Annals of Mathematics
On Iterations of 1 - ax2 on (- 1,1)
Author(s): Michael Benedicks and Lennart Carleson
Source: Annals of Mathematics, Second Series, Vol. 122, No. 1 (Jul., 1985), pp. 1-25
Published by: Annals of Mathematics
Stable URL: http://www.jstor.org/stable/1971367 .
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122 (1985), 1-25
AnnalsofMathematics,
ax2 on (-1,1)
On iterationsof 1
By MICHAEL
BENEDICKS
AND LENNART CARLESON
Chapter I
1. Introduction
- < x < 1, where a is a
We study iterationsof F(x; a) = 1-ax2,
parameterin theinterval(0, 2). Let the Pth iterationof F be denotedby Fv(x; a)
and in particularset ~(ja) = F'(O; a). We shallconsidertwo relatedproblems.
cycle?It is known,see Section4 below,that
1) Does F have an attractive
thisis not the case iflim, pooI
dxFv(1;a) I = xo.
2) Suppose thatfora certainvalue of a thereare no attractivecycles.In
of { (^ }?= I
thiscase, the questionarisesas to whatcan be said ofthedistribution
invariantmeasureofthemap
and in particularofwhattypeis thecorresponding
x -- F(x; a).
The last problemwas treatedby Jakobson[3]. For the firstproblemand
relatedresultssee thebook [1] and the paper [2] by Collet-Eckmann.
In Part II of our paper we provethatfora set of a-values A,,,of positive
Lebesgue measure x -- F(x; a) has no attractivecycles,and in Part III it is
proved that foralmostall a E A,,, x -* F(x; a) has an absolutelycontinuous
invariantmeasure.We even prove that the densityfunctionof the invariant
measurebelongs to LP forp < 2, whichis best possiblein general.The proof
also showsthatfor2 - a0 smallenoughthe set of a E (a0, 2) forwhichF(x; a)
has an attractivecycleis open and dense in A0.
A firstdraftofthispaperwas readyin early1983. In thepresentversionwe
oftherefereeand J.Gheinerforwhich
have profited
verymuchfromsuggestions
we are verygrateful.
2. Notation
We shall study how ~(ja) = F'(O; a) returnsto a (fixed) shortinterval
I* = (- 8, 8), 8 = e-E, and in particularhow close theyare to the origin.We
) forp > 0
divide I* = Uiu >aIA so that 1 = (c,,c,1) where c, = exp{-}
< O.
and 1 = - Imp
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2
M. BENEDICKS AND L. CARLESON
We shallconsiderthreetypesof returnsto (-8, 8), called free,bound and
inessential and denoted by x, y and z respectively.If the free returnsare
x1,x2 ... ., each xi is followedby bound returnsYij, j = 1,... , Yi and these
could be followedby inessentialreturnszij, j = 1,... ,Zi. The sequence
i.e. xif(a)= e(a).
as iterations,
{ mi) 1consists of theindicesofthefreereturns
3. Some lemmas
about the behaviourof
For the proofwe need some initialinformation
calculationfromthe case
iterationsof F. Here thisis obtainedby a perturbation
in the followingsequenceoflemmas.
F(x; 2) = 1 - 2x2, whichis formulated
small 8 > 0. thereexistsaO < 2 such that if
LEMMA 1. For sufficiently
a E [a0, 2] and if qh)e [- 1, 1] and k are such that
j = 0,1,2, .. .,k
lFj(rjo; a) I 2 8,
1,
-
and
|Fk(
%;
a) I<
85
then
dxFk(no; a) I
(3.1)
(1.9)k
Proof:We makethe standardchangeof variablesx = p(O) = sin(7/2)0.
expressedin the new variables;i.e.,
Let F(0; a) denotethe transformation
F(0; a)
=
-arcsin(I -a
sin 2j@).
is
Its 0-derivative
'IT
Cos -
di(0;
2
a) =-2asgn(0)
Icos2
0 + (sin2
)(1-
-
Let Do= Fl'( 0; a) andassumethat
I 1-282,
mlq,
but qjl < 1
-
282
and writeFk
k-1i
(3.2)
,
=
=
p'k-j
0,1,2, ... ,j
0
-
1,
(p-9 o Fi. We have
-
- T -(Ni) H ( -2a^)
dXFk = w'(Ok) H daf(o.; a)
Since Ini-1i 2 3,
v
= 1,2,... ., j, it followsinductivelythat nal ?
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1-
a32,
ITERATIONS OF 1 -
?1
j,...,k and from(3.2) and 1i
to establish.
=
-
3
ax2
2832, V
...,j-
=
1, (3.1) is easy
2. There are constants y > 0 and K = K(8) such that when
close to 2, forall a E (a0, 2] and all IxI > ca+i, there
(Or 2) is sufficiently
ao0
<
K
that
1
such
exists = l(x, a)
i = 1, ... ., -1 1
(i) IF'(x; a) I 2 85
>
d
(ii) logI xF1x; a) I -y.
LEMMA
Proof:Let ok = F'(x;a). When I1 1 2 2 Id F(o0)I = 12aq201 a. We
+ 4e.
< e < 1 implies( < F(t) <-I
observethat =-1+,
(2-a)/a
We chose l so that2'l' 2 'io1 ? 2 -', 1> 22 and notethat
'(d?o;a)
I
22aq0I
?
12amIl
...
12am1_1I
2-l (2a)(1 -2
(2a)
* (2a)(1 -4
=
(a)' .I
-
2)
2
(1
...
The lemmais therefore
provedwithy =
2-21+2)
2-21+2)
(2a) -(I-
*
-2-21+3
log2 and K = 2 + [2log(1/8)].
2
small 8 > 0. any aO < 2 and any positive
3. For any sufficiently
integerN, thereis an integerml ? N and an intervalAOc (a0, 2) such that:
(i) For any a Ec A0, (,,a) = F"(O; a) < for 2 < < ml- 1.
