Continuous measures in one dimension

Commun. Math. Phys. 122, 293-320 (1989) Communications in Mathematical

Physics © Springer-Verlag 1989

Continuous Measures in One Dimension*

Stewart D. Johnson

Mathematical Sciences Institute, Cornell University, Ithaca, NY 14853, USA

Abstract. Families of unimodal maps satisfying (1) T~:[--1,1]~--~['--1,1] with T ( + I ) = - I and ITj(1)I>I, (2) T~v(x ) is C 2 in x 2 and 2, and symmetric in x,

(3) T0(0)=0, TI(0)= 1 with d~ Tz(0)>0

are considered. The results of Guckenheimer (1982) are extended to show that there is a positive measure of 2 for which T)~ has a finite absolutely continuous invariant measure.

The appendix contains general theorems for the existence of such measures for some markov maps of the interval.

(A) Introduction

Jakobson published a theorem in 1978 that states that for the family of unimodal maps T~: [0, 1 ] ~ [0, 1] defined by T~(x) = 2x(1 - x ) , there exists a set of parameters 2 of positive lebesque measure for which Tx possesses a finite absolutely continuous invariant measure.

Since that time there have been several attempts to present a more compre- hendable proof. Rychlik (1986) has produced a cleaner proof than Jakobson using similar techniques. Benedicks and Carleson (1983) use statistical methods and a widey different approach to achieve a similar result. Rees (1985) has pre- sented a proof in the complex case.

In 1982 Guckenheimer proved that for reasonable families of unimodal maps, there is a positive measure set of parameter values for which the corresponding map has sensitivity to initial conditions (a map T has sensitivity to initial conditions if ~ > 0 Vx, 6 > 0 3y, I x - Yl < 6, 3n, [T"(x)-- T"(y)t > ~). Specifically, he

* Supported by the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell University

294 S.D. Johnson

considered the class of one parameter families of maps f# = {fx} which satisfy (t) T~ : [ - I , 1]~-->[-1,1] with T(_+I)= - 1 and IT)I(1)]>t, (2) T~.(x) is C z in x 2 and 2, and symmetric in x,

(3) To(0)=0, TI(0)= 1 with ~ T~(0)>0.

The idea of the proof is a simplification of Jakobson's approach, and relies on a recursive geometric construction. Since detailed estimates need only be carried out for one typical recursive step, such a construction provides a geometric skeleton for the calculations involved. This presents a conceptually simpler proof.

In this paper I demonstrate that for the same parameter values for which T~. was shown to have sensitivity to initial conditions, there is actually an absolutely continuous invariant measure for Tz.

I am indebted to the guidence of John Guckenheimer and to many constructive discussions with Don Ornstein and Yitzak Katznelson.

This proof differs from other proofs of Jakobson's theorem in two ways. Firstly, relying on the geometric construction to organize and simplify allows the reader to have a more clear picture of what is involved in the proof before (or without) going into the details. Secondly, the appendix contains several theorems for the existence of a finite absolutely continuous invariant measure for certain markov maps of the interval. These theorems are tailored to be more easily reached through the technique for inducing, thus the inducing steps can be simpler without losing the conclusion of the existence of an absolutely continuous invariant measure.

A brief description of Guckenheimer's method is as follows. First, the original unimodal map Tz for 2 near 1 is induced on an interval Jo around 0, producing a symmetric map T1, a with monotone branches and a folded piece over an interval J1 (see Fig. A.I).

This is best understood by letting h = Txlr-Xo and considering the preimages h-"(Jo). If Tx(0) e F e h-"(J0), then as 2 is varied so that T~(0) moves from one end of F to the other, the tip T1.4(0) of the induced map will move from the bottom to the top of the induced picture, and the number of monotone branches will remain the same although they will go through a continuous deformation. If 2 is chosen

J°

Fig. A.1

) f

J,

Continuous Measures in One Dimension 295

Fig. A.2

Jo

sufficiently close to 1, T1, g will have a finite number of monotone branches with a uniform bound on their linearity and I T;, hi > L > I, and the central folded branch can be made arbitrarily sharp,

The variables can be renormalized so that the interval Jo can be considered as [ - l , 1] and the tip TI,~.(0) moves from bottom to top as 2 moves from 0 to 1. Excluding from consideration those 2's for which Tl,z(O)eJ1, this map is then induced on the interval J1 producing a map Tz,~(x ) with countably infinite monotone branches and a folded piece over a central interval J2 (see Fig. A.2).

Again, this map can be understood by letting h:, ~ = Tl,~bo_ s~ and considering the preimages hZ,"~(JO. As 2 varies so that the tip 7"1, ~(0) moves from one end of a component F ~ h;"zl(J1) to the other, the tip T2,z(O) of the induced map will move from the bottom of its graph to the top, and the entire graph will go through a continuous deformation. The central folded piece of T2, z is a result of the central folded piece of T~, z being composed with the monotone branches of T1, z n: times. Thus if nl is very large the fold becomes quite sharp, since T~'x(0) increases exponentially in n 1. In this way the relative length of J2 is controlled by the duration of return time. The slope and distortion of the linear branches of T2,~ can be controlled by insisting that if T1, z(O) ¢ F ~ h~"~(J1) then T1,4(0) ties sufficiently far away from F so as not to distort the linearity of the corresponding branch.

Thus by making some careful parameter exclusions, the map T2,~ will have sufficiently linear and steep monotone branches, and a smoothly folded and sufficiently sharp central folded branch.

This map is renormalized, parameter values for which T2, 4(0) ~ Jz are excluded, and the map is induced on J2 producing a map T3, ~ and the recursion continues in this manner.

In order to arrive at a good final picture for this recursion process without excluding too many parameter values certain estimates must be carried through- out the process. This is done by establishing bounds which control the linearity and slope of the monotone pieces, the distortion and sharpness of the quadratic piece, the speed at which the tip moves, and the way the picture deforms with respect to changes in the parameter. Given a map which satisfies those bounds,

296 S.D. Johnson

new bounds are calculated for the next induced map. By excluding certain parameter values at each step Guckenheimer shows that the bounds remain finite or go to infinity at a controlled rate. By showing that proportionately fewer parameters can be excluded at each step, he concludes that the entire recursion can be done for a positive measure of parameter values.

One thing that must be shown in order to exclude proportionately fewer parameters at each step is that the central intervals are proportionately smaller,

l(Jk+ 1) l(Jk) that is ~ ~ l(Kk_ 1)" Letting Sk = Ti;z(0), there is a constant C depending on

the above mentioned bounds for which/(Jk) < CS[ 1/2 Since Sk behaves exponentially with respect to the duration of return time, parameters can be excluded so as to make return times sufficiently long to conclude that Sk + 1 > S~. Thus the Sk grow at a fast double exponential rate and central intervals are proportionately smaller.

The final picture is a sequence of nested intervals J0 C J t C... with the map T~, on Jk--Jk + 1 consisting of branches mapping over Jk with slopes IT', a[ > L > 1 and

I TY I ~,2 distortion ~ Jk - Jk ÷1 < 2kD"

From this picture it can be concluded that the original map has sensitivity to initial conditions.

For a more detailed exposition of these ideas, see Guckenheimer (1969, 1982).

(13) Assumptions, Definitions and Bounds

Definition. For a transformation T ~ C 2 the distortion of T at a point x is defined as

dis(T) (x) = T"(x) (T,(x))2 •

Then dis has the composition formulae:

1 dis(T1 o T2) (x) =< dis(T1) (Tdx)) + dis(T2) (x),

JT;(T2(x))[ n-1 1

dis(T n) (x) < ~ o (r~) ' (Tn-~(x)) dis(T) (T "-~- l(x)).

What is demonstrated by this last formula is that under repeated application of a map to a point, the distortion that is caused under any one application can be reduced through subsequent stretching. In the appendix of this paper it is shown that certain expanding markov maps with bounded distortion and arbitrarily short branches mapping onto neighborhoods of a central point c may possess an absolutely continuous invariant measure by virtnre of having a good ratio of longer branches near c.

It will be assumed that a parameter value 2 is given for which Guckenheimer's construction can be carried out indefinitely. It is assumed that at the i th step of the recursion there is a map T~: Ji~---~Ji consisting of monotone branches mapping onto Ji, and a central folded piece over a central interval Ji+ 1 (see Fig. B.1).

