A NNALES DE L ’I. H. P., SECTION B
J. C. M ARCUARD E. V ISINESCU
Monotonicity properties of some skew tent maps
Annales de l’I. H. P., section B, tome 28, n
o1 (1992), p. 1-29
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1-
Monotonicity properties of
someskew tent maps
J. C. MARCUARD and E. VISINESCU Departement de Mathematiques, Laboratoire de Topologie,
B.P. n° 138, 21004 Dijon Cedex, France
Vol. 28, n° 1, 1992, p. .29. Probabilités et Statistiques
ABSTRACT. - We
investigate properties
ofmonotonicity
ofkneading
sequences and
topological
entropy for some unimodal maps of theinterval,
called skew tent mapsgiven by
the formulawhen the
parameters belong
to theregion
The case
~1
has been studied in[M-V]
and the authorsproved
thatkneading
sequences and entropy arestricly increasing
functions of À and p. Here we prove, when~1,
thatkneading
sequences and entropyare
increasing
and weidentify
someregions
where these functions of Àand p
are constant. We prove also that the set of(À, Jl)
where thekneading
sequences is a
primary itinerary M
is thegraph
of adecreasing
functionÀ= as in
Key words : Unimodal maps, tent maps, kneading sequences, topological entropy.
RESUME. - On etudie les
propriétés
de croissance des itinéraires et del’entropie topologique
de certainesapplications
unimodales del’intervalle, appelées applications
tentes et données par les formulesClassification A.M.S. : 28 D 20.
les
paramètres
appartenant au domaineLe cas 03BB>1 a ete étudié dans et les auteurs ont montré que les itinéraires et
1’entropie
sont des fonctions strictement croissantes de À et jLL Ici nous montrons que si À 1 les itinéraires etl’entropie
sont desfonctions croissantes et nous identifions les
regions
ou ces fonctions sontconstantes. Nous montrons
également
que l’ensemble des(À, Jl)
ou l’iti-neraire
M
estprobablement primaire
est legraphe
d’une fonction decrois- sante 03BB=03B2M( ) comme dans[M-V].
INTRODUCTION
The
problem
of themonotonicity
of the entropy for various classes of unimodal maps of the intervaldepending
on real parameters is very natural. Few answers aregiven today
to thisquestion, only
thefollowing
cases, to our
knowledge,
have apositive
answer.For the tent maps
F~, ~,
the answer issimple,
the entropy islog
p[M-S].
For the
expensive
skew tent maps(~1, ~>1)
the answer is alsogiven
in
[M-V]
and uses thetheory
ofkneading
sequences. The entropy is astrictly increasing
function of Àand Jl,
andrecently
in[B-M-T]
the authors gaveelegant proof
for the case aboveby studying trapezoidal
maps, butthey only proved monotonicity (not
strictmonotonicity).
If~1,
thereexists an attractive fixed
point
which attractseverything
exceptperhaps
one
point
and thekneading
sequence is R~. If~== 1
then there exists awhole interval of
periodic points
ofperiod
2(except
a fixedpoint)
andthe
kneading
sequence is RC. In both cases the entropy is 0.For the
quadratic
maps there exists also apositive
answer as a
corollary
of thestudy
of thecomplex
maps gc(z)
=z~
+ c,[D].
It will be
interesting
to have aproof using only
real technics for this case.Here we
study
theproperties
ofmonotonicity
of the skew tent mapswhen the parameters
belong
to theregion
We consider the relation
(~,
>(À, Jl)
if~,’ _>_ X, ~,’
and at least oneof these
inegalities
is strict. Our results are in the samespirit
as those of[M- V]
and we use some results that areproved
there.This paper is
organised
as follows. In section1,
we state the results. In section 2 we prove theorem 1. Section 3 is verytechnical,
weinvestigate
the
partial
derivative associated tokneading
sequences. In section 4 weprove the main theorems 2 and 3.
1. STATEMENT OF RESULTS
We consider a
partition
of theregion D 1 = { (À, ~)~~1, t~>l} by
the sets
Then each set is
decomposed
into threedisjoint
setsFor
m =1, 2,
3 these sets arepictured
inFigure
1.For m >_ 1 the
graphs
of the functionshave one common
point
with coordinates(Àm" J.1m)
such thatSo the values
Am
aredecreasing to 1,
and the values 11m areincreasing
toinfinity.
