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DISTORTION RESULTS AND INVARIANT CANTOR

arXiv:math/9211215v1 [math.DS] 17 Nov 1992DISTORTION RESULTS AND INVARIANT CANTORSETS OF UNIMODAL MAPSMarco MartensInstitute for Mathematical SciencesState University of New York at Stony BrookStony Brook USAAbstract. A distortion theory is developed for S−unimodal maps.

It will be usedto get some geometric understanding of invariant Cantor sets. In particular attract-ing Cantor sets turn out to have Lebesgue measure zero.

Furthermore the ergodicbehavior of S−unimodal maps is classified according to a distortion property, calledthe Markov-property.1. IntroductionThe work presented here originated in the question whether or not attracting Cantorsets of unimodal maps have Lebesgue measure zero.

This question led to a gen-eral S−unimodal distortion theory. As applications of this theory we got uniformproofs of the basic known ergodic properties: ergodicity, conservativity, existenceof attractors.

But also an answer to the original question.Theorem A. Cantor attractors of S−unimodal maps have Lebesgue measure zero.The strategy for studying invariant Cantor sets is constructing open, arbitrarilyfine and nested covers of them. These covers are constructed in such a way thatan invariance property appears: except for the component containing the criticalpoint every component is mapped monotonically onto its image which is also acomponent of the cover.

Finally all components are transported to the central one,1

2MARCO MARTENSthat is the one containing the critical point. The main question to be answered iswhether this transport has good distortion properties.In fact the covers of the Cantor sets are part of covers of the almost the wholeinterval, having the same invariance property.

S−unimodal maps having arbitrarilyfine covers with uniform good distortion properties are said to have the Weak-Markov-property.Theorem B. Every S−unimodal map not having periodic attractors has the Weak-Markov-Property.The tools developed for proving Theorem B are used to proof Theorem A.

The er-godicity of S−unimodal maps is a direct consequence of the Weak-Markov-Property.For understanding stronger ergodic properties we need a stronger distortion prop-erty.Using the Weak-Markov-Property we see that smaller and smaller intervals aretransported with uniform bounded distortion to smaller and smaller central inter-vals. For getting stronger ergodic properties we need to find almost everywheresmaller and smaller intervals transported with uniform bounded distortion to afixed big interval.

Maps having this distortion property are said to have the Markov-Property.Using this Markov-Property all basic ergodic properties can be proved in a uniformway: conservativity, existence of attractors, etc.The main application of the Markov-Property is an ergodic classification ofS−unimodal maps. In particular it can be used to classify the maps having anattracting Cantor set in the sense of Milnor ([Mi]).Theorem C. An S−unimodal map not having a periodic attractor has the Markov-

DISTORTION RESULTS AND INVARIANT CANTOR SETS OF UNIMODAL MAPS3property if and only if it doesn’t have a Cantor attractor.An appendix is added in which the basic notions of S−unimodal dynamics aredefined.The results presented here are taken from the authors thesis defended at the Tech-nical University of Delft in 1990.The author would like to thank the Instituto de Matematica Pura e Aplicada(IMPA) at Rio de Janeiro in which this work was done.2. Covers and induced mapsThe analytical properties of the construction presented here are based on the fol-lowing fundamental Lemma.

Its proof can be found in different places, for example[MMS].Lemma 2.1 (Koebe-Lemma). Let M, T be intervals in [0, 1] with M ⊂T .

Thecomponents of T \M are denoted by L and R. For every ǫ > 0 there exist δ > 0 andK > 0 such that the following holds. Let f : [0, 1] →[0, 1] be a map with negativeSchwarzian derivative.

If f n|Tis monotone and|f n(L)| ≥ǫ|f n(M)| and |f n(R)| ≥ǫ|f n(M)|then1)|Df n(x)||Df n(y)| ≤K for x, y ∈M(Koebe-Lemma);2) T contains a δ−scaled neighborhood of M(Macroscopic-Koebe-Lemma).Corollary 2.2. For every ρ > 0 there exists δ > 0 with the following property.