Fig. B.1 Hi I J~÷l

Definitions. Let H i = J i - Ji + 1, and define h i = Tiln, and f / = T/l j, ÷1 so that h i is the monotone branch part of T i and fi is the central folded piece (see Fig. B.1). Let si+ 1 = ff(0) (where 0 is the critical point of f~). To exploit the quadratic nature of fi let q(x) = x 2 and define Pi such that f i (x)= pi o q(x) for x s Ji+ 1.

It wilt be assumed that there exist positive bounds Do, L > 2, K1, K2, K3, K4, and K 5 such that for all i:

Bound 1. si + 1 ~> Sz~ '

Bound 2. dis(hi) < 2iDo,

Bound 3. [h'il > L ,

Bound 4. dis(pi)<K 1 ,

1 2p; Bound 5. ~ < < K 2 ,

Si+l

Bound 6. l(J i + 1) < K 3s7+1~ 2 ,

Bound 7. l(Ji+ 1) > K4si+ 1 ,

Bound 8. l(Ji+ 1) <KssF+l/4. l(Ji)

Bounds 1-4 are directly from Guckenheimer (1982), bounds 5 and 6 follow from bound 4, bound 7 is a consequence of a parameter exclusion in Gucken- heimer (1982), and bound 8 follows from bounds 6 and 7.

(C) An Outline of the Proof

Guckenheimer's recursive step can be viewed in terms of a stopping time ni(x). The preimages of J i+, in Ji under H i form a set of intervals NCJ~ such that

1 - 1 o n t o

hT:N ' Ji+ 1 for some n = 1, 2, 3 , . . . . These intervals fill up all of Ji, so it

298 S.D. Johnson

possible to define an integer valued function ni(x ) almost everywhere o n Ji+ 1 by ni(x) = n iff fi(x) s hi-"(Ji + 1).

This makes it possible to write Ti+ 1 explicitly in terms of T i as Ti+ 1 = hTi(x)f~(x) for x e Ji + 1. If F is the preimage of Ji+ 1 under hi that contains f~(0) and n i is defined as ni=ni(0), then fi+t(x)=h~fi(x ) for x e f i - l ( F ) , hence JI-I(F) will be the new central interval J~+2 of Ti + 1. Also, h~+ l(X) = hT'(x)fi(x) for x e Jr+ 1 - f i - I(F) •

1 Since dis(J)(x) behaves like - for x near 0, the parameter exclusions in

Si + 1X 2 Guckenheimer (1982) must be chosen so that ni is sufficiently large as to ensure bounds 1 and 4, and ni(x ) is sufficiently large at all x to ensure bounds 2 and 3.

The idea in this paper is to leave rc i and the definition of f i+t as is, but to redefine hi(X) slightly smaller so as to allow sufficient longer branches to satisfy the hypothesis of Theorem 4 in the appendix. The conclusion will be that a finite absolutely continuous invariant measure will exist for To. By restricting 2 sufficiently close to 1, T O will have a finite number of branches, thus producing a finite absolutely continuous invariant measure for Fa. The proof is outlined as follows.

Consider To on J0, where Jo is assumed to be a symmetric interval around 0 of length less than 1. Let N be the minimal partition of H0 with respect to which ho is continuous and monotone. Let N1 be the partition of J 0 consisting of N and J1. Let ~2 = ho 1(~1)v J1, so that for any M e ~2 either M = Jl , ho(M)= J1, or ho(M ) e Yt. In general, let N , = h o I(N,_ 1)v J1 so that for any M ~ N, either h~(M)= J1 for some k = 0, 1, ..., n - 1 or h~-I(M)e N. The partitions N, form a refining sequence, and since ~ , contains the sets hok(Jo) for k=0 ,1 . . . . , n - 1 , these partitions are increasingly dominated by preimages o f J 1. It is these partitions that will be used to define a new stopping time rather than the partition consisting entirely of preimages of J1.

Define zi sucht that ( - zl, z0 = J1 and consider fo on J1- The distortion dis(fo)

behaves like - - 1 for x near 0. For n = 0,1, 2 . . . . define x, such that dis(fo)(x) S1 X2

<L"-ID0 for x e ( - z ~ , - x . ) w ( x , , z O . It will be shown that x , ~ 0 like (stLn-IDo) -1/2 (If fo were quadratic with fd '=s l then x. would equal (sl L"- t Do) - 1/~.) Let B, = ( - x . _ 1, - x , )u (x , , x , _ 1). These sets B, will give a lower bound on how small the stopping time may be taken and still preserve a reasonable distortion.

As before, let F be the preimage of J1 which contains jo(0), and let J2 = fo-- 1(F) with z2 taken to be such that ( - z2, z2) = J2. Since no is such that h~°(F) = J1, then f l can be defined as f l ( x )= h~°fo(x) for x e J2-

Since fo will have some amount of distortion at zl, let bo be the minimal n such that x, < zl; that is, such that dis(fo)(z~)< L"-1D o. Let eo be the minimal n such that x, => z2; that is, such that dis(fo)(z2)< L"- 1D o. Redefine Bbo = ( - zl , Xbo)W(Xbo, z 0 and B~o=(-X~o_l , --Z2)k-)(Z 2, Xeo-1)- Now H I = J t - J 2 is partitioned into disjoint sets Bbo .. . . . B~o, such that dis(fo)lBo <E ' -1D o.

On H~ a stopping time t/(x) will be defined creating a new function g~(x) = h~o(X)fo(x) for x ~ Hr. This will be done in such a way as to satisfy three conditions:

Markov Conflition. Themapglmusteonsis to fmonotonebranehesmappingontoJo or J1 with a high proportion of them mapping onto Jo.

Linearity Condition. The stopping time q must be large enough to ensure that gl has reasonable distortion and good slope.

Return Condition. The stopping time 11 must be small enough to allow a finite measure to be pulled back through it via a Rohklin argument to another finite m e a s u r e .

In order to satisfy the markov condition, the sets B. will be moved slightly so that fo(B.) fits into the partition ~ ; that is, new sets ~,, will be defined such that the endpoints of the intervals fo(B.) will be division points in the partition ~ . for n = bo, ..., eo. This can be done in such a way that B. will still provide a lower bound for how large t /mus t be in order to satisfy the linearity condition, as well as

preserving the property that 1 (k=~+ , BkUJ2) is°fthe°rder(s~L"-lD°)-l/2which

allows t/ to be taken as small as is reasonably possible to satisfy the return condition.

The stopping time t/will be defined as follows: For x E/~,, consider the position of x in ~.. Either fo(x)ehok(J1) for k=O, l , . . . , n -1 or fo(x)eh~-l(M), where

1 - 1 onto M ~ is some interval for which h o : M 'Jo. If fo(x)~hok(J1) for k = 0, 1, ..., n - 1 let ~/(x) = k. Otherwise let ~/(x) = n.

Then gl(x)=h~o(X)fo(X ) is defined on H 1 and consists of monotone branches mapping onto Jo or J~ (see Fig. CA) and compare with Fig. A.2). A branch of g~ that maps onto J1 will be identical with some branch ofh~. Indeed for any x such that ga(x) ~ J~ it will be the case that gl = h I for some neighborhood ofx since both functions represent the first return of fo(x) to J1 under h 0. Consequently the distortion on the branches of go that map onto J1 is bounded by 2Do and their slope is greater than L by bounds 2 and 3 on h o.

On the other hand, ifx lies under a branch of go that maps onto J0, let n be such that x ~ B,, then dis(fo ) (x) n. Hence dis(h~- ~fo) (x) will be of the order Do by the composition rule for distortion and the fact that Ihbl > g. Furthermore, if x e B, and t/(x)> n, then x

Fig. C.1

I I

300 S .D . Johnson

will be of the order (slL"-1Do)-1/2 and g~(x)=(h~(X)fo(x))' (x) will be of the order L ~. 2sl. (siL"-1Do)-1/2> s~/4 assuming s 1 is large compared with L and D o.

For convenience, let go(x) = ho(x) for x ~ Ho = Jo - J1. Then the estimates for gl on Jo--J2 are compatible with the hypothesis of Theorem 4 in the appendix.