(ii) a -* Ftm(O;a) is a one-to-onemap of AO onto 1* = (- , 8).
j = 12,.. ., ml -1.
(iii) IdxFi(1;a) I ? (1.9)h 1;
LEMMA
Proof. Since
(I+-(
=
1
-
=2-
at
_-
I
a + a(t, + I)ti-
-
it followsthatwhile (^ < - , ( 1 + (r } is exponentially
increasing.Furthermore
thatif (^ < 0 for2 < v < j thena
it followsinductively
l(a) is decreasing
j+
-
since
=t~
da
=_
2atj di<
i - ~~da
42
_ (j2
j
(i) and (ii) of the lemmaare therefore
easy to establishand since
j
|dxFj(l;
a) I = I I (- 2a(^)
v= 1
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4
M. BENEDICKS AND L. CARLESON
= 1 and - 1 <
where
4j
evident.
-2,
v=
2,...,j,
j
m - 1. (iii) is also
We next turn to an importantgeneral principle,which will be used
of Fk
the x-and a-derivatives
thispaper: Undersuitableassumptions
throughout
are comparable.
For the derivativesdx and da we have in our case the two recursion
formulas
a F0= 1
d FV+l =-2aFvdaFV,
and
1 =-
aa F
2aF'd
Fv_
aaF0 = 0.
(F")2
This gives
v-1
(3.3)
dXFP =
v=1,2,...,
H(-2aF'),
i=O
and
aaF =
a F2
daF =-x
(3.4)
U(1+
v=2,3...,
di\)'
2.
These formulasshow that dXF" and daF" grow at the same rate. The
followingversionof thisprinciplewillbe used in the sequel.
LEMMA
1. 1
-
2. ao
4. Thereexist8 > 0 and a0
282
<
<
q
a < 2
3. IaFi-L'(l; a) I
(3.5)
(3.5)
<
2 such that,if
<1
?
ej
/
forj = 8,9, ..., k, then
1
aFkl(,i;a)
aF
~~~16 a k-1(qj;a)
16 <
<
16.
and a simplenumericalcalculationthat
Proof:Firstit followsby continuity
? 1,
< ao < a < 2 and 1 -22<2
if
if 8 is sufficiently
small, 1-82/10
then
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ITERATIONS OF 1 - ax2
5
we verifythat
Furthermore
tI8 (1
ei2,3) 2
When theseestimatesare insertedin (3.4), then(3.5) readilyfollows.
Remark. Withcomputercalculationssimilarlemmascould be provedfora
are then
in smallintervalsAO in a moregeneralposition.Certainmodifications
would seem to be
investigation
also needed laterin theproof.Such a systematic
of interest.
Chapter 1I
cycles
ofattractive
4. The non-existence
Our firstmainresultis
1. Thereis a set A., c (0, 2) of positiveLebesguemeasure,such
- 1 - ax2 from [-1,1]
into [-1,1]
A.. the map F(; a): x
has no attractivecycles(stable periodicorbits).
THEOREM
that for all a
c
From CorollaryII.4.2 of Collet and Eckmann [1] it followsthat if for a
certaina the point x = 0 is not attractedto a stableperiodicorbitthen F(; a)
has no stable periodicorbits.
Thus Theorem1 followsfrom:
2. Thereis a set Ax,c (0, 2) ofpositiveLebesguemeasureand an
such
thatforall a E Art,
integerP0,
THEOREM
dxFv(l;a) I 2 e 2/3
v>20.
of the proofwe can replace e '23 by e 0
Remark. By a minormodification
forany a < 1. What one expectsto be trueis howeverthat IaxFv(1; a) I grows
on OO.
almosteverywhere
exponentially
This is indicatedby the followingargument.In the second partwe prove
the existenceof a "good" invariantmeasureIt. It followsthatforalmostall a
logidxFF(1 a) I vf(logl2axl)dt(x)
and
djL(x) ?
J(logl2axl)
0,
and one can therefore
expectexponentialincreasein general.However,we do
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6
M. BENEDICKS AND L. CARLESON
not have uniformexponentialincreaseforall a considered,and the introduction
of A O.
of a < 1 has allowedus to makeverysimplerulesforthe definition
The restof the firstpartof the paper is devotedto the proofof Theorem2.
.. \
A
The set A is constructedas f=oAk, A.
A,,,, where Ak is
definedat the kth inductionstep.
5. The partition
the conceptof a freereturnto l* and the
We wish to definerecursively
partitionassociatedwitha freereturn.
The indexof a freereturnis denotedby k.
By Lemma 3 there is an integer ml such that a
a) is a one-to-one
map of A0 onto 1*. Also ml is theindexof thefirstfreereturn.The corresponding partitionis
{fi- l(Ia+l)f-
-
Fml(O;
and A, = fI7'(I(a+1))
1(I_ (al))}
U fi7'(Ia+1).
consideration.
The set E1 = A0\ Al is excludedfromfurther
co
k ? 1. We intendto define
partition,
kth
of
the
interval
Let now be an
partition.
the (k + 1)Stfreereturnand the corresponding
=
-*
X
that
a
Fmk(O;a) maps
Suppose
[a,, a2] onto I,. For convenience
we assume that pu> 0. We wish to studythe furtherevolutionof co under
stoppingrule:
mappingsby F. The integerp = p(co) is definedby thefollowing
(5.1)
holds forall a E
(5.2)
|t(a)
-
o,andall 1,? <
P+(aPl
-FP+
Fj(i1;a)
< c
l(q; '' a)
<
Ij(a)
l0j2
, forj = 1,2,...,p but
>
(OPR+a1(a)
+ 1)2
10~~~(p
forsome a E X and some q,0 < X <cam. For certainj's, 1 ? j ? p,
{ Ykjki I 1 5mk~j(a)
may returnto 1*. Those are the boundreturns
Let i be the smallestintegeri ? p + 1, such that Fmk+i(O; a) n I* = Ji
0. Let Si be theset of v so that(i) I, c Jiand (ii) IPI < a + k. The setsof
E I, withIjl ? a + k + 1 arenowexcluded
andare
a E Ak forwhich{mk+i(a)
denoted Eki.
butgo on to
ofa partition
(a) If Si = 0 we do notmakeanyconstruction
=
i
of the first
index
is
the
+
mk
+
i
1.