Let{Mi|i ≥1} be a pairwise disjoint collection of subintervals of the interval T . Fori ≥1 let Li and Ri be such that

4MARCO MARTENS1) Li is a component of T −∪j≥1Mj next to Mi;2) Ri is a component of T −Mi next to Mi;3) Li ∩Ri = ∅;4) |Li| ≥ρ|Mi| and Ri| ≥ρ|Mi|.If g : T ′ →T is monotone, onto and has Sg(x) < 0 for x ∈T ′ then for i ≥1|L′i| ≥δ|M ′i|where L′i and M ′i are the preimages under g of respectively Li and Mi.The Koebe-Lemma assures bounded distortion if there are big extensions of mono-tonicity on both sides.The next step is to develop topological instruments forstudying the maximal extensions.In this section we will fix an S−unimodal map f : [0, 1] →[0, 1] with critical pointc and we assume that f does not have periodic attractors.For x ∈[0, 1] denote the interval (x, τ(x)) by Vx (the involution τ is defined in theappendix). Furthermore defineN = {x ∈[0, c)|Vx ∩orb(x) = ∅}.The points in N are called nice.

Observe that every periodic orbit contains nicepoints. Hence N is not empty.

Moreover since the critical point is accumulated byperiodic orbits N also accumulates on c. Clearly N is closed. Fix x ∈N.Lemma 2.3.

For i = 1, 2 let Ti ⊂[0, 1] be two different intervals such that f ni :Ti →Vx is monotone and onto for some n1 ≤n2 . If T1 ∩T2 ̸= ∅then T2 ⊂T1 andn1 < n2.proof.

Suppose that ∂T1 ∩T2 ̸= ∅. Then n1 ̸= n2.

And x ∈f n1(T2) which impliesf n2−n1(x) ∈Vx. Contradiction.□

DISTORTION RESULTS AND INVARIANT CANTOR SETS OF UNIMODAL MAPS5The set of points who visit Vx is called Cx ⊂[0, 1]. Observe that it can be describedas being the union of all intervals T such that f n : T →Vx is monotone and ontofor some n ≥0.

Furthermore let Λx = [0, 1] −Cx and Dx = f −1(Cx) ∩Vx.Lemma 2.4. Let I ⊂Cx be a component.Then there exists n ≥0 such thatf n : I →Vx is monotone and onto.

Furthermore {I, f(I), . .

. , f n(I) = Vx} is apairwise disjoint collection.

In particular there exists only one such n ≥0.proof. Lemma 2.3 implies easily that for every component I of Cx there exists n ≥0such that f n : I →Vx is monotone and onto.

Observe that this number n ≥0 isdefined uniquely: c ∈f n(I) which implies that f n+s|I is not monotone for s > 0.To proof the disjointness of the orbit it suffices to proof that f j(I) ∩Vx = ∅for allj < n. Suppose f j(I) ∩Vx ̸= ∅with j < n. By Lemma 2.3 we get f j(I) ⊂Vx.Let j < n be minimal such that f j(I) ⊂Vx. Let H ⊂[0, 1] be the maximal intervalcontaining I with f j|H is monotone and f j(H) ⊂Vx.

Suppose that a component Lof H −I is mapped in Vx: f n(L) ⊂Vx. Then by maximality there exists s < j suchthat c ∈∂f s(L).

The minimality of j implies f s(I) ∩Vx = ∅. Hence x ∈f s(L).Because f j(L) ⊂Vx this implies orb(x) ∩Vx ̸= ∅.

Contradiction.So we conclude that f j : H →Vx is monotone and onto. Furthermore H containsthe component I of Cx.

The definition of Cx implies that H = I. In particularj = n. Contradiction.□Lemma 2.4 states the invariance property of the covers Cx discussed in the intro-duction.

It enables us to define the Transfer mapTx : Cx →Vxand also the Poincar´e mapRx : Dx →Vx

6MARCO MARTENSby Rx = Tx ◦f.The next lemma shows that these induced maps are defined almost everywhere.Lemma 2.5. If x ∈N then1) |Cx| = 1 (and |Λx| = 0);2) |Dx| = |Vx|.Furthermore3) Λx is invariant;4) if y ∈Λx is such that orb(y) ∩V x = ∅then Λx accumulates from both sides ony.proof.