At the i th step of this process it will be assumed that there is a map gi defined on J o - J~+ 1 consisting of monotone branches each mapping onto one of the intervals J o , J 1 , - - - , Ji- Let ~* be the minimal partition with respect to which gi is continuous and monotone and for j = 0,1 . . . . . i let ~ be the collection of intervals in Ri over which g, has a branch mapping onto Jj. It will be assumed that ~ C J j, and that every branch of g~ in ~jc~Hj is identical with some branch of hi. The following bounds will be assumed for g,:

Bound a. For k < j s)/4 and Ig)l I~j~Jj > L.

A claim will be made that the D~ remain bounded in the recursion. A new map gi+ 1 will be defined for x e H~+ ~ by using a stopping rule ~h(x) and

defining g~+ 1 = gT*(x)f~(x) . This will be done in such a way that, firstly, g~+ 1 consists of monotone branches each mapping onto one of the intervals Jo,--., J~+ ~ with a large proportion of them mapping onto one of J0, --., J~, secondly, q~ must be large enough to ensure that g~+ ~ satisfies bounds a and b; and finally q~ must be small enough to allow a finite Rohktin pullback of a finite measure.

This process will utilize the following partitions; let ~ = V ~ v {Ji+ 1} and k=0

define ~ ,~=g?~(~_ t ) v {J~+~} for n = 2 , 3 . . . . . Then for any M e . ~ either g~(M) =J~+l for some k=0 ,1 . . . . , n - 1 or g ' ] - l ( M ) e ~ for some j = 0 , 1 .. . . . i. These partitions form a refining sequence and since ~ contains the sets gi-k(Ji+l) for k=O, 1 . . . . , n - 1 , they are increasingly dominated by preimages of Ji÷ 1.

The set Ji+ ~ - J , + 2 is then divided up into pairs of intervals B~ for n = b,, ..., e~

such that dis(g,)lB,<L"-a2'Do and l ( ~ Bin~Ji+ 2) is of the order \ k = n + l /

(s~+ ~L ~- ~2~Do) - 1/2. These sets will give a lower bound for qi. The sets B~ will be moved slightly into sets/~n such that f~(/3~) fits into the

partition ~ . This will be done in such a way tha t /~ will still provide a lower bound

for q~ and I B~kWJi+2 is still of the order (s~+ ~L"-~2iDo) - ~/2. k= 1

Then q~ will be defined as follows: For x e/3~, consider the position of f~(x) in N.~. Either f~(x) e g7 k(Ji+ ~) for some k = 0, 1 . . . . . n - 1 or f~(x) e g7 ("- 1)(M), where M is a component of N} for some j = 0,1, ..., i. If f~(x) e g7 k(j~ + ~ ) for k = 0,1, ..., n-- 1 let ~/~(x) = k, otherwise let r/i(x)= n.

Then gi+~(x)=gT'(~)f~(x) is defined on H~+~ and will consist of monotone branches mapping onto the intervals Jo, J~ . . . . , J~ + ~. Any branch ofg~+ ~ that maps onto J~+ ~ will be identical with some branch of h~+ ~, hence will have distortion bounded by 2 i+ ~Do and slope greater than L by bounds 2 and 3.

On the other hand, i fx lies under a branch ofg~+ a that maps onto Jk for some k < i + 1 (that is, x e ~ ,+ ~), let n be such that x e /~ . Then ~h(x) wilt be greater than n and x will be of the order (st + ~L " + ~ 2~D0)- 1/2, hence g~ + l(x)= (gT,(x)f/), (x) will be of

C o n t i n u o u s M e a s u r e s i n O n e D i m e n s i o n 301

the order L "'- 2s~+ 1" (si+ ~L"- 12~Do)- 1/2 > ~i+°1/41 using bound 1 and assuming that s o is large compared with L and D 0. The distortion will be shown to be bounded by something of the order 2*Do. This is done by analyzing the distortion formula

~(x) 1 dis(gi+~)(x)< E = ~ = 1 (g~- 1), (gT~X)-~ + lf~(x)) dis(g3 (g7 '~x~-~f/(x))

1 + (gT'~X)) ' (f/(x)) dis(f~)(x).

This formula demonstrates that under repeated application of the map gf, the distortion that is caused by f/ or any application of gf is reduced through subsequent stretching.

The first term of the sum, e = 1, contains the distortion ofg~ at g7 '~)- If/(x) and is already bounded by something of the order 2kDo since g7 'tx)- lf/(x)eNik. The last

1 term, ~gT,(~)) , (fi(x)) dis(f/)(x), is controlled by analyzing the slope of gi along the

trajectory f/(x), gif/(x),..., g7 ~ - lf/(x). In controlling the remaining terms, poten- tially large distortions of the order 2JDo withj > k are balanced against the amount of stretching that must occur on the type of trajectors that f/(x) must have in order to pick up such large distortions and then land on an ~ . A final bound of the order 2kD0 is established. This is the manner in which it is shown that g~+~ satisfies bounds a and b.

As mentioned before, the partitions ~ are increasingly dominated by preimages of J~+ 1 as n grows large. It is precisely these preimages that give rise to branches ofgi+ 1 that map into J~+ 1. It will be shown that Ji+ ~ is sufficiently small so as to make the domination occur so slowly that such branches have relatively small measure, something of the order s~+ ~ times the measure of H~÷ 1, which is sufficient for the hypothesis of Theorem 4 in the appendix.

Since points x e/~/are of the order (si ÷ 1L"-12iDo)-1/2, bound 7 can be used to establish that e i is bounded above by something of the order log s~÷ 2- For x ~ H~+ ~, t/i(x) is uniformly bounded by ei. With ni such that f/(0)E gF~'(Ji+ ~) the details of the construction will yield that ni<e v If m~+l(x ) is defined such that gi+l(X)

i

=To~'+~)(x) for x~Hi+ 1 then mi+ l(x)< [-I (e~+l). k=O

The limit of the recursive process will be a map goo : Jo~-*Jo which, satisfying the hypothesis of Theorem 4 in the appendix, posesses a finite absolutely continuous invariant measure #. Using a Rohklin argument this measure can be pulled back to an absolutely continuous invariant measure for To iff the integral

m~(x)d# is finite (where moo(x) is such that goo(x) = T~"°~(~)(x)). It will be shown that i --1

this integral is finite if the sum ~ I-[ (ek+ 1)-#(Hk) is finite. This sum converges i k=O

since ek_ 1 is bounded by something of the order logsk+l, and the results of Theorem 4 imply that ~(Hk) is bounded by something of the order Sk-~ r

302 S.D. Johnson

(D) Detailed Proof

1. The i th Recursive Step

1.a. Assumptions on gv To begin the recursion, let go = ho on H o. At the i th step of the recursion it will be assumed that there is a funct ion g~ defined on J o - J , + l consisting of branches ~ mapping onto the intervals Jk for k = 0, 1 , . . , i. It will be assumed that ~/k ~ Jk for k = 0,1 . . . . . i. The function gg will be assumed to satisfy bounds a and b. The functions hi on Hi and fi = p~(q(x)) [where q(x) = x 2] on Ji + 1 will be assumed to satisfy bounds 1 through 8. It will be assumed that the branches N~nJk of gi are identical with some branch of hk.

1.b. Distortion of f/. Let zi+l be such that J i+l=( - z i+ l , zi+l). Define x, = inf{x e Ji + 1, x > 0; dis(f/) (x) < L " - 12iDo}, and let b, = rain{n; x~, < z,+ 1}. Let B~, ~ ( _ _ Z i + I ~ i i B . - ( - - x . _ I , --x.)u(x. , 1)-For --Xb,)W(Xb~, zi+l) and for n>bi, let ~_ ~ i i i X n - convenience, redefine xib, = z i + 1.

Thus J~+~ is par t i t ioned into sets B~, such that dis(f/) over B~, is at most L,-12iDo"

Lemma 1, 3C 1 independent of i such that

1 ~ - (si + 1L ~- 12iDo)- 1/2 < x~ < Cl(si+ 1L"- 12iDo)- 1/2.

