(T) y0 ; y = f (x x0)g
where the symmetric (or tail) cocycle f : T ! G is dened by
G,
f (x x0) :=
Consider f : 0 G
1
X
(f(T n x) ; f(T n x0 )):
n=0
! 0 G dened by
f (x y) := (x y + f (x))
P (f(T i x) ; f(T i (x))): It is easy to see that the
where f (x) = f (x x) = 1
0
orbits of f are exactly the equivalence classes of T(Tf ) \ (0 0).
In this section we identify the f -invariant locally nite measures for certain
f : ! G which we now proceed to describe.
There is a norm kk = kkG generating the topology of G which is Lipschitz in
the sense that 8 2 Gb 9 M such that j(x) ; 1j M kxk 8 x 2 G . It is not hard to
show that 8 continuous homomorphism : G ! R 9 M j(x)j M kxk 8 x 2 G .
For f : ! G and k 1, let
vk (f) := sup fkf(x) ; f(y)k : x y 2 xj = yj 8 1 j kg:
The collection of Holder continuous functions on is
HG := ff : ! G : 9 0 < 1 vk (f) = O(n) as n ! 1g
and the collection of functions on with summable variations is denoted
FG := ff : ! G : X vk (f) < 1g
k=1
and a function f : ! G is said to have nite memory if 9 N 1 such that
f(x) = f(x1 : : : xN ) (equivalently vN (f) = 0). If : G ! R is a continuous
homomorphism, then HG HRand FG FR.
1
INVARIANT MEASURES AND ASYMPTOTICS FOR SOME SKEW PRODUCTS
11
A measurable function f : ! G is called periodic if 9 2 Gb z 2 S1 and
g : ! S1 measurable, not constant, such that f = zgg T . In case f 2 HG,
necessarily g 2 HS 1 . The function f is called aperiodic if it is not periodic.
2.1 Theorem
Suppose that A is topologically mixing, and that f 2 HG is aperiodic. For every
continuous homomorphism : G ! R:
1) there is a unique (e;( f ) )-conformal probability 2 P (0 )
2) is non-atomic
3) f is ergodic with respect to the Maharam measure on 0 G dened by
dm (x y) = e; y d(x)dy.
Theorem 2.1 is essentially known (although we indicate the proof). Our main result
in this section is
2.2 Theorem
Suppose that f : ! G is aperiodic and has nite memory.
If m is an ergodic, f -invariant locally nite measure on 0 G , then m = m
for some continuous homomorphism : G ! R.
Remark
The collection of all locally nite, -invariant measures on 0 G is identied
by theorems 2.1 and 2.2 as the collection of mixtures of Maharam measures. This is
because by the ergodic decomposition (see e.g. A1]), any locally nite, -invariant
measure is a mixture of ergodic ones.
Conditions for aperiodicity based on Kow] were given in x3 of A-D1]. We'll say
that the mixing almost onto if 8 a b 2 S 9 n 1 a = s0 a1 : : : sn = b 2 S
such that Tsk ] \ T sk+1 ] 6= (0 k n ; 1):
2.3 Proposition
Suppose that is mixing and almost onto, and that : ! G satises (x) =
b 2 S 1 such that : In
(x0), then either is aperiodic, or 9 2 G
particular, if Group(() ; ()) = G , then is aperiodic.
Some of the proofs use the theory of non-singular equivalence relations and we
provide some background.
Let (X B) be the standard Borel space. An equivalence relation R X X is
called standard, if R is a Borel subset of X X, that is R is in the product -eld
B B. For any x 2 X R(x) := fy : (x y) 2 Rg is the equivalence class of x, and
for a subset A X R(A) = fR(x) : x 2 A)g is called the saturation of X. The
standard equivalence relation R is called countable if R(x) is countable for any x.
For a countable, standard relation R, A 2 B =) R(A) 2 B. If G is a countable
group of automorphisms of X then RG = f(x g(x)) : x 2 X g 2 Gg is a countable,
standard equivalence relation, and conversely, any countable standard relation R
is generated in this way by a countable group of automorphisms (see theorem
1 in F-M]). A -nite measure is called non-singular for R if (R(A)) = 0
whenever (A) = 0 it is called ergodic if, in addition, either (R(A)) = 0 or
(X n R(A)) = 0 for every A 2 B.
By a holonomy we mean a Borel automorphism : A ! (A) (some A 2 B)
graph ;() := f(x (x)) : x 2 Ag R. A - nite, absolutely continuous measure
12
J. AARONSON, H. NAKADA, O. SARIG, R. SOLOMYAK
is invariant for R if (A) = (A) for any holonomy . By corollary 1 in FM],
is invariant under R i is invariant for the action of any G with RG = R.
The following proposition appears in P-S].
2.4 Proposition
Suppose that A is topologically mixing, and f 2 FR. There is a unique (e; f )conformal probability f 2 P (0 ).
There exists M > 1 such that
Pm;1
()
(x1 : : : xn]) = M 1e;Pm+ k=0 f (T k x) 8 x 2 n 1
R
where P = maxfhp (T) + fdp : p 2 P () p T ;1 = pg.
The property () is known as the Gibbs property. A T -invariant probability with
the Gibbs property is known as a Gibbs measure.
As is shown in Bo] and R1]:
9 a unique probability f 2 P () such that ddf fT = e;f for some > 0
T is exact (whence is ergodic) with respect to 9 a T -invariant probability pf f such that k log ddpff k1 < 1
and
9 M > 1 such that
P
pf (x1 : : : xn]) f (x1 : : : xn]) = M 1 e;Pm+ mk=0;1 f (T k x) 8 x 2 n 1
where P is the topological pressure of f given by the variational principle
P := maxfhp (T) +
Z
fdp : p 2 P () p T
;1
= pg = hpf (T) +
Z
fdpf :
The probability pf is known as the equilibrium measure of f (being the unique
maximizing T-invariant probability) and is a Gibbs measure.