For m >_ ~
thegraphs
of functionshave one common
point
with coordinates(A~
such thatand
So the values
X£
are alsodecreasing to -.
As in
[M-V]
we shall denoteby
K(À, p),
h(À, Jl)
thekneading
sequence and the entropy of the skew tent mapFÂ.,
~.We shall also use for
kneading
sequences the notations of[C-E].
A maximal admissible sequence
M
is calledstrictly primary
if(1)
(2)
If M = A * B with then B = C.We shall denote
by P
the class ofstrictly primary
sequences.A maximal admissible sequence
M
isprimary
if(1)
(2)
then A = R*m for some m > 0.THEOREM 1. -
(h, if
andonly if RLm-1 CK(À,
More
precisely:
(~,, Jl)
Eif
andonly i. f ’
K(X, Jl)
= RL m - 1 * R 00 =(RL m) 00 . (X, Jl)
E~m if
andonly f
K(X, Jl)
= RL m-1 * B with B(X, Jl)
Eif
andonly if
K(X, ~,)
= RLm R... and K(X, ~,)
E EP.THEOREM 2. - Consider
(X, (À’,
ED1
such that(À’,
>(X, Jl).
(1) If (X, ~,)
and(À’, belong
todifferent
sets thenK
(À’,
> K(À, Jl).
(2) If (À, (À’, Am
then K(~,’~ Jl’) =
K(À, Jl) =
°(3) If (À, (À’, Bm
UCm
then K(~,’~ Jl’) >
K(À, Jl).
COROLLARY. - Let
(À’, ~,’), (X, ~,)
EDi such
that(À’,
>(X, y).
(1) If (X, ~,)
and(À’, belong
todifferent
setsØ/m
thenh
(~’~ Jl’) >
h(À,
°(2) If (À, (À’,
EAm
UBm
then h(~,’ ~
= h(À, Jl).
(3) If (À, (À’, Cm
then h(~,’~
> h(À, I~)-
This
corollary
is an immediate consequence of theorem 2 becausewe know the relations between
kneading
sequences and entropy([J-R], [M-S]).
Inparticular
the entropy is constant on the intervals[RL m - 1
* RLm - 1 *THEOREM 3. - For each
M (except
there exists a numbery
(M)
and a continuousdecreasing function [iM : [y (M),
00[~] 0,1]
such thatK (~,, if
andonly
-_
The
graphs of
thefunctions fill
up all the sets UCm
andB1.
And-
m=1 i we have lim = 0.
In fact for each M these functions are the continuous extensions for
~, 1 of the functions
given
in the theorem C of for ~, >_ 1.2. KNEADING
SEQUENCES
OF SKEW TENT MAPS We set xn =F~, ~ ( 1 ).
Thekneading
sequence ofFÀ, ~
is denotedby
K (À, ~)=Ao...An_1
C ifx~~0
fori=0,
..., n-1 andif
where
Ai
= L or R when xi 0 or 0respectively.
If A
=Ao...
1 is a finite word withalphabet (L, R)
wedefine A|I
as the number of letters in A and
81 (A), 82 (A)
as the numbers ofL,
Rrespectively
in A. We set also E(ð.) = ( -1)82
(~).We
will say that a unimodal mapf
has a n-renormalisation if there exists an intervalIo containing 0,
invariant
by
such thatfn|I0
istopologically conjugate
to an unimodalmap g. If a skew tent
FÂ, ~
has an-renormalisation,
it is clear that the map g is also a skew tent map withslopes given by
thefollowing
lemma.LEMMA
2 .1. - K(03BB, )=A*B if
andonly if Fl.,!!
havea
renormalisation. Moreover
)=A*B
the map islinearly conjugate
to the skew tent map ~~, IJ)r
Proof --
Theequivalence
isproved
in[C-E],
p. 147.Now if
K (~,, Jl) = ~ * ~
the mapFl~J.11 +
1 has an invariantsegment Io containing
0 such thatFl~’ + I, 10
is unimodal andpiecewise
linear.By
the linearconjugacy ~r (x)
=Fl~,}
+1 (0)
x, we haveagain
a kew tent mapgiven
by
By
the definition of the *product,
ifE (A) =1
thenFl~J11 + 1 (0)>0
andby
conjugacy
we haveIf E
(A~
= - I then + 1(0)
0 and theconjugacy
reverses theslopes.
