1
Decay of Correlations for Certain Quadratic Maps1
L.-S. Young
2
Abstract. We prove exponential decay of correlations for ( ), where belongs in a positive
measure set of quadratic maps of the interval and is its absolutely continuous invariant
measure. These results generalize to other interval maps.
f
f
Consider a dynamical system generated by a map f : M ! M preserving a probability
measure , and let ' : M ! R be observables. Mixing properties of the dynamical
system are reected in the decay of correlations between ' and f n as n ! 1. More
precisely, we say that (f ) has exponential decay of correlations for functions belonging
in a certain class X if there is a number < 1 s.t. for every ' 2 X , there is a constant
C = C (' ) s.t.
Z
' ( f n )d
;
Z
'd
Z
d
C n
8n 0 :
The main result of this paper is the following:
Theorem. Consider fa : ;1 1] de ned by fa (x) = 1 ; ax2 , a 2 0 2]. Then there is a
positive Lebesgue measure set in parameter space s.t. if f = fa for a 2 , then
(1) f has an absolutely continuous invariant measure (this is a well known theorem
rst proved by Yakobson J])
(2) (f ) has exponential decay of correlations for functions of bounded variation
(3) the central limit theorem holds for f' f n gn=1 2 ::: ' 2 BV .
1
2
The results in this paper are announced in the Tagungsbericht of Oberwolfach, June 1990.
The author is partially supported by NSF.
2
These results generalize to certain open sets of 1-parameter families of unimodal maps.
Exponential decay of correlations has been proved for primarily two types of dynamical
systems: piecewise uniformly expanding maps of the interval with their absolutely continuous invariant measures, and Axiom A dieomorphisms with their Gibbs states. (See e.g.
HK], Ry1], Ru1], Ru2]). These are by no means the only results. (See e.g. BS], Z].)
The main feature that is new here is that the maps we consider have singularities, and
that these singularities return arbitrarily close to themselves. We reduce the problem to
the expanding case by constructing a uniformly expanding map f~ that factors over f f~
acting on a set J that is the disjoint union of a countable number of intervals. We then
consider a function space on J with a norm that is a weighted combination of the L1,
L1 and total variation norms, and prove the existence of a gap in the spectrum of the
Perron-Frobenius operator associated with f~.
The maps for which our results hold are those studied by Benedicks and Carleson in
BC2]. They have a very simple description, and are non-uniformly expanding in a controlled way. We will recall in detail all of the relevant material from BC2] ; except the
proof of the theorem which says that these maps form a positive measure set in parameter
space.
This paper is organized as follows. Precise statements of our results are given in Section
1. Section 2 contains some background material for 1-dimensional maps. In Section 3 we
prove the existence of absolutely continuous invariant measures and their mixing properties.
Sections 4 and 5 contain the proof of the decay of correlation result.
The author thanks M. Rychlik for helpful conversations.
2
3
x1. Statements of results
We rst state our results for the quadratic family x 7! 1 ; ax2 before proceeding to discuss
generalizations to other 1-parameter families of interval maps.
Consider fa : ;1 1] dened by fa (x) = 1 ; ax2 , a 2 0 2]. Let > 0 be a very small
number that remains xed throughout, say = 1016 . For " > 0, let
" =
fa 2 2 ; " 2] : jf n0j e;n
a
and
Benedicks and Carleson proved that Leb(") > 0
to which our results apply.
j(f n)0(f 0)j (1:9)n 8n 0g :
a
a
8" > 0 BC2]. These are the parameters
Theorem 1 (Existence of invariant measures). 9 "1 > 0 s.t. 8 a 2 "1 , f = fa has
an invariant probability measure with a density that can be written as
= 1 + 2
where 1 has bounded variation and
0
2 (s) const.
1
X
j =1
;j
p(1js:9); f j 0j :
The existence of absolutely continuous invariant measures under similar conditions has
been proved many times. See e.g. J], BC1], CE] plus N], Ry2], BY] etc. The nature
of the density can be understood as follows. The dynamics of f consists essentially of an
expanding part and a contraction due to the quadratic singularity. The expanding part
gives rise to 1 , while the contraction together with its subsequent iterates account for the
inverse square-root singularities in 2 .
3
4
Theorem 2 (Properties of the absolutely continuous invariant measure).
9"2 > 0, possibly < "1 , s.t. 8a 2 "2 f = f
has the following properties:
a
(a) in Theorem 1 is the only absolutely continuous invariant measure
(b) (f ) is exact
(c) supp = f 2 0 f 0] and
inf (x) > 0.