Then
nki
considerthe nextinteger
(5.1)
bytherelations
defined
returns
inessential
return.
Now"bound"inessential
p.
=
=
return
At
the
follow.
may
where
and (5.2) with
IPOI min,5sIlvi
vo,
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ITERATIONS OF
1 - ax2
7
case (a) mayoccuragain. In thatcase this"free"inessentialreturnis
thereafter,
by "bound" inessentialreturnsand thena
nk2'whichin itsturnmaybe followed
"free" inessentialreturn
etc. The set of all inessentialreturnsfollowingXk
up to Xk+1 is denoted {Zki }Zk . These are thus all returnsbetween nk1 and
(nk3
mk+?1
(b) If Si / 0, let xiv be the subinterval
of X mapped to I, v e Si. Then
>
Let
K
be
one
of
the
1
or
2
intervalsof the set
a.
possibly0,
lvI
U Iv) u(-
FMk +i(0;c)\((
Ca+k+l1Ca+k+1))'
If K c IV,,forsome Ii,, whichis then adjacentto one of the selected I 's we
co so thatFmk+i(O; &v) = Iv + 1 U K. Observethatif
expandthe corresponding
v = + a thismeansthatwe go slightly
outside(- 8,3).
The construction
meansthat
(5.3)
C F Mk + i(0;
cV
CoiV) C
Iv U
IIl, 1
forsuitablev and suitablesign +. For a E wi, the nextfreereturnmk+l(a)
mk + i is defined.
=
We have now constructedxi>, v E S, in our partitionP and have obtained
0, 1 or 2 remainingintervalsof w. On each of thesewe repeatthe construction
by consideringi + 1,i + 2,. .. untilwe get an intersection
with 1* as above.
We continuethe construction
and get a partitionP(C; Mk) = { wiV
}. The set o
maybe expressedas the disjointunion
(Au) i) UR U(UEEi)
whereR consistsof thosea whichneverreturnto I* inside[
By Lemma 2 and Lemma 3, (iii),
tdxFv(l; a)I 2 ely,
a E R.
y > O,
ca+ 1' Ca +1
v > v0,
and hence Theorem2 holdsfora E 1R.
The set Ak+1 is nowdefined
as
Ak+1 = Ak\
U
WCAk
UEiv
i,
v
We maketwocomments.It followsfromSection12 thatthemeasureof R is
zero. From the presentpoint of view this is irrelevantsince it is obviousthat
attractivecycles do not exist for a E R. The second commentconcernsthe
partitionP. For the intervalsa, the associatedintervals c I* are precisely
intervalsIA except at the ends wherewe can have the situation(5.3). In what
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8
M. BENEDICKS AND L. CARLESON
in orderto avoid cumbersome
followswe are going to ignorethis possibility,
notation.
6. The maininductionstep
We assume that for all free returnsof order k and for all a E
followingare true:
(i) If mk = mk(a) is an indexof a freereturnof orderk > 1,
dxFmk(1(;
?
a)
A?k the
e2m k
(ii) If 4 < i ? mk(a), then
> ei3
| d~i-(l;a)
1< j
(iii) If.(a)I >
< mk.
Note firstthatLemma3 impliesthat(i), (ii) and (iii) hold fork = 1.
We shall now turnto the proofof the main inductionstep and intendto
prove(i), (ii) and (iii) forindex k + 1.
Let co be an intervalof Ak and let p = p(w), mk and yI be as definedin
Section5. We also assumethatthe inductionstep is provedin the case whereco
is mapped onto an intervalI. with IvI < It if such v exist.We firstuse the
inductionassumption(i) and (5.1) to give an upper bound forp in termsof I.
By the mean value theorem
(6.1)
-(a)-FP(;
a) = FP-1(1; a)
-
FP-'(1
= aq2dxFP- 1(1
where 0 < 71' < q
<
-
a2;
a)
ar'2; a),
-
C, 1.
From (3.3) (the productformulafordXF) and (5.1), it followsthat
2-1 < C-l < dXFv(1
(6.2)
v = 1, 2, ...
p
-
-
)
an2;
< C< 2
1, since C maybe chosenas
0
i} ?2
exp(7ji
(1
Xn
and the same estimate holds forall q', 0
<
a'
<
Hence it followsfrom(5.1) and (6.1) that
q.
1 ?e2V-.
|dxFj- (; a) I < e22
10j 2a
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1 - ax2
ITERATIONS OF
As long as j
9
1 < Mk, the inductionhypothesis
(ii) impliesthat
-
<
e2/3
e2.
By (5.1) and Lemma3,
j
(6.3)
k)3/4? 23/2(a3/4 + k3/4)
? 23/2y2/4? 23/2(a +
m1 + 2mk) < 15-3Mk <Mk
? 15-3(
providedthat 8 is smallenoughand a0 is close enoughto 2.
Hence the inequalityj < 23/273/4will be violatedbeforethe inequality
j
mk as j increasesand it followsthat
p < 23/213/4
(6.4)
Let co= (a, b). Next we give a lower bound for the length of 52=
w) whichmaybe expressedas
F111k+ P +1(0;
(6.5)
121 =I |Fmk+P+l(0;
-
a)
m(a);
FP+?(
-
a)
F
b) I
(0;
MkP
- FP+ ((m
(b); b)
Since
(mka)
mk(b)
=
la - bl aaFmk(0; a')
2
la -bl
*16-em
a
<
<
a'
b)
the explicitdependenceon the parametersa and b in (6.5) is inessential.