This proves formulas
(2 . .1 ).
Moreover K
(X,
= K+ 1 I0)
= B.LEMMA 2 . 2. -
If
K(X, p)
= A * B then we haveProof - By [M-V],
lemma3.1,
ifK(~)=A ~ B,
theslopes (X, Jl)
belong Using
formula(2~1)
weobtain
(2.2).
Now if ...
I is a finite word with
alphabet (L, R)
wedenote
by A~
the word-
where
Ao = 0
and therefore s(A~)
=1, 0 1 (Ao~
=82 (Ao)
= 0.To each finite word we associate the
following
characteristicpolynomial
If K
(X, p)
= A ... is akneading
sequence where A =Ao At
...An
we haveSo
by
iteration we obtain thatFor the finite
kneading
sequence K(À, Jl)
= AC we haveRemarks 2 .1. - In fact the two-variable
polynomials
xA(X, ~,) generalise
those introduced in
[D-G-P]
for thestudy
of tent maps. Sothey satisfy
the functional
equation:
LEMMA 2 . 3 . - Assume
(X,
EDl
and m E 1’~T* .Proof - ( 1 ) Suppose I~ (~,,
then we haveNow suppose then either or
For the first case
K (~,,
L... thenFor the other case K
(À, Jl)
= RLp R... with 1_ p _ m -1 (j~
1 because~ > 1).
So we have
Since
we have
(7~, y)
> O. This proves( 1 ).
(2) By
lemma(2 . 1 )
theequalities
are
equivalent
toi (X, p)
0 and there exists a renormalisation withkneading
sequenceApplying
formula(2 . 1 )
we obtainAs we obtain the
equivalence
ofK (~,
RCwith
This proves
(2).
(3) By
lemma(2 . 1 )
theequalities
are
equivalent
to xRLm-1(Â, P)0
and there exists a renormalisation withkneading
sequencei (X,
We obtain theequivalence
ofK
(Â, Jl)
= RL 00 withThis proves
(3).
(4) By
lemma(2 .1 )
theequalities K (~,,
areequivalent
to(X, Jl) 0
and there exists a renormalisation withkneading
sequence Ki (X,
=As the
kneading
sequence R 00 is associated to1, by
lemma(2 . 1)
and formula
(2 . 3)
we obtain theequivalence
ofK (~,, J.1)
= * R°°with
This proves
(4).
Proof of Theorem 1
( 1 )
Assume thatJl);£
RLmC,
then K(X, y)
= RLm R...or K
(~, Jl)
= RLm C. This isequivalent
toTaking
into account that the firstinequalities
areequivalent
tothis proves the first assertion of theorem 1.
(2)
The second assertion followsimmediatly
from part(4)
of lemma(2. 3).
(3)
Assume thatK (~,,
* B withNecessarly
we get(À,
and there exists a renormalisation withkneading
sequence Ki (X,
= B such that B _ RL 00. From formulas(2 . 2)
and(2. 3)
we obtainSO
(h,
EBm.
Conversely,
if(X, p)
EBm,
asBm
cRm
we have K(X, p)
= RLm ...Therefore
ÀJl2;£ À + Jl,
so the maps withslopes (~,m -1
À mJl)
nearthe
turning point
have an invariant segmentIo.
From lemma 2. 1 we get Notice thatJl>l>À,
so then from([M-V],
theo-rem
B)
we have B E .A. This proves the third assertion.(4) Suppose that
for(À, Jl)
ECm
thekneading
sequence is notstrictly primary.
Then there exist A and B such thatK (À, ~) = A ~ B
withA
= RLm RLk R ... andk __
m -1.By maximality
of thekneading
sequence there cannot be more than m consecutivessymbols
L after the third R. InCm
we have Àl1,
for/== 1,
..., m, so we getand
Lemma
(2.2)
shows that this isimpossible
and thiscompletes
theproof
of theorem 1.