2
x2f 0 f 0]
We remark that (f ) being exact in this setting is equivalent to (f n ) being ergodic
8n > 0. It is also equivalent to the natural extension of (f ) being isomorphic to a
Bernoulli shift. (See L].)
Theorem 3 (Decay of correlations). Let f be as in Theorem 2. Then 9 2 (0
s.t.
8' : ;1 1] ! R with bounded variation, there is a constant C = C (' )
Z
' ( f n )d
;
Z
'd
Z
d
C n
1)
s.t.
8n 0:
Exponential decay of correlations has been proved for various classes of interval maps.
The piecewise uniformly expanding case is treated by Hofbauer and Keller HK] and extended to allow for inntely many branches by Rychlik Ry2]. Ziemian Z] proved the same
result for a class of maps satisfying what is sometimes called \the Misiurewicz condition"
M]. Our aim is to relax this condition suciently so that our results hold for a positive
measure set of parameters in the quadratic and other families.
Our proof of Theorem 3 consists of constructing an extension of f : (;1 1] ) and
analyzing the spectral properties of the Perron-Frobenius operator associated with this
extended map. As noted in K], this knowledge of the spectrum gives us immediately
Theorem 4 (Central limit theorem). Let f be as in Theorem 2, and let ' : ;1 1] !
4
R be
5
R
a function with bounded variation and 'd = 0. Then
2Z n;1
!2 3 12
X
:= p1n 4
' fi d 5
i=0
is well de ned, and if > 0, then 8x 2 R
(
as n ! 1.
n;1
1 X
pn
i=0
'
)
fi < x
!p
1
2
Zx
;1
e
; u22
2
du
Theorem 4 is proved in a similar setting in an unpublished work of Collet C] using the
approach of BS]. C] also contains a slightly weaker version of Theorem 3.
We now mention some generalizations of our results. Let I be an interval, and let
f : I be a unimodal map satisfying the Misiurewicz condition. That is, f has negative
Schwarzian, it has no sinks, and if c is the critical point, then n>
inf jf n c ; cj > 0. Let End3 (I )
0
3
denote the space of C endomorphisms of I . Then there exist constants > 0 and > 1
(depending only on f ) s.t. for \most" smooth 1-parameter families : (;1 1) ! End3(I ),
if
" ( ) =
fa 2 (;" ")j
if
f = (a) then jf nc ; cj e; n
and j(f n )0 (fc)j n 8n 1g then Leb("()) > 0 8" > 0. (See TTY] for details.) Our results carry over directly to .
That is, Theorem 1 above holds (with dierent constants), and if f satises (P3) in x2,
then Theorems 2, 3 and 4 are valid also.
In the rest of this paper we will consider only the quadratic family x 7! 1 ; ax2 . This
avoids carrying around constants that depend only on f .
5
6
x2. Some properties of 1-dimensional maps
In this section we try to isolate those properties of f = fa that are relevant to our discussion.
These are essentially the only properties that will be used.
The specic formula of f is immaterial to us, but we will use the fact that it is a C 3
unimodal map with negative Schwarzian and nondegenerate critical point. The fact that
f is symmetric about 0 simplies our notation a little, but this is a totally unnecessary
assumption. What is important is that f has certain expanding properties, which we
summarize below as properties (P1) and (P2).
Let (; ) be a small neighborhood of the critical point. We distinguish between the
dynamics of f outside of (; ) and that of orbits beginning in (; ). (P1) concerns
the former:
(P1) 9M 2 Z+ and > 1 s.t.
(i) if x fx : : : f M ;1 x 2= (; ), then j(f M )0 xj M (ii) for any k 2 Z+, if x : : : f k;1 x 2= (; ) and f k x 2 (; ), then j(f k )0xj k .
(P1) holds for fa for all a suciently near 2. First notice that it is satised by f2 . One
way of seeing this is to conjugate f2 to its piecewise linear model. (For a more general
approach see M] and the appendix of CE].) The rest follows because (P1) is an open
condition.
For x 2 (; ) f 0 x can, of course, be arbitrarily small. (P2) guarantees a period during
which (f j )0 x recovers. Suppose = e;k0 . Let Ik = (e;(k+1) e;k ) for k k0 , and ;I;k
for k ;k0 .
(P2) There is a function p : (; ) ; f0g ! Z+, constant on Ik and increasing with jkj,
6
7
s.t.