Proof of proposition 2.4
Existence
We claim that f is (e; f )-conformal. To establish this, we show that if
a = a1 : : : an] b = b1 : : : bn] are both nonempty with an = bn, and : a ! b is
dened by (a x) := (b x) then
d (a x) = e;f ((ax)(bx)):
d
Recalling that va : Tan] ! a is dened by va (x) := (a x) we have that va;1 = T n :
a ! Tan] whence
;1 f T k (ax)
d va (x) = df T (a x) ;1 = ePnk=0
d
f
d
and, since = vb va;1 ,
d (a x) = d vb (T n(a x)) d T n (a x)
d
d
d
n
d
v
d
T
b
= d (b x) d (a x)
= e;f ((ax)(bx)):
INVARIANT MEASURES AND ASYMPTOTICS FOR SOME SKEW PRODUCTS
13
Uniqueness
Suppose that 2 P (0 ) is (e; f )-conformal. It follows that if a = a1 : : : an],
b = b1 : : : bn] are both nonempty with an = bn, and : a ! b is dened by
(a x) := (b x) then
d (a x) = e;f ((ax)(bx))
d
whence 9 M > 1 Kn (s) > 0 (n 1 s 2 S) such that
Pn;1
(x1 : : : xn]) = M 1Kn (xn)e k=0 f (T k x) 8 n 1 x 2 0 :
But
Pn;1 k
e k=0 f (T x) = M 1ePn pf (x1 : : : xn])
and so
(x1 : : : xn]) = M 2Kn (xn)ePn pf (x1 : : : xn]):
It follows that
(T ;ns]) = M 2Kn (s)ePn pf (s])
P
whence s2S Kn (s) e;Pn , (x1 : : : xn]) M 0 pf (x1 : : : xn]) and f .
Writing F := ddf , we see from dd = dd ff that F = F mod f , whence
by ergodicity F 1 and = f .
Proof of theorem 2.1
Let : G ! R be a continuous homomorphism. By proposition 2.4, there is a
unique (e;( f ) )-conformal probability 2 P (0 ) equivalent to the equilibrium
measure p( f ) .
It is shown in G] (see also A-D2]) that if f 2 HG is aperiodic then Tf is
exact with respect to m = p mG where p is some equilibrium measure on . In
particular, Tf is exact with respect to m p( f ) mG, whence f is ergodic
with respect to m .
Now let f : ! G be measurable. If 9 a globally supported, -nite Tf nonsingular measure m on G such that
( G B( G ) m Tf ) is exact, then f is aperiodic.
To see this, suppose otherwise, that 9 2 Gb z 2 S1 and g : ! S1 Holder
continuous, not constant , such that f = zgg T . Consider G 2 L1 ( G )
dened by G(x y) := g(x)(y), then G is not m-a.e. constant and GTf = zG: Thus
Tf is not weakly mixing and hence not exact (in particular, G is Tf;n B-measurable
8 n 0).
2.5 Proposition
Let f 2 FG. Any f -invariant, ergodic locally nite measure m on G
with Hm = G is a Maharam measure, and the existence of such implies that f is
aperiodic.
Proof
Let m be a f -invariant, ergodic locally nite measure on G with Hm = G .
By the Basic Lemma, m has the form dm(x y) = e(y) d(x)dy where : G ! R
is a continuous homomorphism and is (e f )-conformal. By proposition 2.4,
is equivalent to a Gibbs measure. Such a measure is globally supported on ,
whence m is globally supported and so as shown above, f is aperiodic.
14
J. AARONSON, H. NAKADA, O. SARIG, R. SOLOMYAK
By possibly changing the state space, we may assume that f(x) = g(x1 x2) in
the assumptions of theorem 2.2. The proofQof theorem 2.2 uses lemma 2.6 below.
For u : ! S1 and ` 1, set u` (x) := `j;=01 u(T j x).
2.6 Lemma
Assume u : ! S1 is Holder continuous, then either:
(1) 9 z 2 S1 g : ! S1 Holder continuous, such that u = zgg T or
(2) 9 > 0 `0 1 such that 8 ` `0 x 2 9 y 2 satisfying
x1 = y1 T ` y = T ` x
and
ju`(y) ; u`(x)j :
Proof
Let be the measure of maximalPentropy on and let P : L1 () ! L1 () be
the transfer operator, then Pf(x) = a2S 1Ta (x)e;h(T ) f(a x) and P1 = 1. Dene
Pu : C() ! C() by Pu(h) := P(uh), then Pun h = P n(unh) and (see G-H]) either
9 z 2 S1 g : ! S1 Holder continuous such that Pu(g) = zg (which implies (1)),
or kPunhk1 ! 0 8 h 2 C().
If (2) fails, then 8 > 0 9 x(k) 2 `k 1 (k 1) satisfying `k " 1 and such
that if
y 2 k 1 x(1k) = y1 and T `k x(k) = T `k y
then
ju`k (x(k)) ; u`k (y)j < :
By possibly passing to a subsequence, we can ensure that x(1k) = s 8 k 1. It
follows that
kPu`k 1s]k1 jPu`k 1s](T `k x(k))X
j
u`k (y)1s] (y)j
= e;`k h(T ) j
y2 T `k y=T `k x(k)
X
e;`k h(T )
(1 ; ju`k (y) ; u`k (x(k))j)1s] (y)
y2 T `k y=T `k x(k)
`
(1 ; )P k 1s]
! (1 ; )(s])
If u : W2 () ! S1 u(x) = u(x1 x2) and a 2 Wn+1 is a path a Q
= (a1 : : : an+1)
of length n, then un is constant on a. We denote un(a) := un ja = ni=1 u(ai ai+1 ):
In lemma 2.6, when u(x) = u(x1 x2), (2) has the combinatorial form:
(2') 9 `0 such that 8 ` `0 , paths a = (a1 : : : a`+1 ) 2 W` 9 a path
b = (b1 : : : b`+1 ) 2 W` such that a1 = b1 a`+1 = b`+1 and u` (a) 6= u` (b).
Proof of theorem 2.2
By the Basic Lemma and proposition 2.4, it suces to show that Hm = G .
Suppose otherwise that H 6= G , then 9 2 Gb 6= 1 such that jH 1.
Since m is -invariant, it is also T(Tf )-invariant and if : A ! (A) (A 2
B( G ) is a T(Tf f )-holonomy, then m((A)) = m(A).
INVARIANT MEASURES AND ASYMPTOTICS FOR SOME SKEW PRODUCTS
15
Using aperiodicity and lemma 2.6, we x ` 1 so large that 8 paths a =
(a1 : : : a`+1 ) 2 P` 9 a path b = ba = (b1 : : : b`+1) 2 P` such that a1 = b1 a`+1 =
b`+1 and f` (a) 6= f` (b), equivalently f` (a) ; f` (b) 2= H.
SetPJ := ff` (a) ; f` (ba ) : a 2 P`g, then J G n H and J is nite. Set
m := j 2J m Qj , then m ? m and 9 K G compact such that m(K) >
0 m(K) = 0.