Take y ∈Λx and assume f(y) /∈Λx, say f(y) ∈I ∈Cx. If c1 /∈I thenf −1(I) ⊂Cx.

Hence c1 ∈I. Because x ∈N we get easily that f −1(I) ⊂Vx.

Thisgives the contradiction y ∈Vx. So Λx is invariant.To prove the first two statements it suffices to proof |Λx| = 0.

Because Λx is aclosed 0-dimensional invariant set not containing the critical point a well-knownlemma in [Mi] states that Λx has Lebesgue measure zero.Take y ∈Λx with orb(y) ∩Vx = ∅. Suppose that y is not accumulated from bothsides by Λx.

Hence y is a boundary point of a component of Cx. Lemma 2.4 statesthat the orbit of y passes through the boundary of Vx.

Contradiction.□As a consequence of the previous lemma the setR = {x ∈[0, 1]|c ∈ω(x)}has Lebesgue measure 1.The branches of the Transfer map are monotone. To apply the Koebe-Lemma wehave to know how much the monotonicity can be extended.

DISTORTION RESULTS AND INVARIANT CANTOR SETS OF UNIMODAL MAPS7Suppose c1 ∈Cx.So there exists a component Sx ⊂Cx with c1 ∈Sx.Letψ(x) = ∂f −1(Sx) ∩[0, c) and define Ux = Vψ(x) = f −1(Sx).We get Sx ⊂(f(Vx), 1) because x is nice. Hence Ux = Vψ(x) ⊂Vx.

Using lemma2.4 we get orb(f(ψ(x))) ∩Vx = ∅. In particular ψ(x) ∈N.To finish the definition of the increasing function ψ : N →N let ψ(x) = x ifc1 ̸= Cx.The pair (Vx, Ux) is called a transfer range.

If Vx contains a δ−scaled neighborhoodof Ux then the pair is called a δ−transfer range.Proposition 2.6. Let x ∈N and I ⊂Cψ(x) be a component, say Tψ(x)|I = f n|I.Then there exists an interval TI containing I such thatf n : TI →Vxis monotone and onto.Combining the Koebe-Lemma with this proposition we get that the branches ofTψ(x) : Cψ(x) →Ux have uniformly bounded distortion.

The bound just dependson the space in Vx around Ux.A pair (T, I) of intervals with the property that for some n ≥0, called the transfertime,1) f n : T →Vx is monotone and onto;2) f n(I) = Uxis called TI-pair for (Vx, Ux). As in lemma 2.4 observe that the time n ≥0 isdefined uniquely.

As we saw above every TI-pair for a certain transfer range hasuniform bounded distortion on the middle part.proof of proposition 2.6. Let I be a component of Cψ(x), say Tψ(x)|I = f n|I, andlet H ⊂[0, 1] be the maximal interval containing I with f n|H is monotone and

8MARCO MARTENSf n(H) ⊂Vx. Suppose by contradiction that the component L of H −I is mappedin Vx: f n(L) ⊂Vx.

Then by maximality there exists j < n such that c ∈∂f j(L).We know from lemma 2.4 that f j(I) ∩Ux = ∅. Hence ψ(x) ∈f j(L).

Becausef n(L) ⊂Vx this implies orb(ψ(x)) ∩Vx ̸= ∅. Contradiction.□The intersection behavior of the TI-pairs is formulated inLemma 2.7.

For i = 1, 2 let (Ti, Ii) be TI-pairs for (Vx, Ux) with transfer timesrespectively n1 and n2. If T2 ∩T1 ̸= ∅and n2 ≥n1 thenT2 ⊂T1 −I1 or T2 ⊂I1and n2 > n1.proof.

From lemma 2.3 we get T2 ⊂T1 and n2 < n1. Suppose, by contradiction,T2 ∩∂I1 ̸= ∅.

Then ψ(x) ∈f n1(T2). Hence f n2−n1(ψ(x)) ∈Vx.

Contradiction.□We finish with the definition of the last type of induced maps. Let Ex be the unionof all intervals I which are part of a TI-pair (TI, I) for (Vx, Ux) with positive transfertime.Again lemma 2.7 implies that the connected components of Ex are intervals whichare part of a TI-pair for (Vx, Ux).