3. ESTIMATES OF SOME PARTIAL DERIVATIVES
We set
F=F03BB,
andxn=Fn(1)
and~xn ~03BB, bn=~xn ~ xi~0
for allWe have the recursive formulas
If
An
=Ao A i ... A,~ -1,
tosimplify
the notations we shall denoteei (n)
=9i (~n~ (i
=1 or2)
and En =(
-1)92
~~‘~. If En = + 1respectively (- 1),
we shall write
A i (respectively A;)
for thefollowing symbol.
Moreover for
(X, p)
E~m
we shall set X = ~,’~ ~,. As~,m ~~
X + y, we getX 1 +-
andX2-1 =-(X+ 1»0.
Y J.1
In the whole section
(X, p)
E~m
with m >_ 1 fixedThe aim of this section is to prove the
following proposition.
PROPOSITION 3. 1. -
If (X, ~m
and K(X, J.1)
=Ao AI,
.. is anin fin ite
sequence
belonging
to then there exists no such thatAno
= R ~ andfor
this no we have:
The
proof
is very technical andrequires
ananalysis
ofkneading
sequences for
(À, ~)
ECm.
Taking
into account theordering
betweenkneading
sequences andusing
theorem 1 we have(À, p)
ECm
if andonly
ifK
(X, p)
E *RL 00, Q
and K(X, p)
is astrictly primary
admissible sequence.
The interval
1~
can besplit
intodisjoint
intervalswhere
To
simplify
the notations we setAp
= RLmR)P...
Again,
each interval can besplit
intowhere
Now each interval for
~==0, 1,
..., m -1 can besplit
intoThese
splittings
arepictured
inFigure
2.The next lemma
gives
thebeginning
of infinitekneading
sequences whichbelong
to these intervals and itsproof
is an obvious consequence of theordering
and themaximality properties
ofkneading
sequences. Lett, V, Ù
will denote the interior of thecorresponding
intervals.LEMMA 3. 1. - Let K
infinite kneading
sequence inCm.
then with
i=0...m-2,
then
K=ApLi+1
R- ...If K~i, np, m
then(1)
For p > I we have ...(2)
For p =1 we have twopossibilities
with h = 0, ... , iwith
k = 0,
..., n.To prove the
proposition
3.1 we shall need estimates ofpartial
deriva-tives at the ends of the intervals of
kneading
sequences definited above.As K = RLm ... we
always
havexm --1 + ~, +
...+~~-~’~0
andWe shall denote
by #o
andperiodic points
of F withperiod (m + 1 )
and with itinerariesrespectively
and(0 03B20 03B21 1 ).
We haveHence
Moreover as
we get from the definition of the order of
kneading
sequences that Let be thepositive
functions definedby
thefollowing
recursive relationsIt is easy to
verify by
induction that these are such that We setand
LEMMA 3 . 2. -
( 1 )
Assume that K =Ap ... ,
then we have(2)
Assume that RLi...for p >_ 1 ,
0 _ i _ m -2,
then we have__ ï __ -
Proof:
( 1 )
Estimateof b2m+2for K=~l’ .. - By
iterations of(3 . .2)
we getUsing (3 . 4)
and 1=Pi + Jl Po
we obtainSince B = ~,m we find
From this it follows that
and that .
(2)
EstimateAp... (/?~2). - By
iterations of(3 . 2)
we get
By (3.4)
we obtainNow
using (3 . 5)’
and thefollowing
recursive propertywe obtain
for p >_ 2,
(3)
Estimateof bm +
1 + + ~for ~o By
iteration of(3. 2)
we getThe numerator is the
expression
of("~2~+2)
obtained in the first part(*).
Soby
the same steps we get(recall
that~,-_ 1)
(4)
Estimateof for
RLi...(p >-_ 2). -
We haveThe numerator is the
expression
of( - b~m + 1 ~ ~ p + 1 ~)
in the second part(*, *).
Soby
the same steps we getand this proves the lemma 3 . 2.
Now we shall prove a similar lemma for the
partial derivatives an
with(~,
We shall denotewhere
.xl,
x2, ..., xm are allnegatives.