Set M = jW`j. Approximating K by larger precompact open sets, we see that
9 U G open , U compact such that K U and m(U) < m2(MK) .
For each z = (x y) 2 K 9 a set W (z) = C(z) V (z) of form cylinder open
such
z 2 W(z) U. By compactness of K 9 z1 : : : zN such that K V :=
SN that
W(z
). We claim that V is a disjoint union of sets of form cylinder open.
k
k=1
To seeS this, let L beSthe maximum
length of the cylinders C(z1 ) : : : C(zN ), then
S
V = Nk=1 W(zk ) = Nk=1 c2WL cC (zk ) c V (zk ) { a disjoint union. Thus K V
and m(V ) < m2(MV ) .
It follows that 9 a set C W of form cylinder open such that m(C W) > 0
and m(C W) < Sm(C2MW ) , otherwise V would not have these properties.
Since C W = a2W` (C a) W, 9 a 2 W` such that m((C a) W ) m(CMW ) .
Next, 9 b = (b1 : : : b`+1) 2 W` such that a1 = b1 a`+1 = b`+1 and f` (a) ;
f` (b) 2 J.
Dene : (C a) W ! C G by ((C a x) y) := ((C b x) y + f` (b) ; f` (a)).
Evidently is a T(Tf )-holonomy and so by assumption, m(((C a) W )) =
m((C a) W) m(CMW ) .
On the other hand, ((C a) W )) Qf` (b);f` (a) C W whence
m(C W )
m(C W ) m((C a) W) m(Q
f` (b);f` (a) C W) m(C W) < 2M
M
and 21 > 1. This contradiction establishes theorem 2.2.
Remark
The proof of theorem 2.2 establishes the (stronger) statement:
Suppose that f : ! G is aperiodic and has nite memory.
If m is an ergodic, T(Tf )-invariant locally nite measure on G , then m = m
for some continuous homomorphism : G ! R.
We conclude this section with an application of theorem 2.2 to the \MarkovPascal-adic" transformations considered in P-S].
Let = A be a mixing subshift of nite type and let f : ! G . Recall from
P-S], the equivalence relations:
SA+ A A dened by
1
SA+ = f(x y) 2 A A : 9 n 1 x1
n = yn (y1 : : : yn) a permutation of (x1 : : : xn)g
and SAf A A dened by
1
SAf := f(x y) 2 A A : 9 n 1 x1
n = yn fn (x) = fn(y)g:
Evidently SA+ = SAF # where F # : ! ZS is dened by F # (x1 x2 : : :)i :=
ix1 (i 2 S).
16
J. AARONSON, H. NAKADA, O. SARIG, R. SOLOMYAK
Suppose that G is discrete. Evidently if f : ! G then
(x y) 2 SAf () ((x 0) (y 0)) 2 T(Tf )
whence
(x y) 2 SAf \ 20 () 9 n 2 Z (y 0) = nf (x 0)
and SAf \ 20 is generated by the induced transformation ( f )0 f0g .
We claim (as in P-S]) that if f has nite memory and : G ! R is a homomorphism, then is SAf -invariant, ergodic.
To see this, recall from theorem 2.1, that m is f -invariant, ergodic whence
m j0f0g is ( f )0 f0g-invariant, ergodic whence our claim (since m (Af0g) =
(A)).
2.7 Corollary
Suppose that f : ! Zd (d 1) is aperiodic and has nite memory.
If 2 P () is SAf -invariant and ergodic, then = for some homomorphism
: Zd ! R.
Proof
We'll deduce this from theorem 2.2. To do this, we show rst that ( n 0) = 0.
We claim that all SAf -equivalence classes are innite (this implies that is
nonatomic, whence ( n 0 ) = 0 as this set is countable).
To see this we'll need the symmetrization F of f dened on the mixing SFT
by F (x y) = f(x) ; f(y) (F : ! Zd). Evidently F has nite memory.
We claim that F is aperiodic. If not, then
)
e2iq(f (x);f (y)) = z g(gTxTy
(xy) (x y 2 )
for some q 2 Z q 6= 0 z 2 S1 g : ! S1 and then
N N y)
8 N 1 x y 2 :
e2iq(fN (x);fN (y)) = z N g(Tg(xT
xy)
Choosing N 1 and periodic points y = T N y y0 = T N +1 y0 , we have for all x 2 0 ,
N +1 )
e2iqfN (Tx) = e2iqfN (y) z N g(Tg(Txyxy
)
0 ) N +1 g(T N +1 xy0 )
2
iqf
(
x
)
2
iqf
(
y
N
+1
N
+1
e
= e
z
g(xy0 )
)
whence (!) e2iqf (x) = Z GG((Tx
x) contradicting the aperiodicity of f.
Let be the measure of maximal entropy on and let
P : L1 ( ) ! L1 ( ) be the transfer operator. By the local limit theorem of
G-H], 9 c > 0 such that 8 cylinders a b ,
n d2 P n(1(ab)\Fn =0] )(x y) ! c(a)(b) uniformly on as n ! 1:
Now x x 2 and N 1, then 9 nN such that
n d2 P n(1(a]b])\Fn =0] )(T n x T nx) 2c (a])(b]) 8 a b 2 WN n nN
whence
jfy 2 X : (x y) 2 SAf gj jfy 2 X : T nN y = T nN x FnN (x y) = 0gj
jWN j ! 1
as N ! 1 and establishing our claim.
INVARIANT MEASURES AND ASYMPTOTICS FOR SOME SKEW PRODUCTS
17
As mentioned above, ( n 0) = 0 and the probability on 0 f0g dened
by (A f0g) = (A) is ( f )f0g-invariant and ergodic. Dene the measure m
on 0 Zd by
m(A) :=
Z 'X
;1
0 k=0
1A kf d:
The measure m is evidently locally nite. By Kac's formula, it is f -invariant, and
by Kakutani's tower theorem it is f -ergodic (see e.g. A1]). Thus, by theorem
2.2, m = m for some homomorphism : Zd ! R. It follows that = .
2.8 Corollary
Suppose is a mixing, almost onto SFT.
If 2 P () is SA+ -invariant and ergodic, then = for some homomorphism
: Zd ! R.
Proof
As mentioned above, SA+ = SAF # where F # : ! ZS is dened by F # (x)i :=
ix1 (i 2 S). Since evidently Group(F # () ; F # ()) = ZS, F # is aperiodic by
proposition 2.3. The result follows from corollary 2.7.