Proof

1 i - 2 _ _ " i " i " i t i (Xn) -- dis(q)(x,) > (dls(gi)(Xn)-- dls(pi)(q(x,)))pi(q(x.) )

1 1 >[L"-12iDo-Klfsi+l ~ > ~7~(s~+lL"-12iDo)

for some C] independent of i. Similarly,

½(x~)- 2 < (L"-I 2'D 0 + K1 ) s~ +12K2 < . . . . _ = t~ltsi+ 1L"- 12iDo)

tt ( ( C f l ~ 112 ) for some C1 independent of i. Let C 1 = m a x \ \ 2 ,] ' (2C~)1/2 " [ ]

Let ~ - ~ ' A , - f/(B,) and y~ = f(xi).

Lemma 2. ~ C 2 independent of i such that

1 C--2 (L"- 12iDo) -1 < lye-f/(0)[ < C2(L"-12iDo) -1

Proof.

i~ f/'(t)dt x~ 1(2 x~, If/~x~)--f/(O)[= = ! p~(q(t))q'(t)dt < si+l ! q'(t)dt

K2 - 2 si+l(x/")2< (C1)2(L~-12iD°)-I

Similar ly ,

1 ~2 1 ]fi(g~)--fin(O)l> - ~ 2 si+ l(xn) > 2K2(C1) g (Ln-12iD°)-l" []

i

1.c. The Partition. Define the partition ~ = V ~ v {Ji+ 1} and recursively define k = 0

~ / = g/-1(~/_ 1)v Ji + 1- Then {~}~= 1 is a refining sequence of partitions, and ~ / contains g:~k(ji ÷ 1) for k = 0,1 ... . . n - 1, hence these partitions increasingly appro- ximate the set of preimages of Jz+ 1 under g~.

Define re, to be the return time of f~(0) to J,+ 1 under gi; that is, gTf/(0) ¢ J,+ 1 for n < n i and g~'f/(0)~ Jz + 1. Let Ni be the component of gT~'(Ji+ 1) containing fi(0).

In order to ensure that g~+ 1 is markov, the sets A~ will be redefined so that they /I--1

fit into the partitions ~ . For n>b~, i fy~e ~ gi-k(Jf+l), then let y~ be such that k = 0

n - 1 n - 1 i ~i (y~, y~) C U g-k(j,+ ~) and lYe-- f~(0)[ is minimized. Ify~ ¢ ~ g-k(j,+ ~), then let M

k = O k=O

be the component of ~ + 1 containing y~ and let yi be such that (y~,i y~)-i E M and lye-f~(0)l is maximized (see Fig. D.1). It is quite possible for many of the y/~ to be identical.

Let a~ be the endpoint of N i covered by f/(Ji+ 1) and let a~ be the remaining endpoint. With ~i as defined, 3el such that - i _ i y~ - a 2 for n__> e i and yi ~ [a~, a/2] for n < ei. Let -~ - Yb,- 1 -- Yb,- 1, and redefine -i s o that -i _ i y~, y ~ - a l . Only the points 37b, - 1 . . . . , y~, will be considered in the following arguments.

For n -- bi, ..., e i let .4~ = (~_ 1, Y~)- The endpoints of ~ tie in the division points of the partition ~ , and ,4~ is divided up by that partition into intervals each of

y~

• ] yL 1 yh y~.l

1 ( 1 1

g-k(Ji+l) ...... fi (0)

>

yL1

I

y~

y~ y~.l

I ) I

M > Fig. D.1 fi (0 )

304 S.D, Johnson

which is either a component of fi-k(Ji+l) for k=0,1 ..... n - 1 or of

Lemma 3. 3C 3 independent o f i such that for bi < n < ei,

lY~ - A ( 0 ) I < CsL-C"-b'~I;~,-1 - f , ( 0 ) l .

Proof lyb,- ~ -f ,~0)l = lYe(z, + 1)-f~(O)l > C~(L ~'- ~2'Do) -1

by Lemma 2. By bound 6, l(Ji+i)<K3s~+~ 2, hence the maximum length of a component in ~ i is less than L- iK3s~+li/2 and the maximum length of a component in ~ c ~ J , is less than sZ i/4 for n < i. With the assumption that L- 1K3si-+~/2 < s:~ i/4

i an upper bound ofsi- 1/4 can be used for the lengths of components in ~/ ~ . Then

n=O either [y~ -- f~(0)[ < lY~ -- f~(0)l or IY~ - Y~{ < L -c"- ~s~- , 4 Hence

lY2 -f.<O)l < C2(Z ~- 12'Do)- ~ + L - ~"- " s ? , 4 < L-C.-b.~(C2(Lb,- ~2,Oo)- + L-Cb, - 1)Si- 1/4) < L-C,- b,)C3 If~(zi + 1) - - f/(0)l

for some C 3 independent of i, assuming s71/4 is small compared to C2(2iDo)- 1 []

Let x i-" =f~- i(yi)c~ [0, 1] and Bn-fi~i _ -l(An) .~~

Lemma 4. 3C~. independent o f i, such that for n=b~ . . . . . ei,

n-bi I ~ I < C 4 L 4 Iz~+d.

Proof

~" ~" p'l(q(t))q'(t)dt 1 s ¢.-.i~z IA(~L)-Z*(°)I= ! A '(t)dt = ~ > ~ '+lt'~") •

n--bi Hence by Lemma 3, 3C4 independent of i such that I~1 < C~L 2 Iz~+ 11. []

The following will allow the measure to be pulled back in a finite fashion at the final section of the proof.

Lemma 5. 3C 5 independent of i such that bi < Cs logsi+ 1 and ei < Cs logs~+ 2.

Proof. [xe,-it > [zi+2[ implies by Lemma 1 and bound 7 that

Cl(si+iLe~-22iDo)-l/2>JXe~_i[ i - i > [Zi+2[ >-~K4si+2,

hence

Lei-2 2C1 < s~+ 2 si+ i2iDoK4 < C's~+ 2

for some C' independent of i, thus ei<C' 5 logsi+l for some C~ independent of i. Similarly, b~ being the minimum such that ]x~,[ < [zi+ 11 yields

Cl(si+ xLb,-Z2iDo) - i/2 > [zi+ II > ½ K , s ~ ,

hence Lb,_ 1 <s2+ 1 2Cl 2C1

st + 12iDoK4 = st + 1 2iDoK----~¢ < C" si + 1

for some C" independent of i, thus b~ < C~ logs~+ 1 for some C; independent ofi. Let C5 = max(C;, C;). [ ]

1.d. Definition of gi+ i. Hi+ 1 is divided into sets/3~ for n = bi,..., ei and each/~i is further divided by f~-l(g~,~) into intervals. Take any M~/3i, nf~ - l(g~i) for any n = b~,..., ei. Then either fi(M) is a component ofgF k(Ji+ 1) for some k < n or fi(M) is

a component of g - " ~/ N} . For x ~ M define q~(x) = k if fi(M) is a component of \ j = 0

g[k(j~+ 1) for k < n and ~(x)= n otherwise. For x $ J i + l let gi+l(x)=g~(x) and for x E H i + l let gi+l(x)=gT'(~)fi(x ). For

x s Ji+ 2 let fi+ 2(x)= gT'fi(x), where ui is the return time of f/(0) to Ji+ ~ under gi- For x e J o - J ~ + 2 then, g~+ 1 consists of monotone branches mapping onto Jo,

J t , - . - , J~ + 1- For k = 0,1, ..., i + I, let ~ + 1 3 ~ be the collection of intervals that map onto Jk under an entire branch of gf + 1. Note that g~ had no branches that map onto Ji + 1 but gi+ i might, hence N[ + ~ C Ji + 1.

In the definition of ~~ ~~ ~ A,, either A, contained y,_ a or else Yn- 1 was in a component M of gi-~(Ji+ 1) for k < n - 1 in which case .4~ and M are disjoint and share a common endpoint (see Fig. D.2). Thus is established the essential fact that

N A 1

k ,k,,

Yd Y.~ 1 I '~,,

Y

Ai,

Fig. D.2

f

y~

I

y~

I x,

> fi ( 0 )

A~

Y

A~

>

fi (0)

306 S.D. Johnson

for x e B~, either th(x ) ~ n or ~/i(x) is the first return of f/(x) to Ji+, under gi- If rh(x ) is the first return, then gi+1,(x)=h~+l(x), and hence the distortion and slope is controlled by bounds 2 and 3. Otherwise the distortion and slope is the subject of the following calculations. 1.e. Verification of Recursive Bounds. Recall the recursive bounds a and b stated earlier

Bound a. For k <j__s)/4 and Ig)ll~j~jj>L. Proposition. Under the recursive definition of gi+ 1, bound a and bound b are satisfied in such a way that the D~ remain bounded.