LEMMA 3 . 3 . -
( 1 )
Assume that K =Ap
..., then we have(2)
Assume thatfor
and then wehave
(1)
EstimateK=Ai... - By
iterations of(3.1)
we getBy (3 . 4)
we have henceand
Xm+3= 1 +~~+2>
1 and so on. Therefore It follows thatUsing
1=Pi
+ we getAs
Pi 1,
it is easy to check thatSo that
and
Therefore
It follows that
and
using (3 . 3)
we get(2)
Estimatesof
+ 1) tm + 1)for Ap ... (p >__ 2). - By
iterations of(3 . 1)
we get
By (3 . 4)
weeasily
check that so thatUsing (3.9)’
and the recursive property we get(3) By
iterationsof
(3 .1 )
we getAs we can write
Using
and we getFrom the
inequality (*)
we obtainHence
As all the
for j= 1 ...
i arenegative
we get(4)
Estimateof
ap ~yn + 1 ) + i + 1for
K =A~ p _ 1 ~
RLi...(p >_ 2). - By
iter-ations of
(3 . 1)
we getAs ~+1,+1 = - F ap ~+1) we can write
Using for j= 1...
i we getFrom
(3 . 9)’
and as Fj(Pi)
0 we have alsoso
by
the same steps as in(2)
we prove the lastinequality.
Remarks 3.1. - From the
equalities
we obtain
By
lemma 3.2 and 3.3, this polynomial in and A IS strictly increasing.If we have and hence
03B1p
(
-xm)
> 1.m- 1
On the other hand if K E U we have
i=0
Now the next lemma
gives
estimates ofpartial
derivatives forkneading
sequences
and such that Tosimplify
we shall denote
by Bn (An)
thepartial
derivatives associated toAp
Li ~.LEMMA 3.4. -
Suppose
that(n > o),
then wehave
the following inequalities
(1) n=O. We have
As for and we get
(3.13).
For the other derivative we have
As 1 and
~+i)~+i)0
we get(3.14).
(2)
Now let us assume that(3.13)
and(3.14)
hold for n -1 and prove them for n.We shall first prove the
following inequality
As X-I
>2:,
we get= >__
1. Theinequality (3 . 15)
is obvious if’
g X-I~,‘ p >_ n + 1. Now we consider the case ~,‘ ~. n + 1
(o -_
i _m -1).
We denoteby Qn
the characteristicpolynomial
ofAp
L~(LRL m - 1)2
n.By
inductionwe see that 1 is
strictly decreasing
andnegative.
and
Using
0 we find"(X-l)(-~)
and
According
to remark(3.1 )
we have 1 so thatBut we have
(-~J=~’~-~’...1A/~-1~
henceand this proves
(3. 15).
To prove
(3 . 13)
we shall compareBk + and Bk.
We haveTherefore
with xj corresponding
to R +.This Xj
satisfies the induction formulasUsing
the relation(*)
of theproof
of lemma 3.2 we obtainThe
partial
derivativeof ð-p Li
is~,‘ bp ~m + ~ ~,
henceby
iterations we getRecalling
formula(3 . 6)
in lemma 3.2 we obtainFor p = 1
Hence for each
p >
1 weAs
~L+20,
from(3.15)
and(3.16)
we deduce(3.13).
To prove
(3 . 14)
weproceed analogously. Comparing Ak+
1 andAk
weobtain
Using
the relation(*)
in theproof
of lemma 3.3 we obtainBy
iterationUsing (3.10), (3.15)
and a2 m + 2 0 we get and this proves(3 .14).
Proof of the
proposition (3.1)
Remarks 3.2. - Recall that m >_ 1 is fixed and we have set B = -
bm+
1,_
Let
K = Ao A1...
be a infinitekneading
sequence inCm.
Assume thatfor n 1 we have
A nl
= R + and>B X2-1,
an1>A X2-1,
then it is easy to infer that for the first such thatAno = R -
we haveWe shall now
study
all casesgiven by
lemma 3.1.( 1 )
Assume that then Li R + ... and setUsing (3 . 8)
and(3 . 5)
we get forp >_ 2 (~ 1 ),
so
As a
=X2014 1, by (3 . 7)
thisinequality
is also true forp =1.
In the same way
by (3 . 12), (3 . 9)
and(3 . 11 )
we get forp >_
1Recall that in
t;, m
we have.ap ( - xm) > 1 (Remark 3.1).
Therefore asThe remark 3.2
gives
the no of theproposition
3.1.(2)
Assume that thenK = Ap Li + 1 R + ...
and setno =
(p + 1 ) (m + ~)+~+
1. We denoteby
yo theperiodic point
of F withperiod (m + 1 )
anditinerary (LRLm -1 ) °° .