< 0 < 1,
From the mean value theoremit followsthatforsome 0, 1 -a2
(6.6)
2
1521
>
(a)
2a|(m
3acjIj
I |m(a)-
I I dxFP(O;
m(b)
a)|
IaxFP(0; a)
From (5.2) and (6.1) with p replacedby p + 1, it followsthatforsome 71and
? < 71/ < 71 < C1,-1,
71/f,
(6.7)
a q2 a FP(1
X
- a(q ")2; a) |
10 __v_)_
10 ~~~~(p
+ 1)2
From (6.4), (6.5) and the uniformity
estimate(6.2) we concludethat
(6.8)
(6.8)
?
1~21I
?
YIj
Cl I
1
I
2 e
L_2 10
3_2
By the induction assumption (iii),
,
+?
____(a)_I
(p + 1)2
-P
?
1. It follows from(6.4), (6.8)
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M. BENEDICKS AND L. CARLESON
10
and [ ? a
=
(log 1/8)2 that
(6.9)
1
u12
e - (2?+3/4)1/2
2?e
2i3/8
providedthat 8 is smallenough.
Proofof (i) at thenextfreeretum.We firstprove:
(6.10)
axFmk+P-
?
1(1; a)
a E Ak.
e2(mk+P)2/3+(1/1o)P2/3
By the chain rule
a)
xFn'k+P-P(1;
a)2am
= aXFmkkl(1;
dxFP-
k(a)
(Mmk
l(a);
a)
estimate(6.2) and (5.2),
By (6.7), the uniformity
(6.11)
mk
2atmk(a)axFP-1(
2_2e-
-
2
(a); a)
1
2!L-
1
1
40 (p +12
1
40 (p + 1)2IPI
expt 2/23
xf
p+
p+1
1
e
p2/3
small.(Note that(6.4) implies ?2 1p22/3 and that(5.2)
if 8 is made sufficiently
impliesthatthereis a lowerboundforp ofthetype ((log 1/8) so thatp is large
forsmall 8.)
Hence,sinceby(6.3) mk ? 153p,
|adFmk+P-1(1;
a) ?
e2(mk+P)2/3+(1/1o)P2/3
fora E a, and (6.10) is verified.
of the partitionwe now followthe
Accordingto the rulesof the definition
image of X until Fmk+P+i(O; co) hits I* fori = il. From Lemma 1 and (6.10) it
followsthat
|a Fmk
p?ii-1(1; a)
?
e2(mk?P~iI)2/3
If the return ?mk+P+ l(a) is free,theproofof (i) is complete.If the returnis
w) hits I, I with v < p. The
precedingargumentforI. maybe used to concludethatat the nextreturn
inessential it followsfrom(8.9) that Fmk+P+il(O;
n8Fk2(la) a)
dxFnk2(1;
? 2en2.
We keep on until the return nk is free.This happens eventuallysince the
lengthsof the imagesof coat the returnsare rapidlyincreasing.
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ITERATIONS OF 1 -
ax2
11
Note thatwe have also provedthat
(6.12)
? e2nk3
a)
axFnkj F(l;
holdsfortheinessential
returns
(nkj.
Proofthat (ii) holds for mk < i < mk~l. We give the proofin the case
< nk1l The case nk' < i < nk j?1 is similar.Assumefirstthat i - mk <
Pk < mk, where p = Pk is definedby (5.1) and (5.2). From the induction
assumption(i) combinedwith(ii) and the chainruleit followsthat
mk < i
- 'l-te~i1Mk)3
axF'-1(1; a) >2 e~'e
+ i1-Mk
e m2k/3
2
123emk/3-
i
? e.
i2/3
In the last inequalitywe have used the factthat mk ? ,. For i
use (6.10) and Lemma 2 and notethat
2(mk
+ Pk)2"3
+ Y(i
-
1
- Pk
Mk)
-
Klog4
-
> Pk + mk
we
? i 2/3.
This completesthe proofof (ii).
Proofof (iii). We have to prove Ijl ? e-i forMk < j < Mk+1. Note that
by Lemma 3, ml maybe chosen ? 2a. It followsthat
mk ? 2ink +
for 2
-
2ml
small, since mik ?
a0 sufficiently
j
formk ?<
2 e- a >k
?
k+ 1 + a
m1and m1
- e-V
2
xo,
as a 0
-
2. Hence
eVJ,
<Mk+?
Estimateof theexcludedset. To completethe proofof Theorem2 we have
to show that the set Ak+l definedin Section5 satisfiesIAk+lI ? 1AkI(1 - ak),
where0 < ak < 1 and Ht' 1(1 - ak) > 0. We assumetemporarily
that
C-1 ?
aaF
Mk+1(o;a
a,)
aaFmk+I(0; a2)1
fora1, a2 e w, wheretheconstantC is independentof k. This willbe provedin
the nextsection.
It followsfrom(6.9) thatthe portionof cwexcludedwhen Ak+l is formedis
less than
e - a ~+k+ 1
Const e
ay3/8
e- 2,13/
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12
M. BENEDICKS AND L. CARLESON
From thiswe see that
ak
and Hl1(1
-ak)
>
< Conste(2(k+a)3/8
0. Hence
UI
- a
+k
I Uk=lAkI
=
+ 1)
> 0.
7. A uniformestimateforthe a-derivative
To finishthe proofof Theorem2 we mustprove:
5. Suppose that xk(a) E I,] is a free returnaftermk iterations
and that FMk(O; a) C I . Then thereare constantsC and C' such that for
a, b e w c Ak and j < Mk+1
LEMMA
axFj(l; b)
(7.1)
C and
axFj(1;a)
| aFi(l ; ab)|
(7.2)
<<
aFC(1;b)
aaFi(l; a) I
Proof.It is enoughto provetheestimate(7.1), since(7.2) followsfrom(7.1)
and Lemma 2. Note that(7.1) is equivalentto
(b )iIi
((b
)
<C.