(i) 12 k p(x) 4k for x 2 Ik (ii) j(f j )0(fx)j constj(f j )0(f 0)j const(1 9)j 8j < p(x)
(iii) j(f p )0xj p for some > 1 (independent of x).
(P2) holds for all fa with a 2 " " suciently small. This property is due to Benedicks
and Carleson, and the assertion in the last sentence is proved in BC1] and BC2]. Since
the main construction in our paper is based on this property, let us recall the ideas in their
proof:
For x 6= 0 2 (; ), let p0 (x) = maxfp 2 Z: jf j x ; f j 0j < e;10j 8j < pg. First we claim
that 9C0 (independent of ) s.t. 8y 2 fx f 0] and j < p(x),
1
C0
(f j )0 y
(f j )0 (f 0)
C0 :
This is true because the quotient in question is exp k=1(k) where
j
j(k)j =
f 0 (f k;1 y) ; f 0 (f k 0)
f 0 (f k 0)
;10k
ee;k
:
Using this distortion estimate, we see that
ax2 C1
0
(1 9)p0 ;1
jf p0 0 x]j 1
giving p0 (x) 4k for x 2 Ik provided k is suciently large. The lower bound for p0 (x) is
obtained from
ax2 4p0 ;1 e;10p0 :
Finally, at time p0 , we must have
ax2 j(f p0;1)0 yj e;10p0
7
8
for some y 2 fx f 0]. This gives
(f p0 )0 x
j
j 2ajxj 1 e;10p0
C0 ax2
p
2 40 ;10p0
C0 e
which is p0 for some > 1 provided p0 is reasonably large. For x 2 Ik , let
p(x) = yinf
2I p0 (y):
k
The number p(x) is called the \bound period" of x. In x4 we will use the notion of \total
bound period", which is dened as follows: For x 2 (; ) p^(x) is the smallest j > 0 s.t.
8i with 0 i < j , if f i x 2 (; ), then p(f i x) < j . It is easy to see that p^(x) 6k for
x 2 Ik , because if f i x 2 (; ) for some i < p(x), then
jf ixj jf i0j ; jf i 0 ; f i xj
jf i0 ; f ixj 91
19 1
1
2
i
;
1
ax C0 (1 9) x 4 so p(f i x) k.
(P1) and (P2) together are sucient for proving Theorem 1. For Theorems 2-4 we need
an additional topological mixing condition:
(P3) For every interval I ;1 1] 9n = n(I ) s.t. f n I f 2 0 f 0].
Lemma 2.1. (P3) holds for all fa with a 2 " " suciently small.
Proof. Let f = fa a 2 ". First we prove that 8I ;1 1] 9n0 = n0 (I ) s.t. f n0 I Ik0
or I;k0 . In light of (P1), some f j I must intersect (; ). If f j I is completely contained
in two adjacent Ik 's, keep iterating, and note using (P2) that jf p (f j I )j >> jf j I j p = p(x)
8
9
for x 2 f j I . After nitely many returns to (; ), there must exist j1 and k1 2 Z+
s.t. f j1 I Ik1 or I;k1 . Consider f j Ik1 j = 1 2 : : : and let j2 be the rst time (after
the bound period of x 2 Ik1 ) s.t. f j Ik1 some Ik . Since jf j2 Ik1 j >> jIk1 j f j2 Ik1
must contain some Ik2 or I;k2 with 0 < k2 < k1 . We then consider f j Ik2 and repeat the
argument until some f j Ikn Ik0 or I;k0 .
Next we argue that there is an n1 2 Z+ s.t. for all a suciently near 2, if x^a is the
xed point of fa in (0 1), then fan1 Ik0 3 x^a . This is obviously true for a = 2 and is an
open condition. Finally, observe that for f = any fa if I^ is an interval containing x^ , then
9n2 = n2 (I^) s.t. f n2 I^ f 2 0 f 0]. This completes the proof.
x3. Existence and properties of absolutely continuous invariant measures:
Proofs of Theorems 1 and 2
Our rst goal in this section is to prove Theorem 1 assuming (P1) and (P2). The proof is
straightforward: Extend p to p : ;1 1] ! Z+ by dening p(x) = 1 for x 2= (; ). Consider
the return map R : ;1 1] given by R(x) = f p(x)(x). Then some power of R is piecewise
uniformly expanding. Piecewise uniformly expanding maps have absolutely continuous
invariant measures (acim) because their Perron-Frobenius operators do not increase total
variation by very much. An acim for f is then constructed from that for R.