Proof Bound b is satisfied by

Ig'i+ I (X) ] = ] ( g ~ f / ) ' (X)] = [g'y(piq(x))p~(q(x))q'(x)[

2 i 1.2 ,

i assuming that o1,/4-~c' r,- ,~Ynl/2 o s + i . . ~ . J l * X 2 ~ L., 0 ,

Bound a is more involved. Recall the distortion formula:

n(*) I dis(g~+ 0 (x) =< Z =~ (g~- 1), (g~(~) - ~ + 1,f/(x)) dis(gi) (g7 ~(~) -'f/(x))

1 + (g~'(*))' (f~x)) dis(f/)(x).

First it is possible that g~(x) all lie in NI~H~ for e = 0, 1 . . . . . t / - 1. Then

dis(gi+0(x)< ~ LJ-~_12'Do+ I . L + I i • =i ~/2-1'2iDo < L-~-_ 1 2 Do .

In particular this starts the recursion for i= 0. Otherwise one of g~f(x) for c~=0,1, . . . , t / -1 lies in N~nHi for some k s~/4, and so for the last term in the distortion formula,

1 - - dis(f/) (x) < s? 1'/4L- "- t . L ~- 1, 2iD o v < s71'/42iDo. gTZ(x)

With j taken to be such that gT- 1, e ~} (that is, x ~ ~}+ 1,), consideration of the sum from ~=1 to ~=~/ divides into two cases; either gT-1'f/(x)~l~c~H~ or gT-1 ' f / (x )~}nJ j+ 1 (note that if j = i then the latter possibility is vacuous).

Suppose that gT- *f/(x) ~ ~ n H j . Then for ~ = 1, dis(g~) (gT- ~f/(x)) < 2JDo and g'JgT-1'f/(x)) > L. For any ~ = 2, 3,..., t/consider the term

1 (g~' - ~)' (gT- ~ + ~f/(x)) dis(g~) (g7- ~f/(x)),

and let k be such that gT-%(x)e ~ .

If k < j then

1 ~-, , ,-~+1 dis(g')(gT-'fi(x)) < ~ 2ko' < ~ 2JD'"

(g, ) (g, f,(x))

If k > j then one of gT-" + lf~(x), ..., gT- 2fi(x) must land in ~ c ~ Jk for some m < k, because gT- ' e ~t~ implies g7 - '+ lf~(x)~ Jk and since gT-if/(x) e Hj with k > j the trajectory of g7-" + lfi(x) must bounce out Of Jk sometime. Then (g~- 1), (gT-,+ lfi(x)) > L~-Zs 1/4 making

1 1 (g~- 1), (gT-~+ lf~(x)) dis(g/) (gT-%(x)) < U - 2s; 1/4 2kDi < L-t~- 1)2JD,,

assuming that 2ks£ ~/4 < 2i. Under these circumstances then,

dis(g,+l)(X)<2JDo+ i_~12 ~ 2JD~+sT~/42'Do<2JDo + L l l 2JD,+sf-~/42'Do.

Suppose now that gT- %(x) ~ ~ n J , + ~ and let w > j be such that gT- ~f~(x) ~ ~ ! ?/--

~Hw. Then for 0~= I, dis(gi) (gT-~fi(x))<2'D~ and g,(g/ ~f~(x))>s~/4. For any c~ = 2, 3 ... . . t/-- 1, consider the term

1 (g~- 1), (g7-~+ l fi(x)) dis(g/)(g7-afi(x)),

and let k be such that g T - % ( x ) ~ . If k < j then

1 1 dis(g/) (g7- %(x)) < _ 2kDi < L- ~- 2)s~ 1/42JDi. (4 1)t (g7 +

l f / (x ) ) U 2S1{4

Ifk > w then one of g7 -~+ lfi(x),..., gT- 2f/(x) must land in ~ n J k for some m > k making

1 (g~ - 1), (gT- ~ + lf~(x)) dis(g,) (gT-'f~(x))

1 <( ~ S w 1[2S k l142kDi < L- (~- 2)S w lt42JDi ,

12-

assuming that Ls~ U42k < 2 j. If j < k < w then

1 1 . . . , - , + 1 . . . . dis(g/) (g7 - ~fi(x))

gi ) (gi Ji{,x).J

1 < U - 2S1~4 2kDi "< L- t~- 2)s~ 1/42WDi < L- (~- 2)s~ 1/S2iD,,

assuming that sU, 1/82w < 2 ~.

308 S.D. Johuson

Under these circumstances then,

dis(g~ +, ) (x) < 2iD~ + ~ L- (~- 2)s~, 1/82iDi + s~ ,/42~Do

< 2JDw + L ~ s~ x/S2JDi + s71/42~Do •

Thus for x ~ B / and "- 1 i gi fi(x) ~ it is shown that:

2JDo L + 1 L - 1

1 dis(gi + 1) (X) < max. 2JDo + ~ 2JDi + s 71/42iDo ".

Z i s " max 2JD~+ ~ s~ / 2~D,+sT '/22'D o

Thus for recursive bound a, define

Do L + 1 L - 1

1 Di+ 1 =max, Do+ -L--,~ Di + si-1/gDo

L max D~ + ~ s~ 1laD i + si- 1/4Dc w 0 and 0 < V < 1, define { i}i=o by the recursion

Pi+l = max {Pw+yw+lPi}. w ~ i

Then i

Pi+l=Po I] (1+~"+1) • n = 0

Proof. Assume that

for m < i, and check that

m - 1

Pz=Po 1-[ (I+Y "+1) n = O

i

Pm+T"+*Pi~Po I-I (1+7 "+1) n = O

for m =< i. By assumption,

m - 1 i - 1

Pm+ym+iPi=Po H (l+?~+1)+Tm+iPo H (l+y "+I) n = O n = O

=Po I] (I+7 "+* ) 1+'f '~+* (I+Y "+*) . 1l=0 ~ m

Hence it suffices to show that

( ifil ) ~i m lq-~) ra+l (1-1-~, n+l) ~ ( lq - 'yn+l ) , n=/Tl //=

or

i -1 t q • n=m

Let v-- log2 ~ - , and - l(rn+ 1) so that ½z<=jNz. Let

i -1 / ' F = 1-[ (gi+l) and i f=

n = m fi (1-~) 2r(m+l)) r=O

Note that ff > 1. Then

i - 1

''÷1) 13 (g,÷l) /3 =/,/I =(1 -7~÷l)r+7'+1r=(1 - ~m+ 1)(1 .~_ ~)m+ 1) (~ .~_ ~)2(m÷ 1))

x (t + 7 4('n + 1))...(1 + ~,2~(,n+ ~))/~ + 7~+ 1F

=(I +o]~)ff +y~+IF=F-7~F+?~+IF>_F> I. []

This concludes the proof of the proposition.

I f Density of Long Branches. In the definition ofgi + 1, points x e/3~ were assigned a stopping time of th(x ) = n unless g~fi(x)~ Ji + 1 for some k = 0, 1 . . . . , n - 1 . Thus the

n--1 subset of/~/, where th(x ) = n is given by ~i B , - U f i - lg i -k(J i+l ) -Sincep°intsxE/~,

k=0 where t/,(x) = n are exactly the points of ~J++ 1,1 this will be used to estimate the proportion of N~+ 1 branches in Ji+ 1 with k 0 independent ofi such that

l ( H i + 1 - ~ i + 1~ ~ i + 11 > 1 - - C7s[-q f l I .

Proof. Let r be maximal such that l(,=U+ 1B")>2si-+~/Sl(j'+l)" Then 1£~1

> sT+~81zi+ 11 (see Fig. D.3). Lemmas 4 and 5 then imply

2 r < ~ (~logs,+l + logC4)+b ,

2 1 < ~ (~logsi+ 1 + logC4)+ C5 logsi+ 1 < C logsg+ 1

for some C independent of i.