Remark
Theorems 2.9 and 2.11 in P-S] both follow from corollary 2.8. In both cases,
S = f0 1g d = 1 and is almost onto.
x3 A logarithmic ergodic theorem
As in x2, let S = f0 1 : : : s ; 1g where s 2 N and let A : S S ! f0 1g
be an irreducible and aperiodic matrix and let A S Nbe the corresponding
(topologically mixing) subshift of nite type and let T : ! be the left shift.
In this section, we consider the asymptotic properties of f , where f : ! Rd
an aperiodic Holder continuous function, with respect to Maharam measures. It will
be convenient to use the supremum norm on Rd k(x1 : : : xd )k := max1kd jxk j.
Fix some 2 Rd and consider the Maharam measure m : B( Rd) ! 0 1]
;y
(f )
dened by dm(x y) = e d(x)dy where = is the (e )- conformal
measure.
As mentioned above, the aperiodicity of f implies that Tf is exact with respect
to m . It follows that f is ergodic with respect to m (generating the the tail
relation for Tf ) and also conservative (being invertible, ergodic and preserving a
non-atomic measure).
We prove the
Logarithmic ergodic theorem
P
log n;1 F k
(z)
k=0
log n
f
;!
h (T) m -a.e. as n ! 1
htop (T) 8F 2 L1(m)+ where is the equilibrium measure of f .
18
J. AARONSON, H. NAKADA, O. SARIG, R. SOLOMYAK
It will sometimes be convenient to denote
( )
Sn (F) = Sn f (F) :=
nX
;1
k=0
F kf :
The proof the logarithmic ergodic theorem is based on the following two reductions:
Firstly, it is sucient to establish (z) for a single F0 2 L1 (m )+ since then, by
the ratio ergodic theorem, SSnn((FF0)) ! ssXX FFdm
0 dm a.e., whence log Sn (F ) log Sn (F0)
a.e..
Secondly, in order to establish (z) for F0 2 L1 (m )+ , it is sucient to nd:
(random) subsequences Mk Nk " 1 such that log Mk log Mk+1 log Nk logNk+1 as k ! 1
sets A B 2 B( Rd) with m(A) m(B) > 0 and
subsequences Mk : A ! N Nk : B ! N such that Mk Nk " 1 log Mk logMk+1 log Nk log Nk+1 as k ! 1
satisfying
SMk (F0) h (T) on A
lim sup loglog
(z)
Mk
htop (T)
k!1
and
logSNk (F0 ) h (T) on B:
(z)
lim
inf
k!1
log N
h (T)
k
top
To see this, note that 8 n large 9 k = kn 1 such that Mk n Mk+1 , whence
log SlogMkM+1k(F0) and it follows from logMk log Mk+1 that
log SMk (F0 )
Sn (F0 ) lim sup
lim supn!1 loglog
k!1 log M
n
k .
SNk (F0 )
log
S
(
F
)
n
0
Similarly lim infn!1 log n lim infk!1 loglog
Nk :
log
S
(
F
)
Sn (F0 ) are T-invariant,
n
0
The functions lim supn!1 log n and lim infn!1 loglog
n
whence so are the sets
Sn (F0 ) h (T ) ] B := lim inf log Sn (F0 ) h (T ) ]:
A := lim sup loglog
n
htop (T )
htop (T )
n!1 log n
log Sn (F0 )
log n
n!1
By ergodicity, both sets (containing sets of positive measure by (z) and (z)) are of
full measure and (z) is established for F0 .
In the Main Lemma (below), we'll establish (z) and (z) for F0 = 1BM (0) and
A = B = BM 0 (0) (for some M M 0 > 0 where BM (0) := fy 2 Rd : kyk M g)
using the local limit theorem of G-H] and large deviation techniques.
The subsequences Mk Nk are related to counting functions.
We dene the counting functions %n : A ! N by
%n(x) := min fN 1 : f( k x)n1 : 0 k N ; 1g = Wn g
where Wn denotes the collection of admissible words of length n (as in x2). The
reader may easily verify that in case is a full shift, %n sn = jWnj and consequently k 7! ( k x)n1 denes a bijection f0 1 : : : sn ; 1g $ Wn 8 x 2 . In
other words, generates T-equivalence classes eciently. For a mixing topological Markov shift, as shown by the counting proposition below, the situation is
analogous.
INVARIANT MEASURES AND ASYMPTOTICS FOR SOME SKEW PRODUCTS
19
3.1 Counting Proposition Suppose that A is a mixing topological Markov shift,
and that L 1 is such that AL > 0, then for x 2 0 :
jWnj %n(x) < 3jWn+Lj
Proof. The left hand inequality follows directly from the denition of %n(x). To
see the right side, assume by way of contradiction that %n(x) 3jWn+Lj, then
there is a word a 2 Wn+L and 0 k1 < k2 < k3 %n(x) ; 1 such that kj x 2 a]
for k = 1 2 3. Set kj x = (a z (j )), then z (1) z (2) z (3) . For every " 2 Wn
choose some point of the form x(") = (" w0L;1 z (2)) where w0L;1 is some word
which makes x(") admissible. Clearly, k1 x x(") k3 x. Thus Wn is spanned
by j x for 0 j k3 in contradiction to the minimality of %n(x). The right hand
inequality is thus proved.
Set := exp htop () and assume without loss of generality that L > 2. For every
x 2 0 and n large enough set
un (x) := minfu > n + L : xu;1 < Pmax(xu )g u0n := un ; L
`n (x) := maxf` < n + L : x`;1 < Pmax(x` )g `0n := `n ; L
where
Pmax is as in x2. By possibly adding a constant to f, we may assume that
R
fd = 0 (note that neither f nor change when a constant is added to f).
Set
n := (n + L) ; `n n := un ; (n + L):
3.2 Lemma 9 M0 2 R+ such that
n lim sup n M a.e. :
lim sup logn
0
log n
n!1
n!1
Proof We prove this only for n, the proof for n being essentially the same. Set
P := Ptop ( f) = 0.Recall that 1 consists
of at most s points, all of which are
periodic. Set = x(1) x(2) : : : x(r) and let p be the least common multiple
1
of the periods of x(i) , then r s and for every x 2 1 , T p x = x. By the denition
of n, if n(x) > b then
b (xn+b+L) : : : Pmax(xn+b+L) xn+b+L ]
T n+L x 2 Pmax
b (xn+b+L ) : : : P b;s(xn+b+L )) is made of a repeating
For b > s the word (Pmax
max
period, hence is the prex of a maximal point. Applying this argument to bn :=
bM0 log nc, using the invariance of and the structure of 1, we have
n > bn ] r
h
X
i=1
x(0i) : : : x(bin);s
i
Since is a Gibbs measure and since for every i T p x(i) = x(i)
h
i b
(i)
(i)
x(0i) : : : x(bin);s = O efbn (x );bn P = O e pn fp (x );bnP
whence
(1)
n > M0 log n] = O
X
r
i=1
fp (x(i) )
nM0 ( p ;P )
!