First,by the inductionassumption(i) and Lemma 4
<
Ij|
Cemk/3
Thus the firstfactoris inessential.It is sufficient
to estimate
Mk
= E
vS=
1
- (Vjb)|I
ltva)
(
Let { tj} N be the indices of the freeand the freeinessentialreturnsto I*
arrangedin increasingorder.We shallsubdividethe sum
Nk-1
S=
j=1
tj+1
L
v=tj
1
Nk-1
E
Sj
j=1
At time tjItj c I~j, and we denote aj = (tj(a), t(b)) C Ilj. We shall first
estimatethe sum S . The contribution
fromthe bound part tj < i < t1 + pj can
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ITERATIONS
1
OF
13
ax2
-
be estimatedby an absoluteconstanttimes
IjaTj Pi C,,Ilt Ida Fs -'(l; a)
~~~~Iaj
lajl + laj
M
(7.3)
(7.3)
E
(We have used (6.2) and themean value theorem;compare(6.6).)
The sum is splitintotwo parts
pi
Pi
pi
=E+
s=1
s=1
where P=
E,
s=p?+l
j. In the first
we invokethe elementary
estimates
(
IdxFsl< 4s
lj(a)
?e-
and in the second we use the stoppingrule(5.1) and (6.1). The resultis thatthe
term(7.3) may be estimatedby
ConstII
1
Aftertime tj + pj the intervalmovesup to time tj+ outside1*. Note that
by Lemma 1 and (3.2),
la
<
) - .^(b)
? [I
tj+ - l
/ 19\2
-
X+
(l)I
2
t+,-
tl(a)
-
t~(b)
|
where at 1-1 is veryclose to 1/ r4. Therefore
t.?1
-S
v=tj+pj+l
~
jbII
(b)
t+(a)-\
< Const
jib
fil)l
lt(a)I
so thiscontribution
maybe absorbedin Et
Nk
j=1X
1
Therefore
+2 -1.
ij+I
Ia1I
IIL1
This sum is estimatedas follows.By an argumentequivalentto thatin Section6
(cf. (6.9)) it followsthat
(7.4)
FPi(IM;aa')
e-2e ,
a' E [a,b].
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14
M. BENEDICKS AND L. CARLESON
During the bound period 1 < i < pj, consider
I2xF(X~a)
dXF'(x2;
=
l
1F
j
i=1
i
1
a2
a2)
(xl; a)
F(x; a 1)
1
v=l
Jp
for x1, X2
E
a,,
a2E2ec
jF"(1
-
a2)
(X2;a2)1
F"(x2;
jF"(xj;aj)-F
{a, i
. Since
-
ax2;a)
FP(1;a)
I ?<
1
^ +Aa)
10(v + 1)2
x e I, a e , this quotient is uniformlybounded by an absolute constant C2.
From this uniformityand Lemma 1 it followsthat
(7.4)
la1+?1 ? |?
>
A
IIjI
2 C27
p(b)
- (?
1(a)
1a
C- leVe I2j8ila.
small S. (Note that L ? a = (log 1/8)2.)
Hence, Iaj +1 ? 51ajI for sufficiently
Because of this exponential increase
Nk
1
Ia I
1
1a
IJ()I
ConstE
where J(s) is the last j forwhich [j = s.
This amounts to the fact that the [ j's can be assumed to be distinct. The
sum
A
1
II
is then split into two parts. Let J, be the set of indices such that
(1/ !Lj)(lajl/I IM I) < [L-2 and J2 is the set of remaining indices. The sum
1 IaI<
jeIh
II~jL'
1
=1 V2
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1
ITERATIONS OF
-
15
ax2
bounded.If j E J2
and is therefore
uniformly
1
1
lajl
/
I yI
M1~
-
H2
41L1P~
and by (7.4) it followsfork > j that
~i78-2iL/~/1
IakI ? C~12e e-2'ij e-uF _>
if 8 is sufficiently
small.Since I
2DUk,
therefore
verylacunaryand hence
ja12
1
ki I I1T
1 IEI
e-33/
<
Ik 9?3/4.
a21
The set {
1i}ie2
is
M1
is uniformly
bounded.The proofof Lemma 5 is complete.
Chapter III
8. The distribution
problem
Let again ~(ja) be theimagesof 0 underthe nthiteration.Let
(8.1)
MPn=
1
n
= Shiv
n
1
__1
We shall studythe asymptoticbehaviourof Mnand more preciselyprove the
followingtheorem.
3. For almostall a E A., definedin Section6, theweak-*limit
is
}n=
absolutelycontinuous,dI = gdx, and g E LP forall p < 2.
IL of {(ILn
THEOREM
offree
The strategyoftheproofis as follows.We firststudythe distribution
returnsxn on 1*.
It turnsout thatxn behavesin a pseudo-random
way as a "pseudo-Markovin
process"and fromthiswe deduce in Section11 thatthe xn have a distribution
I* witha bounded density.
We next provein Section 10 thatthe inessentialreturnsto I* are so few
stillgive
that-althoughwe have no controlof the locationof thesereturns-they
riseto a bounded density.
reflectthe tendencyof the processto repeatits
The bound returns,
finally,
early historyand can thereforebe describedin termsof the precedingfree
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16
M. BENEDICKS AND L. CARLESON
s in g, whichhave the form
returns.These returnsproducesingularities
fxi-x41 -1/2
X> u (resp. x < u)
x< u (resp. x> u),
=1
wherethe u are the freereturnsto 1*. From thisit followsthat g 4 L2. These
is knownin I* it can be
resultsare provedin Section11. Whenthe distribution
transferred
to (-1, 1) (Section12).
9. Distribution
ofthefreereturns
In the previoussectionwe defineda set A = A0 c AO,wherethe mapping
is aperiodic. Let us renormalizethe Lebesgue measureso that mA = 1 and
measure.We can thenspeak of expectationsE and
considerit as a probability
in the usual way. We are goingto studyreturnsto some
conditionalprobability
In
on returns,
fixedintervalX 0 0.
doingthatwe can disregardthe restrictions
of A.
whichwere the basis of the definition
Let / be a subsetof A0whichis definedby conditionson the freereturns
x, ..., xX. More precisely,let 1 < v1 < v2 <
< v1< n and 1M,2,' . NIIl
= 1,...,1}, or W= A. Sets of this
be given. Then _= {a E AnIxj EIej)
to the different
type can be subdividedintosubsetsdV(i) corresponding
itineraries i goingthroughthesedifferent
intervalsIj at the vlhfreereturn.