310 S.D. Johnson

B~ B~+I

,A,, A,

I I 1 1 x ~.~ x ~ 7~+,

"/Sl z I Fig. D.3 " S i+l i+l

~. l ( N - Ji + 1 ) Let M = U /3~ and N = f i ( J ~ + t - M ) = 0 A',. Note that

. = , + 1 ,, = b~ I ( N ) r--1

>l--2Kasi-+l~/2. Let U = U g~k(Yi+O. Then H i + ~ - - Y l ~ + ~ ) f i - l ( N - - U ) . k=O

L Using the composition rule for distortion it follows that dis(g~) (y) < 2~Di L - I

whenever g~ is defined at y. Thus if g~(x) and gT(Y) lie under the same branches of gi

g~'(x) < st/+sl by assuming that for n = 0, 1,..., k - 1 then ~k,(y-~

. ._~_ e l / 8 )>e2'D~L l oi+ 1

r - 1 To estimate l ( N - U ) , first remove di+l from N, then remove D gi-k(Ji+l)

k = l from whats left. This yields

I ( N - U)< l(N)(1 -- 2K3s~+~/2) (1 -- l(Ji+ 1)~ r - 1

i11/8

< I(N) (1 - 2K3&-+~2) (1 --el/8oi+l*xSOi+l ~ e-- l[4"lCl°gsi+l!

< l(N)(1 - C's[-+~4) °°g~`+~ +~

. , ~ t( v ) for some C' independent of i. This implies m a t / ~ < C'(logsi+ Os/-+'l/4 for some C"

independent of i. For x e Y i + 1 - M it follows that

1 1 t ~/s If'(x)l = tp;(q(x))q'(x)l > - ~ 2 s~ +~" 2(s~+~ 8 lz, +~ l) = K22 s,+~ lz,+~ l,

and similarly

hence

Ke 2 ]f/'(x)l < --2-- si + l Izi + 1t = K 2si + i ]gi + 11,

max f[(x) <(K2)Zst/+sl ~ , y ~ J , + , - M f ; ( y )

or equivalently,

max fi-~'(u) <(Kz)2S~/+ 8 . u, v e N f / - lt(V)

The sets H~ + ~ and ./~ + ~ - M are reasonably comparable:

l(Ji +1 - M) l(J~ + x) - 2sf-+~/Sl(ji + 1) > = 1 - 2 s T s ~ s .

l(Hi+ t) l(Ji+ 1)

Therefore,

lg Ll" d-hi+ . u , ~ + 1 -

l(Hi+ 1) l(J, +~ - M) l(H, +1 - ~21 + ~) > (1 - 2s~-+ ~/8) l(f~ - I(N - U))

= I(H,+ 1) l(f~-l(N)) l(fi-~(N))

> (1 - - 2si-+ ' / s ) (1 - (Kz)Zs~/+s 1C"(log s, + 1 )s/-+tl/4) ~> (1 - C7s?? 1)

for some C7, 6 > 0 independent of i. []

(2) The Invariant Measure

2.a. The Measure for go. The limit of this recursive process is a map g~ : Jo~--~Jo consisting of monotone branches ~ ; mapping onto intervals Jj. This map satisfies the bounds:

Bound aoo. dis(go0)[~ < 2kD~.

, .~1/4 and [g'[ [ ~ s j > L. Bound boo. For k <j, [g~lls~sj>-j This, along with Lemma 7, bounds I and 8, and the markov properties of goo, is

sufficient to conclude the existence of a finite absolutely continuous invariant measure # for go by Theorem 4 in the appendix. This also yields the property that #(Ji)<Gls~ -~ for some G 1, e>0.

2.b. Pulling # Back to To. The argument here is the typical Rohklin type reasoning with special care being taken to account for the map being many to one and for the use of a stopping rule rather than a return time.

Let ~ be the minimal partition with respect to which g~ is continuous and monotone, and for a component M e ~ define q~(M) so that g~(x)= T~=(M)(x) for x e M . Thus r/o~ is an integer valued function on ,~.

With # as the absolutely continuous invariant measure for g~, then/2 is defined for any measurable set E by

t/~(M)- 1 /~(E) = Z Z #(To"(EnTO"(M))c~M)

Me-~ n=0

312 S.D. Johnson

is an absolutely continuous invariant measure for T O as the following calculation indicates:

r /~(M)- 1

ft(To I(E)) = ~ ~ #(To ~(To I(E)n T~(M))c~M) M~.~ n = 0

T/~(M)- 1

= E E I~(To-(n+I)(E~Tg+I(M)) riM) Me.~ n=O

~l~(M)- 1

= Z Z p(T~(EnTg(M))nM) M ~ n = 0

+ E [/z(To~(U)( Ec~ T~o(M)(M)) nM)-#(EnM)] Me.~

= fi(E) + #(g ~o ~(enJo))-/~E) = fi(E).

Proposition. /~ is finite.

Proof. By definition, q ~ ( M ) - 1

/2(Jo)= ~ ~' #(M)= ~ qoo(M)#(M). M~,.~ n=O Me.,~

Consider the sets {B,}k=b, and the stopping time r/~ defined on the ith step of the recursion. It follows from the construction that max ~h(x)= e~. Thus if M e/~,

x~Hi + 1

then ~- 1 rl~(M)~ YI (ej+ l).(n+l).

j = 0

Hence, fi(Jo) = E q~(M)" #(M)< #(Ho)

M e..~

+ ~ ~ifll (ej+l)j. ~b(n+l)#(B~). i = 0 L j = o

From Theorem 5 in the appendix and the definition of # it follows that if U e Hi + 1, then #(U)

G #(Ji+ 1) ~(J~+l) or / ( u ) < l(U).

Hence, ei ei -- bi

(n+ 1)#(B/.)=bd~(Hi+ 1)+ ~ n#(/3~+.) n=b~ n=O

=b;#(H~+l)+ E # U /~ n=O j = b ~ + n

< bi#(Hi+ t) + Y, u2 ~ 1 U n=O \ j = b i + n f

#(Ji+l) e-i-b, _n <b'#(H'+a)+G2 l(J,+O ,=lZ C4L 21(j,+O

D/2 < b~(H~+ 1) + G2 C4#(Ji + 1) L . 2 - 1

<Cbi#(Ji+l)

C o n t i n u o u s M e a s u r e s in O n e D i m e n s i o n 313

for some constant C. Therefore,

/2(So) < #(Ho) + i=l < #(Ho) +

i=1

< #(Ho) + i = l

_;=-' filo (e j+ 1)]" [ Cbi#(Ji + 1)]

:;~_I~10 (ej-I-l)]'[CbiG4s:~-:a]. 3r 3e 3e 3~

This sum converges since by bound 1, s/-+E1 <s/+~ "si 4. s/_4 ...sl 4, hence

i-- 1 3e 3e i-- 2 3e

bisT-~l [I (ej+ 1)<(b/s/+S, )(si+, s (e,_, + 1)) l-I (sj+~(ej + 1)) j = o j = o

3~ 3e i - - 2 3e

<(C5 logs/+x. 8 s si+ 1 ) (si+ 1 (Cs logs/+ 1 + 1)) r l sj+~(c5 logsj+ 2 + i)), j = 0

and this product approaches 0 faster than an exponential rate. []

Appendix

In this appendix I give some background material on sufficient conditions for the existence of finite absolutely continuous invariant measures for some Markov maps on an interval.

Continuous Invariant Measures and the Folklore Theorem. The simplest non-trivial example of a continuous invariant measure is for a linear map consisting of branches that map onto an interval. Let the unit interval I be divided into a countable union of disjoint intervals, I = U Hi, and T:I ~--~I maps each Hi linearly onto I. Then for any measurable set E C I,

l(E) l(T- I(E))= ~/ l (T- l(E)nHi) = L l ~ l(H,) = l(E),

hence lebesgue measure is invariant. ~'(x)

If such a transformation is distorted in such a way that T"'~,) is uniformly

bounded on branches of T", then a continuous invariant measure is given by any Banach limit # = LIM l o T-": By the properties of the Banach limit, # is finite, positive, finitely additive, and invariant under T It remains to be shown that # is countably additive and continuous. Let 2 be the partition formed by {Hi} and 2"

r"',(x) = T-"(2). Suppose that ~ < B for Vn and Vx, y e J e 2". If F is a measurable

1 - 1 onto set in any J e 2" and E = T"(F), then T" : F ~ E, and so

I(E)= y IT"'ldl. F

314 S.D. Johnson

1 - 1 o n t o Tin(X) < B imply 1 l(I) l(I) ,. But T n" J .~ I and T"'(y) / ~ < I T"'(x)[ < n / ~ 1or x ~ J. Hence

o r

1 t(I) S dl< l(I) B I(J) F l(E) < B I~ ~ dI

1 I(E) l(F) l(E) l(I~ < I ~ < B l(I~"

Thus for any measurable set E and any J ~ ~",

1 l(E) l(T-"(E)nJ) _ ,. l(E) -B I(I-~ < - l(J) <.n ~ j .