:
20
J. AARONSON, H. NAKADA, O. SARIG, R. SOLOMYAK
(i)
It follows from the unicity of the equilibrium measure that fp (px ) < P . Thus,
the exponents in (1) are all negative and for M0 large enough,
1
X
n=1
n > M0 log n] < 1:
The result follows.
The next lemma is the main lemma, being the version of (z) and (z)) that we
prove. Let
1
X
B := 2Lkf k + vk ( f)
3.3 Main Lemma If 9M > 2B , then
k=1
(2) lim sup n1 log Su0n ;1 (1BM (0)) h (T ) m - a.e. on BM=2 (0)
n!1
inf n1 log S`0n ;1(1BM (0) ) h (T ) m - a.e. on BM=2 (0)
(3) lim
n!1
The rest of this section is devoted to the proof of the main lemma. Set
UN (x M) := f" 2 WN : 8y 2 "] kfN (y) ; fN (x)k < M g
VN (x M) := y 2 0 : 8z 2 y0N ;1 kfN (z) ; fN (x)k < M
=
"]:
"2Un (xM )
3.4 Lemma
For each M > 2B 9M1 M2 > 0 such that for all (x t) 2 0 BM=2 (0) and n
large enough,
jU`0n (x M2)j and
u0n ;1
X
j =0
`0n ;1
X
j =0
1BM (0) jf (x t)
1BM (0) jf (x t) jUun (x M1)j
Proof
P
j
N ;1 1
Fix some x 2 0 and t 2 Rd. We estimate AN := j =0
BM (0) f (x t)
for N = u0n `0n . By the counting proposition 8j = 0 : : : %N ; 1 , T N +L 0j x =
; P
T N +L (x). Thus jk;=01 f 0k x = fN +L (x) ; fN +L 0j x , whence
n
o
AN = ] 0 j %N ; 1 : fN +L 0j x ; fN +L (x) ; t M
j N +L;1 is 1-1, so A =
1
Since for j < %N (0j x)1
N
N +L = xN +L , the map j 7! (0 x)0
jBN j where
N +L;1 j j
BN = 0 x 0
: fN +L 0 x ; fN +L (x) ; t M 0 j < %N :
INVARIANT MEASURES AND ASYMPTOTICS FOR SOME SKEW PRODUCTS
21
We now prove the required inequalities. Setting N = u0n in the above inequality
we have 8(x t) 2 0 BM=2 (0)
jf(0jx)u0 n;1 : fun 0j x ; fun (x) ; t M 0 j < %u0n gj
jf" 2 Wun : 8y 2 "] kfun (y) ; fun (x)k 32 M + Bgj
Au0n =
and the upper inequality follows with M1 := B + 3M=2.
Using the same argument for N = `0n one shows that for all (x t) 2 0 BM=2 (0)
and n large enough so that `0n is well dened,
A`0n `n ;1
h j `n ;1 i M
0j x
0
0
0
;
B
0
j
<
%
:
8
y
2
(
x)
f
(y)
;
f
(x)
<
`n
`n
`n :
0 0
2
0
`0n ;1
j
Since
0 x
0
: 0 j %`0n ; 1 = W`0n ,
j M AN " 2 W`0n : f`0n 0 x ; f`0n (x) < 2 ; B and this is the lower inequality for M2 := M2 ; B.
The following lemma provides, together with lemma 3.4, the upper estimation
(2) in the Main Lemma.
1 log jUn(x M)j h (T ) m a.e.
3.5 Lemma 8M > 0 nlim
!1 n
Proof Since isn;the
Gibbs measure for f, there exists some constant K such
that for all y 2 "0 1 ,
K ;1 efn (y);nP (f ) "n0 ;1 Kefn (y);nP (f ) :
By the denition of Un , for every "n0 ;1 2 Un (x t) and y 2 "n0 ;1
n;1 fn (y);nP (f ) fn (x);nP (f )
whence
"0
;
e
e
t))
jUn (x t)j efn(U(x)n;(x
nP (f )
nP (f );fn (x) . Recall that according to our assumptions,
Thus,
R jUn(x M)j = O e
fd = 0, so P ( f) = h (T). The lemma follows since by the ergodicity
of , for almost all x 2 0 , fn (x) = o (n).
We now turn to the lower estimation (3) in the Main Lemma.
For every N 2 N and > 0 set
n
o
EN () := y 2 0 : y0N ;1 > e;N (h (T );)
By the denition of UN (x M), 8M > 0 x 2 0 and N > 0,
(4)
jUN (x M)j eN (h (T );) (VN (x M)) ; (EN ())]
22
We prove that
J. AARONSON, H. NAKADA, O. SARIG, R. SOLOMYAK
1 log (En ()) < 0
nlim
!1 n
lim 1 log (Vn (x M)) = 0
n!1 n
-a.e.
-a.e.
Since for almost all x 2 0 , `n (x) n, (3) will follow from this, (4) and lemma 3.4
1 log (En ()) < 0 -a.e.
3.6 Lemma nlim
!1 n
Proof is a Gibbs measure, so 9K such that 8n 8 y
yn;1 < Kefn (y);nP (f )
whence
n
0
o
En () " y 2 : Kefn (y);nP (f ) > en;nh (T ) :
Since ( f) = 0, P ( f) = h (T) so for n large enough,
En () " fy 2 : fn (y) > n=2g :
We will prove that
lim 1 log fy 2 : fn (y) > n=2g < 0:
n!1 n
using large deviations theory (see El]) for the -distributions of fn .
Using the Holder continuity of f and the Gibbs property of , it is not dicult
to prove that the following limit exists for p 2 R (see Bo]):
;
lim 1 log E epfn = P ( f + p f) ; P ( f) =: c(p)
n!1 n
where P() denotes topological pressure and E denotes expectation with respect
to .