We definethe sets
AV= {ae -lXn(a)
AVP(i) = {aaedVxn(a)
E
I>,)
E IV,
xn+1(a)
y
Il
itin.i}
As= UAv(i).
',
I
available(see (6.9)) is thatforeach i, I, is increasedto
The onlyinformation
an intervalU2(i), where
(p(v) = e
L(v, i)
which is then subdividedinto intervalsII. This is the significanceof the
estimates
of the partitionP (Section5). Withregardto the uniformity
definition
in Section7 thismeansthat
(9.1)
|i2(i)
=
< Const
mAVM,(i)
L (v,)
(9.2)
EmA5vk(i)
k
lemma.
We have the following
6. Let _V1=UVA. be given C A or d C An as above. Thereis a
constantCO, independent
of si, so thatifforsome Q > CO
mA. < 2QII5Ims forall v,
LEMMA
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1 - ax2
ITERATIONS OF
17
then
mA' < QItIm-W
forall [t.
Remark. The statementclearly means that we have an estimate of "conditional probability" of transitionfrom v to [t.
Proof From (9.2) we get
mAf= EmA,(i) <
mA,(i)
E
v, i
1?vqi
E
1
+ CI1IMI
E~k'
V>q L(vi)
2mAkkU),
and by (11.1) the second sum may be estimated by
1
E
II Im
< 2Q2
EmAvk(i)
q(
v>q
i,k
qV
v>q
< 22 Mae,
2C
if q is suitably chosen since EZII,/cp(v) < ox. For v < q we have L(v, i) ? Lo,
so that
E
v<q
mA v(i)
<
i
if CO ? (2C/Lo).
C
L 1t1 >EmAvk(j)
0
vk
?<
?L
I
m
f
This implies
mA? <
2IlI
+
Ci1,1i m-{<
QjI, Am'l,
which completes the proof.
We formulatea special case of this in
LEMMA
7. IfA,,,,= {a e A, Ixn(a) e I,}, n = 1, 2,,
mAn
thenforall n
< COV1I.
Proof. For n = 1 this holds by Lemma 3. We then use Lemma
6 induc-
tively.
With these lemmas it is now easy to estimate the distribution function of the
not containing 0, and assume that
c
x1's. Let co be an interval of length
I
o.
Then with the notation
for some v. Let X, be the characteristic function of
above with _/= A
E(X(Xn))
-|XA(Xn(a))
da
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18
M. BENEDICKS
AND L. CARLESON
itinerariesi. For each i the
A. can be decomposed,AV= UjAj(i) fordifferent
distribution
of xj(a) in I,, has a uniform
densityby Lemma 5. Hence
? ConstE
< ConstmA.
E(Xjx(x))
wherethe last inequalityfollowsfromLemma 7. Hence if
l
N
FN(a) =
1=1
X(XJ)
then
(9.3)
E(FN) < ConstE.
We now choose h as a largeinteger(fixedas N -- oo).
(9.4)
(9.4)
h) =
E(F~ NNf
-h
E
ji
A
-, jh <N
...
,(X
jxi)j)...X(
a
h
XII(jh)d
The numberof h-tuples(UjP*. j,)
M forwhichminklIk
< vN is bounded
- ill
fromthesetermsto (9.4) is therefore
by C( h)Nh- (1/2), The contribution
/2
O(N
). Take jl < j2 <
jh< N so thatjk+l - jk 2/ , k = 1,...,
and consideran intervalX C I>. Suppose that x;L X for s < 1. We
choose,in Lemma6, n = j, and s&'=s-, = {a c AIx1 C I,, s < 1).
The assumptionsof Lemma 6 are satisfiedwith 2Q = 11,1-' and
A= {a E--/Ix 1(a) e I}. Let A(k) {a EC-Ix ?+k(a) E II}. It followsthat
h-i,
-
mA(1) <Qm-Vl
and afterk = j
I It,
j, ?
we have
A iterations
FN
k) < 2-k+ 1 QMVl.
mA(
|1,II Clnl
fl
providedthat VN221og(1/lIIp I CO); i.e. IIj > CO-12- FN.Hence,
M-VI + 1 < Co IIVImffll;
i.e.
mJ&'h
?
(CoIIjI)h
forN largeenough.Now, ifforeach js we requirethat xe c
fromLemma 5 in Section7 thatthe measureis decreasedby
X c
I,, it follows
Ch(
(We have used thefactthatx1k has a probability
densityboundedabove.) Hence
E(FN) ? (Coc)h,
N ? NO(h,co),
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ITERATIONS OF
19
1 - ax2
so that
lim 11
FNII
N --+
o
00
<
COE
has a densityboundedby
withprobability1. It followsthatthelimitdistribution
of a pointmassat x = 0 is
COoutsidethe pointx = 0. The remainingpossibility
in n and v, and the fact that
excluded by Lemma 6, whichholds uniformly
xn # 0, forall n.
10. The inessentialreturns
j
=
Zki
As describedabove, a freereturnXk to IA is followedby a sequence Yk;,
1,2,... Y,, ofboundreturnsand, possibly,a sequenceof inessentialreturns
i =
1,
..
Zk'
.,
2 thatwhenXk
Forthesequence{ Yk; } we knowbyTheorem
describesIW,Ykj describesan intervalJ so that
|1
J|< 4NkjI
e Nki2"|I|
| <
1I
where Yk(a)
=
mk+Nk
(a).