Summing over all J ~ ~" yields

l(E) l(T-"(E))= • I(T-"(E)~J)< ~" B I(J)=BI(E).

J~.~n ~ j~,.~n l ~

Combining this with a similar argument yields

1 Bl(E) > l(T-"(E)) > ~ I(E).

1 Thus B" l(E) > #(E) > ~ l(E) implying that # is countably additive and continuous

with respect to lebesgue measure. Weiss, Flatto and Adler (see Adler's afterword to Bowen's paper [1979] for a

discussion of the somewhat nebulous origins of these ideas) used the notion of the

logarithmic derivative d~ log [T'I = T" T' to establish a bound on the quantity

log = k__Z ° log I Z ' ( Z ~ ( x ) ) l - l o g l Z ' ( Z k ( y ) ) l =

for x, y e J e 2". Thus if -~-T" is bounded by B and T is expanding, T' > L > 1, it can

be concluded that

T"'(x) ,,~1 T~(y) T"(u) du log ~ = k=O r~(~)T~

=k=O rk(x) =k=o -- L--1

This approach does not involve checking conditions for arbitrarily large iterates T"

of T. A similar conclusion can be reached if the quantity ( - ~ is bounded. This is a

weaker bound to establish and is independent of scale changes in the domain,

Ttt hence is a more desirable notion of distortion to work with. If < B, then a change of variables w = T(u) in the integration yields

l o g =

n - 1 Tkil(y ) T,,(T-I(w)) 1 dw <-B L = E T ' (T- l(w)) T ' (T - l(w)) - L----i- k = O Tk+l(x)

[where if w e (T k+ l(u), T k+ 1(@ then T-l(w) is taken to be in (Tk(x), Tk(y)).] These results plus a result on bernoullicity was finally stated by Adler [1975],

and due to its nebulous origins was called the Folklore theorem:

Folklore Theorem. Suppose T: I~--~ I where I is a union of disjoint intervals I = U Hi

such that T is C 2 over each interval H i and T: H i 1-1 o, to i" I f SB, L> 1 such that T "

IT'I> L and (T,) 2 <B, then there exists a finite absolutely continuous invariant

measure with respect to which T is bernoulli.

Generalization to Markov Maps. These results are readily generalized to certain Markov maps. Let T: I ~ I be a piecewise monotone C 2 transformation with base partition .~ formed by the intervals over which Tis monotone, and image partition

formed by T(~). Then T is Markov if ~ refines ~ , and is said to have a finite image partition if ~ is finite.

If T is linear over each interval in .~ then the set of measures that have constant densities over the intervals of ~ is a finite simplex mapped into itself under composition with T - 1 and thus contains a fixed point, i.e. an invariant measure with constant densities over each element in ~. ,, T J t~,~

Let .~" = T-"(.~), then T is said to be of bounded distortion if ~ L > 1. As before, T"

T will be of bounded distortion if Tis uniformly expanding and ~ is bounded.

The following theorem is a generalization of the folklore theorem and the proof is left for the reader.

Theorem 1. I f T is a uniformly expanding C z Markov map with finite image partition and bounded distortion then T has a finite absolutely continuous invariant measure.

Proof. Let # = L I M l o T-", then # is positive, shift invariant, and finitely additive. Let E be any measurable set. With {Ri} = ~ let E~=Ec~Ri. Fix i and let J be any component of ~" such that T"(J) = Ri. As in the proof of

the Folklore Theorem,

I l(Ei) l(T-"(Ei)c~S ) < < B l(Ei) B l(Ri) l(J) l(Ri)"

316 S.D. Johnson

Summing over such J's yields

1 l(Ei) I(T-"(E,)) l(E,) l(Ri~-- ~ < l(T_.(Ri)) < B l(Ri)"

Hence

1 I(T-"(R~)) I(T-"(Ri) ) l(E,) l(R,) < I(T-"(E,)) < BI(E,) l(Ri~)

There are only a finite number of R[s and the Banach limit is finitely additive. Hence I(T-"(I))= 1(1) implies that there is some nonempty collection J of i's such

that LIMI(T-"(Ri))>O for i e J . Let K I = mi nLIM ItT-"tRO)-''" and K2 i~J l(Ri)

= max LIM I(T-"(RI)). Then for i ~ J , i~J l(Ri)

g l ~ I(E~) < #(E,) < K2BI(E ~,

hence # is a finite absolutely continuous invariant measure. []

Markov Maps with Infinite Image Partition. If ~ is allowed to be infinite the difficulty arises that points may become absorbed into arbitrarily small portions of I. For example, take 0 < ~ < 1 and let T be the transformation on I = [0,1] that consists of linear branches mapping each interval [1 -~" , 1 - ~ " + 1) linearly onto [1 - a"- 1, 1) for n = 1, 2,... and mapping [0, 1 - ~) linearly onto [0, I) (see Fig. A1). This transformation has a finite absolutely continuous invariant measure iff a < 1/2. If ~ > 1/2 then high iterates of the map become increasingly dominated by arbitrarily short branches.

It is possible to take ~ < 1/2 and very close to 1/2 and perturb T in such a way that it is expanding, has bounded distortion, and yet will not have a finite continuous invariant measure. [The proof of Theorem 1 breaks down in this case in that it is possible for LIM l(T-"(Hi)) = 0 for all i.] Thus expanding maps of this type with bounded distortion can behave quite differently from their linear counterparts and the need for additional criteria is clear.

The remainder of this appendix is concerned with establishing sufficient conditions for the existence of a finite absolutely continuous invariant measure for some expanding markov maps of bounded distortion for which ~ is infinite. These criteria will be used in the main paper and are based on establishing an asymptotic bound on the ratio of longer to shorter branches.

Let T be a uniformly expanding Markov map with base partition 22 and image partition ~ . Let 6 z be the collection of intervals that are images of intervals in ,~. It

/ _ _ k

will be assumed that 6e={Si} is countable, with I ( ~ ) S i } ~ O as k ~ . Let \ -Z k

22" = T-"(22) and let 22~ be the collection of intervals in 22" which map onto Sk under a single branch of T". If 3e > 0 such that 1(22~) > e for arbitrarily large n and k and T

is transitive, then T will not have an absolutely continuous invariant measure. It is

thus necessary that 1 27 ~ 0 uniformly in n as k~oo for the existence of an i

absolutely continuous invariant measure. This turns out to be sufficient.

Theorem 2. I f T is a uniformly expanding C 2 Markov map with bounded distortion / \

and l ( U 27] ~ 0 uni form, inn as k ~ oo, then T has a finite absolutely continuous k i = k /

invariant measure.

Proof Let p = L I M l o T-" . Then # is finite, shift invariant, and finitely additive.

~ l(~) converges uniformly in n, so for measurable sets E, k=O

I(T-"(E))< ~ Bl(~,) l(Ec~sk) ,0 as I(E)~O ~ = o I(Sk)

uniformly in n. Hence # is countably additive and absolutely continuous. []

The property that l (Ui=k ~7) ~ 0 unifOrmly in n as k-* °e inv°lves arbitrarily

high iterates of T. If there are sufficiently good estimates on the ratios of longer to shorter branches for T, this property can be established by analyzing how short branches become longer and long branches become shorter under iterates of T

Pick an arbitrary integer k and call branches that map onto So, S~,..., Sk- long branches and branches that map onto Sk, Sk+ ~ .. . . short branches. Then

l(S~)

is the ratio of points in Si that lie under long branches of T. Since T"(~') = Si, and T is of bounded distortion, this is an estimate of the ratio of points in ~ that lie under long branches of T" + ~.