By standard large deviations theory (see e.g. theorem II.6.1 of El]):
lim sup n1 log fy 2 : fn (y) n=2g ; pinf=2 I (p)
n!1
where I() is the Legendre-Fenchel transform dened by I (p) := supq fpq ; c (q)g.
We outline the (standard) proof that inf p I(p) > 0.
2
By theorem 5.26 in R1], c (p) is C 2 in R (see also G-H]). By aperiodicity, f
is not cohomologous to a constant and therefore (see G-H])
c0 (p) = p ( f) and c00 (p) > 0
where p is the equilibrium measure of (1 + p) f . It follows that I (p) = q0p ; c (q0)
where q0 is the maximum point for q 7! qp ; c(q) satisfying
0 = pq0 ; c (q + 0)]0 = p ; c0 (q0) = p ; q0 ( f)
whence
I (p) = q0q0 ( f) ; P (1 + q0) f] + P ( f)
By the variational principle,
P (1 + q0 ) f] = hq0 (T ) + q0 ( f + q0 f) :
Thus,
;
I (p) = P ( f) ; hq0 (T ) + q0 ( f) > 0
INVARIANT MEASURES AND ASYMPTOTICS FOR SOME SKEW PRODUCTS
23
for p 6= 0, because then q0 6= (q0 (f) = c0(q0) = p 6= 0 = (f)). Since I is
nite and convex (being the the Legendre-Fenchel transform of the convex function
c), it is continuous, whence inf p=2 I (p) > 0.
3.7 Lemma There exists M3 > 0 such that 8 > 0, for -a.e. x 2 0 9 N1 2 N
such that 8n > N1 9 n0 < n " 2 Wn0 satisfying
kfn0 (y) ; fn (x)k < M3 8y 2 "] :
Proof Fix some 0 > 0 (to be determined later). By the Ergodic Theorem, for
-almost all x 2 kfn (x) k = o (n) so there exists N1 = N1 (x 0) such that
8n> N1 kk
f1n (x)k < 0n. Since f is aperiodic and (f) = 0, the -distributions
of f T k=1 satisfy a local limit theorem (see G-H]). Thus, 9k0 2 N and c > 0
such that 8 (!1 : : : !d) 2 f+1 ;1gd k k0
8i 3B < !i(fk )i < 4B] pck
where (fk )i denotes the i-th coordinate of that vector. In particular, for every
! = (!1 : : : !d) 2 f+1 ;1gd , there exists u(!) 2 Wk0 such that
(5)
8z 2 u(!)] 8i 2B < !ifk0 (z)i < 5B
It follows that for every c 2 WL such that u(!)c 2 W and 8z 2 u(!)c] and 8i
B < !i fk0 +L (z)i < 6B
We use u(!) to construct ". Fix some n > N1 and 1 i d. We begin by
constructing words "i 2 Wn0i such that j"i j < 0 n and such that for N = j"i j and
all z 2 "i ]
(6)
jfN (z)j j < 7B for j 6= i
(7)
jfN (z)j ; fn(x)j j < 7B for j = i
We construct by induction sign vectors !k = (!1k : : : !dk ) and words ck 2 WL such
that for all k vk := (u(!1 ) c1 u(!2 ) : : : ck;1 u(!k )) is admissible and such that
(6) holds for all z 2 vk ] with N = Nk := jv k j. Choose !1 arbitrarily. Assume v k has
been chosen and choose some z 2 uk ]. Dene !k as follows: if jfNk (z)i ; fn (x)i j <
7B stop and set "i := v k else set for j = i !jk+1 := sgn(fn (x)j ; fNk (z)j ), and for
j 6= i, !jk+1 := ;sgnfN (z)j . Now set v k+1 := (v k ck+1 u(!k+1 )) where ck+1 2 WL
is some word which makes v k+1 admissible. Since at each step we get nearer to
fn (x)i in steps bounded from below by B, this procedure will stop after less than
kfn(x)k=B 0n=B steps. It can be easily veried that "i satises (6) and (7) for
N = j"i j. Now consider
" := ("1 c1 "2 : : : cd;1 "d )
where cj 2 WL make the above word admissible. The length of " is at most
Ld + 0 dn=B so by choosing 0 small enough and n large enough (i.e. N1 large
enough) we can make this length smaller than n as required. Also, it follows from
the construction of "i that for all z 2 "],
kfj"j(z) ; fn(x)k < 8Bd
The lemma is thus proved for M3 := 8Bd.
24
J. AARONSON, H. NAKADA, O. SARIG, R. SOLOMYAK
3.8 Lemma 9 c > 0 N2 2 N such that 8n > N2
n;1 y 2 : 8z 2 y0
kfn (z)k < 2B
ncd2 :
Proof The probability in question is bounded from below by kfn k < B], and
this in turn is bounded below by by the local limit theorem.
3.9 Lemma There exists M4 > 2B such that for almost all x 2 0
lim n1 log (Vn (x M4)) = 0 -a.e.
n!1
Proof Fix some arbitrary > 0. Fix n > max fN1 (N2 + L) =(1 ; )g where N1
and N2 are given by lemma 3.7 and lemma 3.8, then 9M1 M2 such that for almost
all x 2 0 and all t 2 R 9" = " (x) 2 Wn0 such that n0 < n and
8z 2 "] kfn0 (z) ; fn (x)k < M3
0
0
y : 8z 2 y0n;(L+n );1] kfn;(L+n0 ) (z) k < 2B
> e;(n;(L+n ))
0
Set W := y : 8z 2 y0n;(L+n );1]
Vn0 :=
n
kfn;(L+n0) (z) k < M2
. Consider the set
" c y0n;(n +L);1 ] : y 2 W and c 2 WL
0
o
One checks that Vn0 " Vn (x M4) where M4 = M3 + 3B. We estimate the measure
of Vn0 . Since is a Gibbs measure, there exist a constant K1 > 1 such that
K2 such that
a] b] a b] 6= ) a b] > Knh1;1 a]
b]i and a constant
o
n
;(n0 +L);1
;N
0
8a 2 WN a] > K2 . Set W := y0
: y 2 W , then
(Vn ) > K1;1 K2;(n +L)
0
X
a]2W 0
a] K1;1K2;(n +L) (W)
0
Thus, (Vn ) > K1;1 K2;L K2;n e;n : Since the above is true for all n such that
n > N1 (N2 + L) =(1 ; ),
lim n1 log Vn (x M4) ; (1 + log K2 ) :
n!1
Since > 0 is arbitrary, the lemma is proved.