This defines the Markov mapMx : Ex →Ux.Directly after defining the first two induced maps we where able to show thatthey are defined almost everywhere. For Markov maps the situation is not thatsimple.

Section 4 will deal with the question whether or not the Markov mapsare defined almost everywhere. There we will show that Markov-maps are definedalmost everywhere only in the case of absence of Cantor attractors.

DISTORTION RESULTS AND INVARIANT CANTOR SETS OF UNIMODAL MAPS93. The Weak-Markov-PropertyIn this section we are going to prove a general distortion result for S−unimodalmaps, called the weak-Markov-property.Definition 3.1.

An S−unimodal map is said to satisfy the Weak-Markov-Propertyif there exist K > 0, a set Gw ⊂[0, 1] with full Lebesgue measure and a set D ⊂Naccumulating at the critical point such that for every x ∈Gw and for every y ∈Dthere exist t ≥0 and an interval I ∋x such thatf t : I →Vyis a diffeomorphism with distortion bounded by K.Theorem 3.2. Let f be an unimodal map without periodic attractor.

There existsδ > 0 and D ⊂N accumulating at the critical point such that for every x ∈D thefollowing property holds.If I is a component of Cx, say Tx|I = f n|I then f n maps the maximal intervalcontaining I on which f n is monotone over a δ−scaled neighborhood of Vx.The Koebe-lemma implies directly the main consequence of this Theorem:Theorem 3.3 (The Weak-Markov-Property). Every S−unimodal map with-out periodic attractor satisfies the Weak-Markov-Property.During the proof we will see that the number δ > 0 only depends on the power ofthe critical point and that the set D has a topological definition.During the proof of Theorem 3.3 we will see that in general the monotone extensionsof the branches f n : I →Ux with I ⊂Cx will have images much bigger than thetransfer range.

So Theorem 3.3 does not learn us something about the geometry oftransfer ranges.

10MARCO MARTENSHowever in our study of the Lebesgue measure of the critical limit sets we need abetter understanding of the geometry of transfer ranges. That is why we need thefollowing stronger form of Theorem 3.3.Theorem 3.4.

Let f be an only finitely renormalizable unimodal map having anon-periodic recurrent critical point. Then there exists a sequence {(Un, Vn)}n≥0 ofδ−transfer ranges with |Vn| →0.In this section we are going to prove Theorem 3.2 and 3.4.

Fix an S−unimodalmap f whose critical point c is recurrent. Furthermore we assume f not to haveperiodic attractors.Proposition 3.5.

There exist δ, ρ > 0 such that for x ∈N the following holds.1) Assume c ∈Rx(Ux). Let I ⊂Cψ(x) be a component, say Tψ(x)|I = f k|I, andT the maximal interval containing I for which f k is monotone.

Then f k(T )contains a δ−scaled neighborhood of Ux.2) Assume c /∈Rx(Ux) and |Vx| ≤(1 + ρ)|Ux|. If Tx|Sx = f n|Sx and if T is themaximal interval containing Sx such that f n|T is monotone then f n(T ) containsa δ−scaled neighborhood of [Rx(Ux), c].proof.

Let Tx|Sx = f n|Sx and M = f(Ux).We are going to study the or-bit {M, f(M), ..., f n(M)}. M is contained in Sx.

Hence by lemma 2.4 this or-bit consists of pairwise disjoint intervals.Choose m ∈{0, 1, 2, ..., n} such that|f m(M)| ≤|Df|20|f j(M)| for all j ≤n and f m(M) has neighbors on both sides.This means that both components of [0, 1] −f m(M) contains intervals of the formf i(M) with i ≤n. Observe that by taking m′ in such a way that f m′(M) is thesmallest one we can take m ∈{m′, m′ + 1, m′ + 2} having the described property.Let f l(M) and f r(M) be the direct neighbors of f m(M).