It will be hypothesised that there is a good ratio of points in S~ that lie under branches of T mapping onto So, S~ .. . . , Sk- ~. The ratio of long branches for T" + a will be estimated in terms of the ratio for T" and an asymptotic bound will be established.

Theorem 3. Suppose T is a uniformly expanding C 2 Markov map with bounded

distortion and 3Zk with ~ z k < oo such that k = l

l(Si) < zk

for i=0, l , . . . ,k+1. Then there exists a finite absolutely continuous invariant measure.

318 S.D. Johnson

Proof.

\ i=o : i=o l(Si) / k + l

> Z l(~7)(1-B~,) i = 0

> l 2~ (1 --BZk). \ i=o

Hence by repeated application,

( k ~ m X~ k+m

\i=o / \ i=o

and this quantity approaches 1 uniformly in m as k--* ~ , hence a finite absolutely continuous invariant measure exists for T by Theorem 2. []

The case where the S~ are nested, So D $1D .... and 21 C S~ so that the branches in St-St + 1 map onto St or larger arises naturally in repeated induction of certain unimodal maps. The following result holds.

Theorem 4. Suppose T is a uniformly expanding C z Markov map with bounded distortion and image partition c j = {St} ' where So D $1D ... and ~ C St. I f 3z < 1 such that

< z k and < z k, l(Sk) l(Sk + 1)

then there exists a finite absolutely continuous invariant measure I~ for T with I~(Sk+ 1) < Gt zk for some G 1 independent of k.

Proof. The measure exists by Theorem 3, and is a Banach limit of l o T-" ,

l(T-m(Sk+t)'< ,=~ Bl(S~+'l) l(~)q-I(i=k~)+l l(~i)

<n l(Sk+ l) , l~l~)-k)t(i:~)o2'[* ) + l (i=~+l ~? )

<Bzk + l(So)-l (S)o ~'~ )

< Bzk + l(So)-- l (ku~ ~m) ~Ok (1-- BzJ) , \ i = 0

hence

lim sup l(T-m(Sk+ i)) < Bzk + l(So) -- l(So) ~ (1 - Bz j) rn-'~ oo j = k

< B'ck q- I(So) (1-- jH_ k (1-- B'cJ) )

< Glz k

for some G, independent of k, since (1- S=kI~I (1--BzJ)) is d°minated by S=k ~zs which in turn is domina ted by z k. []

T" If the sum of I(Sm) is finite, then it is possible to allow ~ ) ~ on any b ranch of T

to vary according to the length of the b ranch and still preserve bounde d distortion.

..,Ttt Theorem 5. I f 3{%}, L > I such that iT' I >L, < am, and L %l(Sm)< 0% (T') 2 m=O then 3G 2 such that

sup T"'(x) x,y~M~" T"'(y) < G 2 .

Proof Fix any n and x, y e M e 2". Then for k = 0, 1 . . . . . n - 1, T is con t inuous and m o n o t o n e on [Tk(x), Tk(y)]. Let 7J,, = {k < n: Tk(x) e ~ } and k,, = max {k ~ ~,,}. Then

T"'(x)

,-1 T)(y) T"(t) _ i = Y~ loglZ'(Zk(x))l--loglZ'(Zk(Y))[ = ~ Y~ -Z-y~at

k=l m=O k ~ m Tk(x)

= ~ 2 Tk)~( , )T"(T- i (u) )du o~ m:O k~V'm T~+,(x)(T'(T-t(u))) 2 < rn=o ~ k~'1"m~ ¢Yml([Tk+l'(X)' Tk+I(Y)])

/ 1 \k,,-k _ L < m=OE k~l/tmE (TmtTSL) l(Sm) "¢~ .=0 ~ GmTs L--] ~(Sm)<°O" []

References

Adler, R.L.: F-expansions Revisited. Lecture Notes, Vol. 318, pp. 1-5. Berlin, Heidelberg, New York: Springer 1973

Adler, R.L: Continued fractions and bernoulli trials. In: Ergodic theory. Moser, J., Phillips, E., Varadhan, S., (eds.). Lecture Notes N.Y. Courant Inst. Math. Sci. 1975

Benedicks, M., Carleson, L.: On iterates of 1 - a e z on (-1,1). Ann. Math. (2), 122, t-25 (1983) Bowen, R.: Invariant measures for Markov maps of the interval. Commun. Math. Phys. 69, 1-17

(1979) Collet, P., Eckmann, J.-P.: Iterated maps on the interval as dynamical systems. Boston:

Birkh/iuser 1980 Collet, P., Eckmann, J.-P.: On the abundance of aperiodic behavior for maps on the interval.

Commun. Math. Phys. 73, 115-160 (1980) Collet, P., Eckmann, J.-P.: Positive Lyapunov exponents and absolute continuity for maps of the

interval. Univ. de Gen6ve, UGVA-DPT, 07-302 (1981) Coltet, P., Eckmann, J.-P., Lanford, O.E. III: Universal properties of maps on an interval.

Commun. Math. Phys. 76, 25%268 (1980) Feigenbaum, M.: Quantitative universality for a class of non-linear transformations. J. Star. Phys.

19, 25-52; 21, 669-709 (1978, 1979) Guckenheimer, J.: On bifurcations of maps of the interval. Invent. Math. 39, 165-178 (1977) Guckenheimer, J.: Sensitive dependence on initial conditions for one-dimensional maps.

Commun. Math. Phys. 70, 133-160 (1979)

320 S.D. Johnson

Guckenheimer, J.: The growth of topological entropy for one dimensional maps. Lecture Notes in Mathematic, Vol. 819. Berlin, Heidelberg, New York: Springer 1980

Guckenheimer, J.: Renormalization of one-dimensional mappings and strange attractors. Let~chetz Centennial Conference, Part III, 1984. Verjovsky, A. (ed.). Contemp. Math. 58, 143-160 (1982)

Jakobson, M.: Absolutly continuous invariant measures for one-parameter families of one- dimensional maps. Commun. Math. Phys. 81, 39-88 (1981)

Jonker, L.: Periodic orbits and kneading invariants. Proc. Lond. Math. Soc. 39, 428-450 (1979) Jonker, L., Rand, D.: Periodic orbits and entropy of certain maps of the unit interval. J. Lond.

Math. Soc. 22, t75-181 (1980) Lasotsa, A., Yorke, J.A.: On the existence of invariant measures for piecewise monotone

transformations. Trans. Am. Math. Soc. 86, 481-488 (1973) Ledrappier, F.: Some properties of absolutely continuous invariant measures of an interval. Erg.

Theory Dyn. Syst. 1, 77-93 (1981) Misiurewicz, M.: Structure of mappings of an interval with zero entropy. I.H.E.S. 78, 249-253

(1978) Misiurewicz, M.: Absolutely continuous measures for certain maps of an interval. I.H.E.S. 53,

17-52 (1981) Milnor, J., Thurston, W.: On iterated maps of the interval (preprint) (I 977) Ognev, A.: Metric properties of a class of maps of the interval. Mat. Zametki. (1981) Renyi, A.: Representations for real numbers and their ergodic properties. Acta. Math. Akad. Sci.

Hungary 8, 477-493 (1957) Rees, M.: Ergodic rational maps with dense critical point forward orbit. Erg. Theory Dyn. Syst. 4,

311-322 (1985) Ruelle, D.: Applications conservant une mesure absolument continue par rapport a dx sur [0, 1].

Commun. Math. Phys. tiff, 47-52 (1977) Rychlik, M.: Another proof of Jakobson's theorem and related results. Erg. Theory Dyn. Syst. 8,

93-110 (1986) Shaw, R.: Strange attractors, chaotic behavior, and Information Flow (preprint) (t978) Singer, D.: Stable orbits and bifurcations of maps of the interval. SIAM J. Appl. Math. 35, 260-267

(1978) Ulam, S.M., von Neumann, J.: On combinations of stochastic and deterministic processes. Bull.

Am. Math. Soc. 53, 1120 (1947)

Communicated by J.-P. Eckmann

Received January 14, 1987; in revised form August 22, 1988

Continuous measures in one dimension

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