As mentioned above, lemma 3.6, lemma 3.9 imply via (4) that 9M > 2B such
that
lim
inf n1 log jUn (x M)j h (T) a.e.
n!1
whence (using lemma 3.4) we have (3). This proves the Main Lemma, and the
logarithmic ergodic theorem.
INVARIANT MEASURES AND ASYMPTOTICS FOR SOME SKEW PRODUCTS
25
x4 Bounded rational ergodicity
Recall from A2] that a conservative, ergodic, measure preserving transformation
(X B m T) is called boundedly rationally ergodic if there is a set A 2 B 0 <
m(A) < 1 such that 9M > 0 such that for all n 1,
(?)
n;1
X 1A T k k=0
L1 (A)
M
Z nX
;1
A k=0
1A T k
!
dm:
R P
;1
The rate of growth of the sequence an = m(1A)2 A nk=0
1A T k dm does not depend
on the set A 2 B 0 < m(A) < 1 satisfying (?). This sequence is known as the
return sequence of T and denoted an (T) (see A1]). In this section we prove the
following theorem:
Theorem 4.1
Let be a topologically mixing subshift of nite type, let be the measure of
maximal entropy for and let f 2 HRd be aperiodic, then f is boundedly rationally
ergodic with respect to mRd and
n :
(log n) d2
To prove theorem 4.1, we show that for A = BM (0) M large, 9 0 < c <
C < 1 such that
cn Z S (1 )dm kS (1 )k 1 Cn :
n A L (A)
(log n) d2 A n A 0
(log n) d2
As before, these estimations are rst carried out along counting function sequences
using the local limit theorem. We begin with the upper estimation.
Since f is invariant under addition of constants to f, we can and do assume
that E (f) = 0.
an ( f ) Lemma 4.2
8 M > 0 9 A(M) > 0 such that
kfn() ; bk M] A(M)n;d=2 8 b 2 Rd n 2 N:
Proof Set F := kyk M] " Rd and x some a = a(M) > 0 such that
d sin ay 2
Y
i = ^(y)
1F (y1 : : : yd ) 2
ayi
i=1
Q
where ^ is the Fourier transform of (t) := 2( 2a2 )d=2 1ktk2a] (t) di=1 (1 ; j 2tai j),
then,
kfn ; bk M] = E (1F (fn ; b))
E (^ (fnZ; b))
eibtE (e;itfn )(t)dt
= (2)1d=2
ktk2a]
Z
(2)1d=2 ktk2a] E (e;itfn ) (t)dt =: An(M)
Note that the last term, An (M), is independent of the choice of b.
26
J. AARONSON, H. NAKADA, O. SARIG, R. SOLOMYAK
As shown in G-H], there exists " > 0 and : k k < "] ! C such that
(t) = 1 ; ct2 + o(ktk2 ) as t ! 0 and that for some 0 < < 1 and K > 0,
(
n
n
;itfn
E (e
) = (t)n + O( ) ktk "
O( ) ktk 2 " 2a]:
Making " smaller if necessary, we assume that for all ktk ",
j(t)j 1 ; 21 ct2 e;ct2
Using the above to estimate An(M), we have
An (M)
/
Z
ktk2a]
Z
;itfn E (e ) (t)dt
j(t)jn(t)dt + 2Kn
n Z
( p ) ( p )d + 2Kn
= d=2 2
n
n k k"pn] n Z
nd=2 2 k k"pn] e;c 2 ( pn )d + 2Kn
Z
2(0)
nd=2 Rd e;c 2 d
= 2
ktk"]
The lemma follows from
P this.
Set B := Lkf k1 + k>0 vk (f) where L, as usual is some number such that all
the entries of AL are positive. Fix some M > 4B, set
A := 0 ktk M]
and '(x t) := 1A .
Lemma 4.3 There is some C1 > 0 such that for almost all (x t),
n
jSn(x)(1A)(x t)j C1 nd=2
Proof Let s be the number of states of and set
un(x) := inf fu n + (L + s + 2) : xu;1 < Pmax(xu )g
`n(x) := supf` n + L : x`;1 < Pmax(x` )g:
For almost all x 2 these are nite. For such x we have the following representation:
un;`n ;1 (xu ;1) : : : Pmax(xu ;1) xu ;1 x1 )
x = (x`0n ;1 Pmax
n
n
n
un
Dene kn (x) 2 N by the equation
un ;1(xu ;1 ) : : : Pmax(xu ;1 ) xu ;1 x1 )
kn (x) (x) = (Pmax
n
n
n
un
Let b > xun ;1 be the minimal state such that tbun = 1, then
un ;1(b) : : : P (b) b x1 )
kn (x)+1 (x) = (Pmin
min
un
INVARIANT MEASURES AND ASYMPTOTICS FOR SOME SKEW PRODUCTS
27
We estimate Sn 1A by breaking it into three members
Sn (x) (1A )(x t) = Skn (x) (1A )(x t) + Sn (x);kn (x) (1A )( kfn (x) (x t))
Skn(x)(1A)(x t) + Sn( kn (x)+1 x)(1A)( kfn(x)+1(x t)) + 1
=: I + II + 1:
The inequality follows from the minimality of %n (x) as f( j x)n0 ;1 : 0 j kn(x) + 1 + %n ( kn (x)+1 x)g = Wn.
To estimate I, we begin by noting that the map j 7! ( j x)`0n ;1 is 1-1 for 0 j kn ; 1. To see this note that for such j, x j x kn x in the reverse lexicographic
order whence
k 1
un;`n ;1 (xu ;1) : : : Pmax(xu ;1) xu ;1 x1 )
x1
`n = ( n x)`n = (Pmax
n
n
n
un
j
Thus the dierence between the x's must be reected in the rst `n coordinates.
Since `n n + L,
Skn (1A )(x t) = jf0 j kn ; 1 : kt + (f )j (x)k M gj
= jf0 j kn ; 1 : kfn+L ( j x) ; fn+L (x) ; tk M g
jf" 2 Wn+L : 8y 2 "] kfn+L(y) ; fn+L(x) ; tk M + Bgj
Since , being the measure of maximal entropy, is the Gibbs measure for the zero
potential, there is some constant K such that for every a 2 Wn , K ;1n < a] Kn . In particular, cylinders of the same length are of comparable sizes whence
jSkn(x)(1A)(x t)j Kn+Lkfn() ; fn(x) ; tk M + 2B]
Lemma 4.2 now implies that I = O(n n;d=2 ) uniformly on A.