DISTORTION RESULTS AND INVARIANT CANTOR SETS OF UNIMODAL MAPS 11Let H be the maximal interval containing M for which f m|H is monotone andf m(H) ⊂[f l(M), f r(M)]. We claimf m(H) = [f l(M), f r(M)].Fix a component L of H −M and assume that f m(L) ⊂[f l(M), f r(M)].bymaximality there exists j ≤n with c ∈∂f j(L).

Because f j(M) ∩Ux = ∅we seethat f m−j−1(M) ⊂f m(L) ⊂[f l(M), f r(M)]. But m −j −1 ≤m −1.

Hence[f l(M), f r(M)] contains at least four intervals of the form f i(M), i ≤n. Thiscontradiction finish the proof of the claim.From the definition of m, l, r and the Macroscopic-Koebe-lemma we get a universalconstant δ1 > 0 (only depending on |Df|0) such that H contains a δ1−scaled neigh-borhood of M. Because the critical point of f is non-flat there exists a universalconstant δ2 > 0 such thatH′ = f −1(H) contains a δ2 −scaled neighborhood of Ux.proof of statement 1.

Assume that c ∈f n(M). Take a component I of Dψ(x), saywith transfer time k ≥0.

Denote by T the maximal interval containing I on whichf k is monotone and f k(T ) ⊂H′. We claim f k(T ) = H′ which proves statement 1.To prove this, fix a component L of T −I and suppose that f k(L) ⊂H′.

From themaximality of T we get j < k with c ∈∂f j(L). But f j(I)∩Ux = ∅.

Therefore f k(L)contains f k−j−1(M) in its closure which is contained in H′. Because f m+1 is mono-tone on the component of H′ −{c} which contains f k(L), the iterate f m+1+k−j−1maps the closure of M monotonically into the interior of [f l(M), f r(M)].

In par-ticular, because c ∈f n(M), we have k + m −j ≤n.Again this implies that[f l(M), f r(M)] contains at least four intervals of the form f i(M), i ≤n. Contra-diction.

12MARCO MARTENSproof of statement 2. Let T be the maximal interval containing Sx on which f n ismonotone.

The components of T −M are denoted by L, R, say c ∈f n(R). Becausef n(Sx) = Vx and c /∈f n(M) the interval f n(R) contains a component of Vx −{c}.So using the non-flatness of the critical point we get a constant δ3 > 0 such that|f n(R)| ≥δ3|[f n(M), c]|.So we only have to study f n(L).

We claimL′ ⊂f n(L)where L′ is the component of H′ −Ux which lies on the same side of c as f n(L).Once this claim is proved statement 2 follows by taking ρ > 0 sufficiently small.Indeed, suppose by contradiction f n(L) ⊂L′.Again the maximality of T as-sures the existence of j < n with c ∈∂f j(L). Observe f j(M) ∩Vx = ∅.

So weget f n−j−1(M) ⊂f n(L) ⊂L′.Consider the interval [f j(M), c] (if f j(M) andf n−j−1(M) are on the same side of c. Otherwise consider τ([f j(M), c]).The map f n−j : [f j(M), c] →[f n(M), f n−j−1(M)] is monotone and onto. Becausef doesn’t have a periodic attractor we getf j(M) ⊂(f n−j−1(M), f n(M)).Because f n−j−1(M) ⊂L′ and hence f n−j−1+m+1(M) ⊂[f l(M), f r(M)] we getn −j −1 + m + 1 > n. Hencej < m.Furthermore [f n−j−1(M), c] ⊂[L′, c] and f m+1|[L′, c] is monotone.

Because j < mthe map f j+1 : [f n−j−1(M), c] →[f j(M), f n(M)] is monotone and onto. Further-more we have [f j(M), f n(M)] ⊂[f n−j−1(M), c].

Hence f has a periodic attractor.Contradiction.□

DISTORTION RESULTS AND INVARIANT CANTOR SETS OF UNIMODAL MAPS 13Shortly speaking Proposition 3.5 states that the central branch of the Poincar´e mapRx has a quadratic shape.Corollary 3.6. Let f be an unimodal not having periodic attractors.

There existρ > 0 and K < ∞with the following properties. Let x ∈N and suppose Tx|Sx = f n.If |Vx| ≤(1 + ρ)|Ux| then1) For all y1, y2 ∈f(Ux)1K ≤|Df n(y1)||Df n(y2)| ≤K;2) For all y ∈Ux|Df n+1(y)| ≤K.proof.