More formally, let be the coarsest partition of
constant. Dene g : ;1 1] ! R by
g(x) =
(
1
R0 (x)
0
;
1 1]
into intervals on which p is
if x 2 interior(B) some B 2 otherwise:
The Perron-Frobenius operator associated with R, written PR or simply P when there is
9
10
no ambiguity, is dened to be
P (')(x) :=
X
y2R;1 x
g(y)'(y):
Note that if n := _ R;1 _ _ R;n+1 , and gn(x) := g(Rn;1x) g(x), then
X
P n(')(x) =
y2R;n x
gn(y)'(y):
Lebesgue measure is denoted by m throughout this paper. L1 kk1 etc. in this section
refer to L1(;1 1] m). If I is an interval, let VI (') denote the total variation of ' on I .
If I = ;1 1], then we simply write V ('). Say ' 2 BV if V (') < 1. It is notationally
convenient to assume that each B 2 or n is closed, for this allows us to write
V (P n') X
B2
n
V ('gn) = V ('gn):
Lemma 3.1. (1) kgnk1 ! 0 as n ! 1.
(2) V gn < 1
8n.
Proof. (1) follows immediately from (P1) and (P2). We prove (2) with n = 1. Let
B = b1 b2 ] 2 . Since Sf < 0 gj(b1 b2) has exactly one minimum. So
VB g 2kg1B k1 +
4kg1 k1:
lim
x!b+
1
g(x) +
lim
x!b;2
g(x)
B
Recall that on fp = j g jR0 j j for some > 1. So
V g V;1 ;] g
+
V; 1]g
10
+
1
X
8
;j < 1:
j =1
11
It is easy to check inductively that V g < 1 ) V gn < 1 8n.
The idea that the Perron-Frobenius operator shrinks variation for piecewise uniformly
expanding maps is due to Lasota and Yorke LY]. Assuming the additional condition that
V g < 1, Rychlik Ry1] extended their result to the case where f may have an innite
number of uniformly expanding pieces.
Lemma 3.2. (following Ry1]). Let ' 2 BV . Then
sup V (P n ') <
n
1:
Proof. Choose N s.t. kgN k1 < 101 . Since the local oscillation of gN at each point is < 51
and V gN < 1, we can choose a nite partition 0 s.t.
max V g
B 2 0 B N
< 41 :
Then
V (P N ') V ('gN )
X
=
VB ('gN )
B 2 0
X0V ' kg k1 + k'1 k1 V
B 2
Using the fact that
k'1 k1 B
we get
V (P N ')
B
N
1
m(B)
B
Z
B
g :
B N]
' dm + VB '
Z
V
B gN
VB ' (kgN k1 + VB gN ) + m(B) B ' dm B 2 0
X
11
12
VB gN . Since kP'k k'k , it follows from this
which is < 21 V (') + C k'k1 with C = Bmax
1
1
0
2 m(B)
that
k
V (P kN ') 21 V (') + constk'k1
for all k.
P
;1 P n(') some C 8n, a subsequence
Proof of Theorem 1. Let ' 12 . Since V n1 ni=0
of these functions converges in L1 to some '1 2 BV . Clearly, '1 dm is an R-invariant
probability measure. Let Pf denote the Perron-Frobenius operator associated with f , and
let
1
'0 :=
X k;
Pf '1 1fp>kg :
k=0
P k' k mfp > kg < 1.
Then '0 dm is a nite f -invariant measure, nite because 1
k=1 1 1
(See (P2).) Normalizing '0 dm, we obtain our desired acim . To see that the density of
R
has the properties as claimed, let 1 = '1 = '0 dm. Since
;
constk' k
Pf '1 1fp>1g (s) p 1 1 js ; 1j
(P2) (ii) tells us that for k > 1,
;
Pfk '1 1fp>kg (s)
constk'1 k1 (1 9);(k;1)
pjs ; f k 0j
:
Next we prove Theorem 2 assuming (P1) - (P3). Note that (P1) and (P2) imply that f
has a positive Lyapunov exponent at ; a.e. x. First we recall a theorem of Ledrappier:
Theorem L]. Let h : ;1 1] be a C 1+" piecewise monotonic map with nonat critical
points, and let be an acim with a positive Lyapunov exponent a.e. Then a.e. ergodic
component of h is an acim. Moreover, if (hn ) is ergodic 8n > 0, then the natural
extension of (h ) is isomorphic to a Bernoulli shift.
12
13
Our next proposition probably has some independent interest. It is a corollary to the
proofs in L].