It is easy to prove that70=-~0.
"X+l
Now as
by
theordering
property of itineraries I. ~. ,and
As
(À,
we have x ’f" 1 "+-Hence if K E
Vp, m
we get the conditionNow we have
by (3 . 6)-(3 .10)
And
by (3 . 5)-(3 . 9)
we getUsing (3 . 17)
we prove that(3)
Assume that K E then K =Ap
Li)2
n R - ...As we have
the
condition(3 .17)
becomes(a) Suppose that/?> 1,
then and letWith the notations of lemma 3.4 we have
So
using (3 . 13)
we getUsing X2 n > 1,
we find thereforeX~"~~~p+i)~+p-l>0,
andAs before
using
lemma 3.2 we getBy (3 . 18)
we obtainIn the same way we have
Using
lemma 3.3 we getBy (3 . 18)
we obtainThe remarks 3.2
gives
the no of theproposition
3.1.(b) Suppose
that p =1,
lemma 3.1gives
twopossibilities
for thekneading
sequences. First assume that
We shall denote
by Bn, k (An, k)
thepartial
derivatives associated toWe compare
Bn, k+ 1
and We haveWe denote
by j
the indexcorresponding
to R - in thepreceding formula, by
iteration of(3 . 2)
we getLet
Yi=F(Yo)>0
be theperiodic point
ofitinerary
From the
inequality
we deducethat
x~ _ m + 1 > Y 1,
and thatAnd
by
iterations we getBy (3 .19)
withp =1
we obtainLet
~i=2~+2+~+2~(~+l)+~+1+2~(~+1),
with our notations wehave
bnl
=Bn, k
andby
the sameproof
as in(a)
we obtainNow we compare
An, k + and An,
~.Taking
into account thatby
iteration of(3.2)
we getAs 1 we have
So that
Now we shall prove that
So with our notations we have and
by
the sameproof
as in(a)
we shall obtain
the remark 3.2 will
give
the no for theproposition
3.2.We prove
(3.21) by
induction on k.(i )
For k = 0 we have Li n R - L~.We denote
by j
the indexcorresponding
to R - .By
iterations of(3.1)
we get with the notations of lemma 3.4
But we have x; + 1 > x 1, ~ ..., so that
Therefore, by (3.14)
we getWe observe that
Therefore
hence
(ii )
Assume thatA~> -~2~+2. by (3.20)
we getTaking
into account that yo =,
X + 1 >0,
and(3 . 9)
we getX+l 1
But recall that
If
(X, y)
ECm
we have À m - 1 y > ~m -1 + ~,m - 2 + ... +1,
henceA > I > so
and this prove
(3 . 21 )
(4)
The lastpossibility
to be considered is We set n1 = ~2~+2~+/’+2~~+1~+~+1 1 thenUsing
the same arguments as in theproof
of(a)
we getand then we use remark 3.2 as before
This
completes
theproof
ofproposition
3.1.4. PROOFS OF THEOREMS 2 AND 3
We
keep
the same notations as before. If(~,,
recall that X=~,m > 1+ ~’
andB= -b
+A= -a .
m+
1~ m+ 1LEMMA 4.1. - Let
(03BB, ,)
ECm
and let K(03BB, )
=Ao
A 1... be aninfinite kneading
sequence. We know that there exists no such that___ . -,
Set c = -
BX
b.. = - AX
a
Then
c, d
arestrictly positive
and we havefor
n >_ 0:Ifs,+
= -1 thenthen
where
6~ (n) = 9i (n + no) - 6~ (no) (i =1, 2).
Proof. -
c and d arepositive by hypothesis,
and we prove the other assertionsby
induction.For
n = 0 it is obvious.Assume that the
inequalities
hold for n prove them for n + 1.( 1 )
Assume thatAn+no = L -,
thens~+~+i= -1.
Let I be the first index such that
Ano+n-l=R. By
themaximality
property of
kneading
sequences we haveBy
iterations of(3 .1 )
and
(3.2)
we getTherefore as - Àl = - X
(~ 1)
we obtain(2)
Assume that then + 1.Let I be the first index such that
(0~/~).