The first statement follows directly by applying the Koebe-lemma and theprevious Proposition.To prove the second statement we define a point m ∈Ux.Let m = ψ(x) if|Df n+1(ψ(x))| ≤31−ρ.If |Df n+1(ψ(x))| >31−ρ let m be the closest point inUx to ψ(x) such that |Df n+1(m)| =31−ρ (because Df n+1(c) = 0 such an m exists).First we are going to show|[m, c]||[ψ(x), c]| ≥13.We may assume m ̸= ψ(x). Because f n+1(Ux) ⊂Vx we have|Vx| ≥|f n+1([ψ(x), m])| ≥31 −ρ|[ψ(x), m]|.Hence|[m, c]||[ψ(x), c]| = |[ψ(x), c]| −|[ψ(x), m]||[ψ(x), c]|≥≥1 −21 −ρ3|Vx||Ux| ≥1 −21 −ρ311 −ρ = 13.

14MARCO MARTENSBecause c is non-flat we get for all y ∈Ux|Df n+1(y)| = |Df n+1(y)||Df n+1(m)|.|Df n+1(m)| ≤|Df(y)||Df(m)|K31 −ρ ≤≤B |x −c|α|m −c|α ≤B |ψ(x) −c|α|m −c|α≤B3α = Awhere α is the order of the critical point and B a positive constant depending onρ, K and the behavior of f around the critical point.□Lemma 3.7. There exists δ > 0 such that for all x ∈N with1) c /∈Rx(Ux);2) Rx(c) ∈Vx −UxVx contains a δ−scaled neighborhood of Ux.proof.

Let δ > 0 be given by proposition 3.5 and denote the part of the δ2−scaledneighborhood of [Rx(Ux), c] which lies on the same side of c as Rx(Ux) by H′. Thepair of intervals P ⊂Q is such that1) f(Ux) ⊂P and P ⊂Sx;2) f n(P) = [Rx(Ux), c];3) f n : Q →[H′, c) is monotone and onto (where Rx|Ux = f n+1|Ux);Proposition 3.5 implies that such intervals exist.

Let |Vx| = (1 + ρ)|Ux| and assumethat ρ is small. Furthermore the Koebe-Lemma tells us that f n|Q has boundeddistortion.

We have to show that ρ is not too small. Observe that |Rx(Ux)| = O(ρ).This follows from assumption 1) and 2).Consider U = f −1(Q) as a neighborhood of Ux.

Using the bounded distortion off n|Q and the fact that the critical point of f is of order α it is easy to show thatU is a O(ρ−1α )−scaled neighborhood of Ux. This bound implies that for ρ smallH′ ⊂U.

DISTORTION RESULTS AND INVARIANT CANTOR SETS OF UNIMODAL MAPS 15Suppose that ρ is sufficiently small to assure that U will contain H′. Now let H bethe component of U −{c} containing Rx(Ux).

Thenf n+1 : H →H′ ⊂His monotone. Hence f has a periodic attractor.

Contradiction: ρ is away fromzero.□Lemma 3.8. There exist ρ, δ > 0 such that for all x ∈N with1) |Vx| ≤(1 + ρ)|Ux|;2) there exists p ∈Ux with Rx(p) = p and Vp ⊂Rx(Vp)Vp contains a δ−scaled neighborhood of Up.proof.

Observe that this periodic point is nice: p ∈N. Let M ⊂Vp be the maximalinterval with Rx(M) ∩Vp = ∅.

Easily we get Up ⊂M. Let ρ be given by corollary3.6.

Then we get a universal bound on DRx|Ux: |DRx(y)| ≤A for all y ∈Vp. Thisimplies that both components of Vp −M are bigger than 1A|Vp|.

Hence|Up| ≤|M| ≤(1 −2A)|Vp|.Let δ = 1A.□proof of Theorem 3.2. If there exists a neighborhood V of c such that V ∩orb{c} = ∅then let D = N ∩V .