Proposition 3.3. Let h : ;1 1] be a C 1+" piecewise monotonic map with k critical
points, all of which nonat. Let be an acim with a positive Lyapunov exponent a.e. Then
(1) is the sum of at most k ergodic acim's
(2) each ergodic component of is supported on a nite number of intervals, on which
the density is > 0 a.e.
(3) if h is topologically mixing, then (h ) is measure-theoretically mixing, and hence
Bernoulli.
Proof of Proposition 3.3. Let J0 : : : Jk be the intervals of monotonicity of h. It is
) of (h ) as living on
shown in L] that if we view the natural extension (h
f
= (x a)
2 ;1 1] f0 : : : kg j 9x0 x1 : : : 2 ;1 1] with x0 = x
h(xi) = xi;1 and xi 2 Ja g
N
i
then local unstable manifolds of h are canonically identied with subintervals of ;1 1]
and acim's on ;1 1] with positive Lyapunov exponents correspond to measures on with
smooth coniditional measures on unstable manifolds.
Let i be an ergodic component of . It follows from the discussion above that at i ; a.e.
x, there is an interval Jxi containing x on which the density of i is > 0 a.e. Let J1i J2i : : :
be the maximal intervals on which the density of i is > 0 a.e. By the ergodicity of i ,
there can only be a nite number of these. Moreover, it is not possible for hjJji to be 1 ; 1
for all j , because h has a positive Lyapunov exponent and the Jji's cannot grow indenitely
in length. So at least one of them must get \folded", i.e. 9j = j (i) s.t. Jji contains a critical
point in its interior. This puts a bound on the number of ergodic components.
Assertion (3) follows from what we have said and the last statement in the theorem we
13
14
quoted from L].
Proof of Theorem 2. Since 1 2 BV , there is an interval I ;1 1] s.t. xinf2I
(x) > 0. By
virtue of (P3), we have inf2 (x) > 0. The rest follows immediately from Proposition
x2f 0 f 0]
3.3 and (P3).
x4. Decay of correlations: Main steps in the Proof of Theorem 3
Let f be as in Theorem 2.
Step I. Construction of an extension of f : (;1 1] ) .
Recall that there is a function p : ;1
? p(x) 1
?
?
?
1]
; f0g ! Z+ with the following properties:
8x 2= (; ),
p(x) 4k 8x 2 Ik ,
P (x) " as jxj # 0,
9 > 1 s.t. 8x =6 0 2 (; ) j(f p)0xj p.
1
2k
For reasons to become obvious shortly, we will choose with e; 4 < 1.
1
Our new space J is the disjoint union of a countable number of intervals J1 J2 : : : where J1 = ;1 1] and for k > 1 Jk has length 21 (k;1)mfp kg. Note immediately that
1
X
1
m(Jk) 1 +
X
k>1
n
o
12 (k;1) 2 min e; 4k < 1:
14
15
We now dene a map f~ : J . Let
J1; = ;1 ;] J10 = (; ) J1+ = 1]:
On J1; J1+, let f~ = f in the sense that f (J1) J1. The interval J10 is mapped anely
1
onto J2, with a magnication of 2 . Assume now that f~ has been dened on J1 : : : Jk;1.
We again write Jk = Jk; Jk0 Jk+ where Jk0 = f~k;1 fp > kg. The map f~ then takes
Jk0 anely onto Jk+1, magnifying it by 21 , and takes Jk into J1 in such a way that
8x 2 fp = kg J10 f~px = f px. Note that on Jk jf~0j 12 k .
Clearly, there is a projection : J ! ;1
1]
s.t. f~ = f .
Next observe that lifts to an f~-invariant probability measure ~ on J with density
1 ;
X
~ := Pfk~ 1 1fp>kg k=1
where 1 is as in Theorem 1. We mention a few relevant properties of ~. For a function
' : J ! R or C , let us use the notation 'k to denote 'jJk . Then ~k 2 BV 8k and
k
~k k1 ; 21 (k;1)k
~1 k1 . Since p > 1 on (; ), the support of ~ is contained in
S J . Henceforth let us change J to f 2 f]. Moreover, if I J is an
f 2 f ] k
1
1
k>1
interval, then 9n = n(I ) s.t. f~n I contains the xed point of f~ in J1+. (This follows from
the corresponding statement for f .) So the same argument as in the last section tells us
that xinf
2J ~1 (x) > 0.
1
Lemma 4.1. (f~
~)
is exact.