By
iterations of(3.1 )
and(3.2)
we getFirst,
we haveO aCno + n - l 1
so thatTherefore as B = À m we find
As X _ t we
have 1
~1
then~-’
For the other derivative we use
Hence
We know that x 1, ..., so
A I --...
Then
(3)
The two casesAn+no = R +
andAn+no = R -
are the same as in( 1 )
and
(2)
with 1= 0.This
completes
theproof
of lemma 4.1.Now recall that En =
(-1)~(")
where82 (n)
is the number ofsymbols
Ramong
If the
kneading
sequenceK (À, p)=AC
is finite we setEn = 0
forn>
~A~.
.PROPOSITION 4.1. - Assume that
(X, p)
ECm.
Thenfor
n >__2,
we haveeither or and
Moreover
for infinite kneading
sequences we haveProof -
In theproof
ofproposition (3.1 )
we have studied all thebeginnings
ofkneading
sequences inCm
up to the index no and we control thesign
of thederivatives an
andbn.
If
K (À, Jl)
is an infinitekneading
sequence inCm, by
themaximality
property,
there are at most m successivesymbols
L inK (À, Jl).
Henceand as
~~1
we find~,81 ~n~ ~,e2 ~n~ ? (ÀmJl)92
~"~. Recall that inCm
we haveÀ m J.1 >
1 henceby
lemma 4.1 we findProof of Theorem 2
(1)
Theregions ~m
are boundedby
thegraphs
of thedecreasing
functions
Hence if
(À, Jl), (~’, belong
to differentregions ~m
and if >(À, Jl)
we have m’ > m.
By
theorem 1 we deduce that Therefore we haveK (~,’, ~)>K(~ ~).
(2)
Assume that(À, ~), (~~
Thenby
theorem 1 we have(3)
Assume that(X, u), (~’, J.1’)
EBm
UCm
and(À’,
>(X, Jl).
As the
graphs bounding
theregions Bm
andCm
aredecreasing,
thereare three cases:
(1~ (À, (x" Bm
i(ii) (x, Bm
and(~1,’,
ECm
i(iii) tp (x’, Crn.
(i )
Assume that(x, (x’,
EBm By
theorem 1 we havewith
B,
B’primary.
By
lemma2.1,
the map F admits a m + 1-renormalisation 1 such thatwith 1
~~1’, j)
=(À m - 1 ).12,
À my) .
Notice
that ~ >
1>_ ~,
so À m-l 1. Hence maps theregion
B- into
As the map cpRLm -1 i is
strictly increasing by
theorem 1 of[M-V]
we haveB’ > B. The
application
RLm -1 * is alsostrictly increasing
so we get.
(ii)
Assume that(7~, j)
EBm
and(À’,
ECm.
Using
theorem 1 we find .(iii)
Assume that(À, Jl), (~/,
ECm.
If
(À, p)
ECm by proposition
4.1 we have for n >_ 2.En an > 0 and En
bn
> 0 ifEn =1=
0 and for infinitekneading
sequencesWe have the same results as in
[M-V] (proposition 3.6), , so
in the sameway we prove that
Proof of theorem 3
Let the function
y()
isgiven
in[M-V] (lemma 6.1).
( 1 )
Assume thatRecall that
(Jl) = 201420142014. If Jl>Y ()
we haveby
theorem 2Hence
by
the intermediate value theorem[M]
there exists(~
such thatY " K (#M (i), 1).
-
The
unicity
and thedecreasing
property of the function ~=?M
followfrom theorem 2.
-
If there exists such that and
then follows that lim = 0 . The functions
_
J.I. ~ 00
map into
]0, 1]. By
the sameproof
as in([M-V],
TheoremC),
we prove that the functionsP~
arecontinuous,
andby
theorem1,
theirs00
-
(2)
Assume that M then M = R*.By
the renormalisation construction(lemma 2.1 )
we have(K (À, Jl)
= M if andonly
if K(Jl2, Xp)
= B.As
~>~>1, by
theorem C of this isequivalent
toJl2
=[iB
or À= - 1 ~B 1 (Jl2).
-
Jl -
These functions are continuous and
decreasing
fromoo[ into ]0, 1] ]
and fill up the whole set
B1.
-
ACKNOWLEDGEMENTS
The authors thank M. Misiurewicz for useful conservations and
perti-
nent remarks about this article.
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(Manuscript received January 15, 1990;
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