Rememberthat Xk(a)
=
Jmk(a).
we know from(6.8) that
that p < 23/2t3/4 and furthermore
By (8.4) it follows
> 20
I101
1s120
2 e--VP
p
Let us genericallydenotethisinterval52by I'. When the firstinessential
returnZk1 occurs,PI is mappedintoan intervala, suchthata, C 1 U I +1. It
mustthen hold that pu < 4p 3/4 (cf. (6.9)). Ity(togetherwith I +l ) may now
have a bound evolutionand al mapped into 'a v = 1,2, ..., ZV). Afterthis
time our intervalal may be mapped onto a2, whichintersectsIjA2' The results
in Section7 give
of the derivatives
about the uniformity
Alall
Clalle2>/-?i2
1a21? C0afl
forsome fixedA ? 2 (say). This processis continueduntilthe intervalsa or av
are so long thata freereturntakesplace. Observethatinessentialreturnsoccur
forall a1 and aj .
Let again o be a fixedintervaland consider
(10.1)
1
G(a) = -N
N1?k?N
L
X.(Zkj(a))
N
=
N
E
k=1
Gk(a),
to freereturnsXk, k = 1,2,..., N. We
where Gk are the sums corresponding
i so that Xk E 1p.
shallestimateE(G(a)). Let us fix and consideran itinerary
on I1 with
it's,and Xkis distributed
E is the sum of expectationsEi fordifferent
o,
to
the
corresponda boundeddensity(Section11). For thoseZkj whichreturn
,.
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20
AND L. CARLESON
M. BENEDICKS
ing densityis bounded fromabove and below by fixedpositiveconstants.We
shall makethe worstpossibleassumptionand assumethatall Zkj returnto W.As
to G. When Iaj, I > Ic I we get a
long as IajVI < IwI we get 1 as a contribution
expectationEi( Gk(a)) can
<
X
The
conditional
relativecontribution Cl l/
ll I.
be estimatedby a constanttimes
therefore
(
(10.2)
co
< 1I
l'apvI
and we claim that
) + Co l
< Const jel/2)+
1
E
(10.3)
1 < IcI ElyjvI
IapI > IcoI illjV
Observe firstthat by the mean value theorem(cf. (6.9), (6.2) and (ii) of the
inductionstep)
2l2e| yjvl >
4 tL3/8e ^2/3>21e
- 4tL3/8
the lengthof each boundevolutionofsome a, is less or equal to Ct,3/4. Since Iai
the numberof a 's < Const ,.t and (10.3) follows.
increasesexponentially,
To completethe proofwe mustagain considerhighpowersh of G(a):
...
=
(a).
Gjh
N hL Gjh(a)
By (10.2) and (10.3), the argumentof Section 9 can be repeated when
G(a)h
ik
il ? N.
The numberof remainingtermsis < C(h)Nh-(l/2). For such a choice of
{ hk} observethat
ji
ih
h
li
Note that the numberof termsin Gk forXk E=
LG h(a) da < C IC
A
A
is less than Ct5/44.Therefore
EI,(5/4)he V<
<0
of G(a) is made by dividingby N. i.e. the
We note that the normalization
smaller.
of Sunin (8.1) is therefore
numberof freereturns.The normalization
oftheinessentialreturnsalso have a
We concludethatthelimitdistributions
bounded density.
11. Distributionofbound returns
of the bound returns{ y,},
We are now goingto studythe distributions
forfixedj < J and
=
1,2...... We firststudythe distribution
n = 1,N...,N j
to j > J is small.
thenshow thatthe totalmass corresponding
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ITERATIONS OF
Take
j
? J and let the
{
jth
1 - ax2
21
return of 0 to 1* occur at time q, and set
=
We call the returnodd (even) if the correspondingkneading
sequence is odd (even). Let p and 1 > 0 be givenand let o(a) be the interval
uj(a)
(uj
.~a)
=
-
u -p),
p - 1,
(uj + p,
p + 1),
+
We consider
HNI)
=
HN(a)
oddreturns
even returns.
N
L xco(a)(Ynj(a))
=
n=l
1GI,},. Suppose that uj(a) is the qjth
Let U be an intervalin { a E A0Ixn(a) E
iteration of 0. For a E U. consider the images of a = (0, ? c ,LI - i) D 1,, under
F. i.e. a/(a) = Fqj(a; a). It followsby (6.2), Theorem2 and the mean value
theorem that
|l (a)l 2 Contia(a)12eq2/
(11.1)
Contla(a)12ej/
Note thatthe right(left)endpointof a1 is u; if {
kneadingsequence.
0
qj}i 1 has an odd (even)
uj
w(a)
It followsthatthe "conditionalprobability"p, thata point x, from1,2at
the jth bound returnbelongsto c(a), satisfies
(
p <
Const
lujl
? p
~~if jail
j
if IjaI <p.
t0
withrespectto xn
By conditioning
E(X.(a)(
Ynj( a )))
E
1 and by (11.1), we concludethat
< Const
11,2
_2/3
le
E2GM
CU
where
M = {(.lcp? Const peU273}
(The set M contains{fp 1aj1? p}.) This gives
< Constle( --1/2)j2/3p
E(H()(a))
-1/2
We consideragain highpowers.The sum
N-h
?
EX(a)hYnlj(a))...
Xw(a)(Ynhj(a)))
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22
M. BENEDICKS AND L. CARLESON
I ?< VNand Ins+1 -
intothecasesminshs+ 1 bysplitting
is estimated
11 and 12. The resultis
A as in Sections
BE
NEx
< Constle ( - 1/2)j/p
X (a)(ynj( a) )]
1/2 +
nsI
(Cl( h)N- 1/2) /
of the ynj's to the limit
As beforeit now followsthatthe contribution
g1 so that
has foralmostall a E A, a density
distribution
f g1(x)dx< Constle(-
1/2)j2/3p-1/2
@(a)
foreverychoice of p, l > 0, and j
1,2, ..,J.
withj > J. Let
to returns
We nextturnto thetotalmasscorresponding
=
otherwise.
and
set
We
define
1
exists
the
return
if
0
On=
Onj=
yj
Onj(a)
=
l+ n, and have to estimate
EJ+
N
TN =
L
n= 1
Of
Therenj < 23/2j3/4 ? Itti.
concludethat
foreOn< I I and On:$ 0 onlywhenI 2I J+ 1. We therefore
If x EI p,/3 ? a, it followsfrom(6.4) that
00
E(On)
< Const L
jL=1+ 1
ae-r < e(-
1/2)yl
and the same estimateholdsforE(TN). As beforewe estimateE(TN) and since
E(Onh)
< C.