Proof. Let ~ be the partition of J into fJk 0g. We have H~ (~) < 1 because const e;2k k
~ (Jk0) const e; 4 8k , and ~ (Jk ) is either 0 or has similar bounds. Since f~ is essentially
expanding, ~ is clearly a generator. So if ;~ : (f ) ! (f~ ~ ) is the natural extension
15
16
of
~ ~ ),
(f
and B is the Borel -algebra on J , then
;~ ;1
f~;n B
n0
is contained in the
Pinsker -algebra of (f ) (Ro] 12.3). It therefore suces to prove the triviality of the
Pinsker -algebra of (f ).
We will show that (f ) is measure-theoretically isomorphic to (f ), the natural
extension of (f ). (See Theorem 2.) Let x~ = (x~0 x~ 1 : : : ) be a history of (f~ ~ ) and
dene (x~) = x = (x0 x1 : : : ) by letting xi = x~ i . Clearly () = , since () is
f-invariant and projects onto . So all we have to do is to verify that is 1 ; 1 a.e.
Let x = (x0 x1 : : : ) be a typical element of (f ). We say that xi is a \marker" for x
if xi 2 (; ) and @j > i s.t. p^(xj ) > j ; i p^ being the total bound period dened in x2.
If xi is a marker, and (x~ ) = x, then the only possibility for x~ i is that x~i = xi 2 J1. Hence
;1 fxg contains at most one point if x has innitely many markers.
Assume for simplicity that x0 2 (; ). From Theorem 1 we know that j ;e; 6 e; 6 <
;j ;j +
1, so by the Borel-Cantelli Lemma, 9k(x) 2 Z s.t. xj 2= ;e 6 e 6 8j > k(x). This
guarantees that for j > k(x) p^(xj ) < j . Suppose that x0 is not a marker. Let j1 > 0 be s.t.
xj1 2 (; ) and p(xj1 ) > j1 . Then either xj1 is a marker, or 9j2 > j1 s.t. xj2 2 (; )
and p(xj2 ) > j2 ; j1 . If this process continued, there would be a jn > k(x) s.t. p^(xjn ) > jn ,
contradicting our choice of k(x).
j
j
Step II. The Perron-Frobenius operator.
For the rest of this paper let m denote Lebesgue measure on J , and let k k1 denote the L1 and L1 norms wrt m on J . For ' : J ! C , we dene
V (') =
X
k
16
V ('k )
k k1
17
where V ('k ) is the total variation of ' on Jk . Consider the function space
X = f' : J ! C jV (') < 1 k'k1 < 1 k'k1 < 1g
with norm
k'k := V (') + k'k1 + "k'k1 where " > 0 is a small number to be determined later. (X
k k) is a Banach space.
As usual, the Perron-Frobenius operator P~ associated with f~ is dened to be
P~ (') =
X
y2f~;1 x
'(y)g~(y)
where
g~(y) = ~10
jf yj
except possibly at the end points of ~ = fJk 0g. For purposes of estimating variation it
is convenient to adopt the following convention. For B 2 ~n and a 2 @B, we consider a
as belonging in B if for some j 0 j < n f~j B some Jk and f~j a 2 @Jk otherwise
we say a 2= B. The advantage of this convention is that if we let g~ = 0 on @Jk and
g~n (x) = g~(f~n;1 x) g~(x), then we can estimate V (P~ n') by
V (P~ n') X
B 2~n
VB ('g~n )
(which is clear), and at the same time have
X
B 2~n
VB (g~n ) < 1:
(See Sublemma 5.1 and cf. Ry1].)
17
18
The next lemma contains the main estimates in this paper. Its proof is postponed to x5.
Lemma 4.2. P~ : X ! X is a bounded linear operator whose spectrum
(P~ ) = 0 f1 : : : n g
where 0 fjzj 0 g for some 0 < 1 and i 2 S 1 8i. Moreover, each i is a smiple pole,
and the corresponding projection has a nite dimensional range.
Step III. Finishing the proof.
First we use the exactness of (f~ ~ ) to prove that (P~ ) \ S 1 = f1g. Let ' 2 X be s.t.
P~ (') = ' for some 2 S 1. Since ~ > 0, we can write ' = ~ for some . Note that
2 L2( ~ ), for
1 (k;1)
2
k
'
k
1
kk k1 min ~k k'k1 min
~1 so that
Z
X
2 d ~ const k e; k4 k
which is < 1 by the upper bound we imposed on .