We now estimate II. Set (x0 t0) := kfn (x)+1 (x t).
We have to estimate Sn (x0 ) (1A )(x0 t0). We do this by showing that
(8)
%n(x0 ) kn(x0 )
thus reducing the problem to that which was discussed in the previous step.
There exists n + L + 1 < u0n < un (x0) such that Pmin(xu0n ) < Pmax(xu0n ) since
otherwise, there would be an admissible word a1 : : : ar ] for some r s + 1 with
a1 = ar and Pmax(aj ) = Pmin(aj ) (1 j r). This contradicts the aperiodicity
of A.
Now consider
u0n (x0 0 ) : : : P (x0 0 ) (x0)un ;1 x1 )
x0 = (Pmin
min un
un
un
u0n
0n 0
u
y := (Pmax(xu0n ) : : : Pmax(x0u0n ) (x0)uu0nn ;1 x1
un )
0
k
(
x
)
0
u
1
n (xu ) : : : Pmax(xu ) x )
n x = (Pmax
n
n
un
0
Since un > n + L + 1, for every " 2 Wn there is some w0L;1 such that
x(") :=
(" w0L;1 yn1+L ) is admissible and since u0n < un, x0 x(") kn (x0 )+1 x0 . This
shows that Wn is spanned by ( j (x0))n0 ;1 for j = 1 : : : kn(x0) ; 1, whence (8).
This completes the upper estimation, and we now address the lower estimation.
Lemma 4.4 There exists n0 such that for all x, 90 i1 < i2 n;%1 n+n0 (x) ; 1 such
j
that for every i1 j i2 , ( j x)1
n+L is the same, and f( x)0 : i1 j i2 g =
Wn.
28
J. AARONSON, H. NAKADA, O. SARIG, R. SOLOMYAK
Proof Let L be large enough such that AL > 0 and set n0 := L + n1 where
jWn1 j 3. Choose three dierent aj 2 Wn1 . There are
0 k1 k2 k3 %n+n0 ; 1
(
j
)
(
j
)
n
+
L
k
j
such that z := T ( x) 2 aj ]. In particular, z are dierent. Without loss
of generality, z (1) z (2) z (3) . For every " 2 Wn, construct an admissible word
of the form x(") = (" w0L;1 z (2)). Let x; and x+ be the minimal and maximal
points among the x("). Clearly, k1 x x; x+ k3 x whence 90 i1 < i2 %n+n0 (x) ; 1 such that x; = i1 x and x+ = i2 x. It follows that Wn is spanned
j
; 1
+ 1
(2) j 1
by x for j = i1 : : : i2. Since (x )n+L = (x )n+L = z , ( x)n+L is constant
for j = i1 : : : i2.
Lemma 4.5 There exists
Z C2 > 0 such that for n large enough,
n
Sn (x) (1A )(x t)dm(x t) C2 nd=2
A
Proof It is enough to prove that for some C3 and all ktk B,
Z
n
Sn (x) (')(x t)d(x) C3 nd=2
(the lemma will then follow by integration dt over ktk B]).
By lemma 4.4 for some n0 , for every x 2 and n 2 N there are 0 i1 <
i2 %n+n0 (x) ; 1 such that ( j x)1
n+L is constant for j = i1 : : : i2 and such that
Wn = f( j x)n0 ;1 : j = i1 : : : i2g. It follows that
Sn+n0 (x) (1A )(x t) i2
X
j =i1
= i1
(1A jf )(x t)
jf j i2 : kfn+L( j x) ; fn+L(x) ; tk < M gj
= jf( j x)n0 +L;1 : j 2 i1 i2 ] kfn+L( j x) ; fn+L (x) ; tk < M gj
jf" 2 Wn : 9y 2 "] kfn(y) ; fn(x)k M ; 4Bgj
K ;1nkFn(x )k M ; 4B]
where F : ! Rd is the symmetrization of f (as
in the proof of corollary 2.7)
P
n
;1
given by F(x y) = f(x) ; f(y), and Fn(x y) := i=0
F (T i x T ix). Integrating
with respect to d(x) we have for all ktk < B,
Z
Sn+n0 (x)(1A )(x t) K ;1 n ( )kFnk M ; 4B]:
As in the proof of corollary 2.7, ( T T) is a subshift of nite type, F :
! Rd is Holder continuous, and F is aperiodic. Thus, by the local limit
theorem of G-H] Fn satisfy a local limit theorem:
1 :
( )kFnk M ; 4B j] / nd=
2
Proof of theorem 4.1 We prove that for M > 4B, A := ft : ktk < M g
satises that
kSN 1Ak1 = O(EmSN (1A)) (N ! 1)
By the counting proposition, uniformly in x, %n(x) jWnj n. Therefore, there
exists c 2 N such that for all x 2 0 and n, n;c+1 %n (x) n+c . Fix N > 1+c
INVARIANT MEASURES AND ASYMPTOTICS FOR SOME SKEW PRODUCTS
29
and choose the n such that n N < n+1 . The last estimations imply that for
every x 2 0 ,
%n;c(x) N < %n+c(x)
whence, by the preceding lemmas, for almost all every (x t) 2 A and N large
enough,
n+c
SN (1A )(x t) C1 d=2
(n + c)
Z
2n;c
SN (1A )dm (nC;
c)d=2
A
The theorem follows from this.
References
A1] J. Aaronson, An introduction to innite ergodic theory, Mathematical surveys
and monographs 50, American Mathematical Society, Providence, R.I, U.S., 1997.
A2] J. Aaronson, Rational ergodicity, bounded rational ergodicity and some continuous measures on the circle, Israel J. Maths., 33, (1979), 181-197.
A-D1] J. Aaronson and M. Denker, Local limit theorems for Gibbs-Markov maps,
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(Jon Aaronson) School of Mathematical Sciences, Tel Aviv University, Ramat Aviv,
69978 Tel Aviv, Israel
E-mail address : [email protected]
(Hitoshi Nakada) Dept. Math., Keio University,Hiyoshi 3-14-1 Kohoku, Yokohama 223,
Japan
E-mail address : [email protected]
(Omri Sarig) School of Mathematical Sciences, Tel Aviv University, Ramat Aviv,
69978 Tel Aviv, Israel
E-mail address : [email protected]
(Rita Solomyak) Department of Mathematics, Box 354350, University of Washington,
Seattle, Washington 98195-4350, USA
E-mail address :
[email protected]