Lthe-rt'
theargument
is thesameas in Section10. We concludethat
<
lim TN(a) e?(-
1/2)V
N-.o
foralmostall a E A.
theorem.
so farin thefollowing
We can summarize
theresults
THEOREM
of free and of
4. For almostall a E A, any limitdistribution
of
A limitdistribution
boundeddensity.
inessential
toI* hasa uniformly
returns
g(x) and
boundreturns
has a density
g(x) < Const
E
U,
e(
1/2)j2/3T(u -Ux)
odd
+
E
e(-1/2)i2S/3(x
us even
where
> 0
1
< ~0.
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-u
)]
ITERATIONS OF
23
1 - ax2
12. IteratedpointsoutsideI*
In thepreviousdiscussionwe have onlydescribedthereturnsto 1*. We now
keep a e A fixedand studythe orbitmorein detail.Let
ln
tin
=
E 8tv,(a)'
V=1
For { ji }n 1 the following
resultholds.
{
5. For almostall a E A everyweak-*convergent
subsequenceof
n}?=1 has an absolutelycontinuouslimitgdx and g E L P(-1, 1) for every
THEOREM
p < 2.
Proof.As before,let { u(a)}M'L be the returnsto I* up to time n. Then
/A
M
n
v=1
k=i
E SUV+
kE
n
E
v
6Fk(uv)
M
k)
E jU~nk
=
k=O
The primein the summationsignindicatesthat 8Fk(u ) is includedin the sum if
and onlyif F j(ut,) i 1*, j = 1,2,. . ., k.
Let p* denote the weak-*limitof 0{
}=L . We have alreadyprovedthat
lim (??*
ELP
for p < 2 (Theorem4), and we turnto the {
}0
whichare supportedon
[1 - a82, 1]. Let X be a subinterval
of [1 - a82, 1]. Now F'1 consistsof two
intervals5; and -52 in 1*. We have
symmetric
?=1,
Pn(C)
=
-f(n)(W)
=
t(n)(Q
U
2).
-
By Theorem4,
M1*(@)< |
QU
h(x) Ax,
where
00
h(x) = Const ,
e(-1/2)v2/3
1
By a change of variables
wU-h
F)
L|
}Fa
2
__h(x) dxA=
h(F71(x))(Fj
l)'dx,
where tFi- )j.,= are the two branchesof F-'. We claim that the density
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24
M. BENEDICKS AND L. CARLESON
2 lh( Ft- (x))( Ft- )'(x) E LP forp < 2. We have
(12.1)
i=1
1
~
h(F
(X)) I|I
C(Fo t(x))jh|
~
~
~
dx
p
<?ConstLfIh(x)l p
< Const I , h(x)l
dx
p dx
1
00It
<Const L e(P-'),
Ifh(x)1P dx.
,u= aI
But
l/p
dx
(ph(x)1P
< Const
[L
7/8]
(-p/2)v
i/P
2/3
d
Xf_ dx)Up/
z
V=
00
+
To estimate S1 we use the relation u. I2? e?
Ie-,7/16
V <
[I7/8]
x C
Si + S2.
+
V=V~~~~~~~~~~~~/8]
which implies Ix - u.
v,
. Hence
X
([,L7/8]
S < Const
2
e(/2)23jI11/P
i'=1
< Const exp(,t7/16 -(1/p)
/).
The second sum is estimatedas follows:
00
S2 < Const
/]
/2)
e(
d
23d/3/l
< Const e(-
and the convergenceof (12.1)1/2),I7/12
follows.Hence
(12.2)
y*(1-a321
)
E
LP
Let now cobe an intervalcontained in (-
forp < 2.
1, 1
Lemma 3, (iii), we conclude that for x E (1v= 1.2 ...,k)- 1
(12.3)
|dxF k(x) I 2
- a82)
\ 1*. From Lemma 2 and
a82, 1) such that F'(x) O 1*,
A,5
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ITERATIONS OF 1
- ax2
25
constant.From(12.3) it now followsthat
where X > 1 is a uniform
k = 2,3,-
lM An.I (CO) < Igk(X)dx,
j-o 00
< ConstX-k,p < 2.
where IlgktIp
Let Ek = {x e I*IFV(x) 4 1*, v = 1,2,...,k}. By (12.3), cf. [1], Theorem 11.5.2,part3, thereis some q < 1 such that IEkI = C(9Crk).By thisfactand
of 1(I- a32, 1) it is clear thatforeverye > 0 thereis a
the absolutecontinuity
q = q(e) such that
n
-
Zy
k=q+1
<
II ?e.
it|)
We finallyconcludethat
<
whereZ:k.2gk
ak=2
gk(X) dx,
M*(co)
E LP and Theorem
5 is proved.
ROYAL INSTITUTE OF TECHNOLOGY, STOCKHOLM, SWEDEN
INSTITUT MITTAG-LEFFLER, DJURSHOLM, SWEDEN
REFERENCES
[1] P. COLLET AND J.-P. EcIMANN, IteratedMaps on the Interval as Dynamical Systems,
Birkhauser,1980.
, On the abundanceof aperiodicbehaviourformaps on the interval,Comm. Math.
[2]
Phys.73 (1980), 115-160.
[3] M. JAKOBSON, Absolutelycontinuousinvariantmeasuresfor one-parameter
familiesof onedimensionalmaps. Comm. Math.Phys.81 (1981), 39-88.
of stochasticand deterministic
[4] S. M. ULAM AND J.VON NEUMANN, On combinations
processes,
Bull. A. M. S. 53 (1947), 1120.
(ReceivedMarch1, 1983)
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