The rest of this argument is quite standard (see e.g. HK]). We let U = Uf~ be the
operator on L2( ~ ) dend by U () = f~. Then U = (in the sense of L2) because
Z
~
hU i = ( f~)
dm
=
Z
P~ (
~)dm =
P~ (
~) ~ =
h i
, which means that
for every 2 L2 ( ~ ). From this and jj = 1 we deduce that U = = nU n is measurable wrt f~;n B. Hence = some constant c a.e. by Lemma 4.1.
Thinking of P~ as an operator on L1(m), one sees immediately that = 1.
18
19
We have shown that ' = c
~ a.e. To see that ' = c
~ as elements of X , use Lemma
5.1. This together with Lemma 4.2 proves that P~ = P~0 + P~1 , where P~0P~1 = P~1 P~0 =
R
0 (P~0 ) fjz j 0 g, and P~1 (') = c' ~ for some c' 2 C . In fact, c' = 'dm, because
R P~ 'dm = R P~ n'dm large n R P~ n'dm = R 'dm.
1
1
Finally we return to the original dynamical system f
' 2 BV (;1 1]) lifts to '~ : J ! R with '~
~ 2 X :
V ('~ ~) X
V ('~k )k
^k k1 + V (
~k )k'~ k k1
k h
X
k
1
2V (') ; 2 (k;1)k
~
1
We have thus shown that 8' 2 BV (;1
Z
Z
;
: ( 1 1]
. Observe that
i
1 (k;1)
;
~
2
k1 + V (
) k'k1 < 1:
1
1]),
Z
; 'd d
Z
Z
Z
n
~
~
~ ~
~
~
~
~
=
( f )'d ; 'd d
Z
Z n
~
~
~
=
P ('~
~) ; '~
dm
~ dm
m(J ) kk1 C (' ) n
for some
(
f n)'d
)
0 < < 1:
.
This completes the proof.
x5. Spectral properties of the Perron-Frobenius operator
The purpose of this section is to prove Lemma 4.2. Recall that our norm consists of a
total variation part, an L1-part, and an L1-part. Roughly speaking, the variation part is
contracted by P~ the L1-part does no harm and the L1-part is needed to control what
happens on the Jk 's for k large.
19
20
>From here on we will be working exclusively with f~ : J . So for notational simplicity
let us drop all the 's in f~ ~ P~ etc. The following line of argument is also fairly standard:
Lemma 5.1. (1) P is a bounded operator on X (2) 9N 2 Z+ and R 0 s.t. 8' 2 X ,
kP N 'k 51 k'k + Rk'k1 :
Using Lemma 5.1, and remembering that kP'k1 k'k1 , one deduces immediately that
supkP n k < 1. This gives (P ) fjz j 1g.
n
1.
Lemma 5.2. For N suciently large, there is a nite rank operator Q with kP N ; Qk <
This second lemma tells us that outside of some disk of radius < 1 (P ) consists of at
most a nite number of points 1 : : : ` , and that the projection corresponding to each
i has a nite dimensional range (DS] VIII 8). No i 2 S 1 can be a pole of order > 1, for
that would violate supkP nk < 1. This completes the proof.
n
Sublemma 5.1a. Bsup
V (g ) < 1 8 N .
2 N B N
Proof. Since jf 0 xj 8x and f maps each element of to at most 3 elements of , we
have
X
B 2 N
B J1
VB (gN ) 3N 4;N :
(See the proof of Lemma 3.1.) Next consider B
20
2 N with B Jk k > 1.
If f j B
21
J 8j N , then V
i>1 i
g
B N
= 0.
X
B 2 n
B Jk
If not, then j(f N )0xj N 2 (k;1)8x 2 B, giving
1
VB (gN ) 3N 4;N ; 12 (k;1):
1 and ; N2
We now choose and x an N 2 Z+ with kgN k1 100
101 .
Sublemma 5.1b. For every " > 0 9R1 = R1(N ") s.t. 8' 2 X ,
V (P N ') 101 V (') + R1k'k1 + 101 "k'k1 :
Proof.
V (P N ') X
B 2 N
VB ('gN )
X kg k1 V
B
N
') +
B(
X
B
k'1 k1 V
B
g :
B( N)
The rst term is kgN k1 V ('), and the second term is estimated as follows. For K 2 Z+,
X
VB (gN ) k'1B k1
N
BS2
B kK Jk
1 Z
X
VB (gN ) mB B 'dm + VB (')
! X
max
V (g ) V (' ) +
B
B
;4kg k1 V (') + R1 K k'k1
B
N
B
N
21
X Z !
V (g )
max B N '
B mB
B
B
22
where R1 K is a constant depending on K . Choose K s.t.
X
VB (gN ) < 101 ":
BS2 N
B
Jk
k>K
Sublemma 5.1.c. 8 suciently small " > 0 9R2 = R2(N ") s.t.
"kP N 'k1 101 V (') + R2k'k1 + 101 "k'k1 :
Let us explain the idea of the lemma, assuming N = 1. A formal proof will follow. We
1
wanted to say that kP'k1 < k'k1 . Clearly, k(P') 1 Jk k1 ; 2 k'k1 , but for (P')1
k>1
P
1 (k;1)
;
k'k k1 for some C . Choose K0
all we can say is that k(P')1 k1 C k'1 k1 + 2 2
k>1
P ; 21 (k;1) is small. For 1 < k K , we can write k' k V (' )+ j R ' j=m(J ).
s.t. 2
0
k
const (V (') + k'k1 ) + ; 21 +
small
k>K0
Together this gives us
kP'k1 1
k
k
k
k'k1 which explains the "-weight in the L1 -part of the norm k k.
Proof of Sublemma 5.1c. Since
k
k ; 2 k'k1 101 k'k1 sup (P N ')k 1
k>N
N
we only have to consider (P N ')k for k = 1 : : : N . Let N0 be a large number to be specied
shortly. Dene
0
1
X
k
M1 (') = 2 @
; 2 A k'k1 k>N0
and
M2 (') = maxfk'1 k1 : : : k'N0 k1 g
M = maxfM1 M2 g:
22
23
One veries inductively that 8j k N k(P j ')k k1 Dj M where
D = a1 + 2N + 2
X ; 2k
:
For instance,
k
(P j +1 ')
k a
1 1
1
X ; 2k
(Dj M ) + 2N (Dj M ) + 2
M
j
the three terms being contributions from (P j ')1 2(P j ')i and i>j
(P j ')i respectively.
We have proved that
n
kP N 'k1 max 101 k'k1 DN M (')
o
:
If 101 k'k1 is bigger, we are done. So suppose DN M (') dominates. We choose N0 and "
s.t.
0
1
DN @2
X ; 2k A 1
10
and
k>N0
100"DN
1001
and consider the following possibilities:
Case 1: M = M1 . We have kP N 'k1 DN M1 101 k'k1 .
1 V (').
Case 2: M 100V ('). We have "kP N 'k1 100
Case 3: M = M2 and M2 > 100V ('). Let k N0 be s.t. k'k k1 = M . Since
M = k'k k1 mJ1 k'k k1 + V (')
k
it follows that
99
M
100
mJ1 k k' k1
k
23
24
and
k
k1 PN'
100
99
N0
2 DN
k'k1 :
Proof of Lemma 5.1. We x " acceptable with regard to 5.1c and take R = R1 + R2 + 1.
(Note that if we had dened k k = V () + k k1 , then P could be unbounded. Take
'(k) = mJ1 k 1Jk .)
Proof of Lemma 5.2. Let N 2 Z+ be as in Lemma 5.1, and let E = EN be the
conditional expectation wrt the -algebra generated by N on J . Dene
Q^ (') = P N (E')
and
; ;
Qj (') = P N E ' 1
kj Jk :
We claim that for suciently large j Q = Qj has the desired property.
Sublemma 5.2a. kP N ; Q^ k 101 .
Proof. Let = ' ; E', so that (P N ; Q^ )' = P N .
(i) V (P N ) kgN k1
B 2 N
VB () + Bmax
2N VB (gN ) Bk 1B k1 .
Since VB () = VB (') and k1B k1 VB (), we conclude that
5
V (P N ) 100
V ('):
(ii) kP N k1 N mB
B 2
V
1 V (').
) 100
B(
24
25
(iii) The same argument as in 5.1c (case 3 cannot occur) gives
"kP N k1 max
n1
"kk1 10
o
1
V (')
100
:
To complete the proof of Lemma 5.2, consider an arbitrary ' 2 X . Write
kP N ' ; Qj 'k
;
N
^
(P ; Q) '1
J k j k
+
;
P N '1
k>j Jk :
The rst term is 101 k'k by 5.2a, and the second term is
k'k
1
5
R'1
k>j Jk 1
+
by Lemma 5.1. Since
0
1
0X 1
X
'1
k>j Jk @ mJk A k'k1 1 @ mJkA k'k
" k>j
1
k>j
we have kP N ' ; Qj 'k 21 k'k for suciently large j .
Note added after completion of manuscript. The author has heard that G. Keller and
T. Nowicki recently obtained some related results.
25
26
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and
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