The theorem follows from the fact that the boundary of theimages of the maximal intervals of monotonicity consists of critical values, pointsin the orbit of c. Hence we may assume that the critical point is recurrent.First we are going to define the sequence of closest approach to c. Let cn = f n(c)and defineq(1) = 1;q(n + 1) = min{t ∈N|ct ∈Vcq(n)}.

16MARCO MARTENSBecause c ∈ω(c) and c is not periodic the sequence {q(n)}n≥0 is well defined.Since c is an accumulation point of N there are infinitely many n ≥1 for which(Vcq(n−1) −Vcq(n)) ∩N ̸= ∅. For those n ≥1 definex(n) = sup{(Vcq(n−1) −Vcq(n)) ∩N ∩[0, c)}.Observe that the points cq(n) are not in N: the supremum is in fact a maximum.Furthermore x(n) ∈(Vcq(n−1) −Vcq(n) ∩N) .

Hence for i < q(n) f i(c) /∈Vx(n) andf q(n)(c) ∈Vcq(n) ⊂Vx(n). In particular Rx(n)|Ux(n) = f q(n).Let D = {ψ(x(n))}.

We distinguish two cases.Low Case. c /∈Rx(n)(Ux(n)).

Because f doesn’t have periodic attractors and c /∈Rx(n)(Ux(n)) we get ψ(x(n)) > x(n). Hence, by using the definition of x(n) we getψ(x(n)) ∈Vcq(n) or Ux(n) ⊂Vq(n).

So we can apply Lemma 3.7: Vx(n) is a δ−scaledneighborhood of Ux(n). Use proposition 2.6, the fundamental property of transferranges, to finish the proof.High Case.c ∈Rx(n)(Ux(n)).

The statement follows directly from proposition3.5.□proof of theorem 3.4. The transfer ranges will be defined in terms of the pointsy(n) ∈N.

These points are an adjustment of the points x(n) from the proof oftheorem 3.2 and will be defined below. Let δ > 0 be the minimum of the numbersδ given by lemma 3.7 and 3.8.

We distinguish two cases.Low Case. c /∈Rx(n)(Ux(n)).

The definition of x(n) implies that Rx(n)(c) = cq(n) /∈Ux(n). Hence we can apply Lemma 3.7: Vx(n) is a δ−scaled neighborhood of Ux(n).Let y(n) = x(n).

DISTORTION RESULTS AND INVARIANT CANTOR SETS OF UNIMODAL MAPS 17High Case.c ∈Rx(n)(Ux(n)). Let ρ > 0 be given by lemma 3.8.If |Vx(n)| ≥(1 + ρ)|Ux(n)| then let y(n) = x(n).

Assume that this metrical property doesn’thold. The map Rx(n)|Ux(n) has a periodic point p ∈Ux(n).

Because f is only finitelyrenormalizable we may assume (by taking n large enough) that Vp ⊂Rx(n)(Vp).Let y(n) = p and apply lemma 3.8.Because cq(n) →c it follows that y(n) →c.□Let us finish this section with a simple consequence of the Weak-Markov-Property.It implies directly the ergodicity.Lemma 3.9. Let f be a S−unimodal map without periodic attractor and D ⊂[0, 1]the set given by the Weak-Markov-Property (Theorem 3.3).If X ⊂[0, 1] is aninvariant set with positive Lebesgue measure thenlimD∋x→c|X ∩Vx||Vx|= 1.Proof.

Using lemma 2.5(1) and |X| > 0 we get a density point y ∈X of X withy ∈Cx for x ∈D. For every x ∈D let Ix be the component of Cx containing y.Say f nx : Ix →Vx is diffeomorphic with distortion bounded by K. Furthermorethe contraction principle implies |Ix| →0 if D ∋x →c.

ConsiderlimD∋x→c|Xc ∩Vx||Vx|≤limD∋x→c|Tx(Xc ∩Ix)||Tx(Ix)|=limD∋x→cRXc∩Ix |DTx(t)|dt|DTx(βn)||Ix|for some βn ∈In. ThuslimD∋x→c|Xc ∩Vx||Vx|≤limD∋x→cK |Xc ∩Ix||Ix|= 0.

18MARCO MARTENSThis finishes the proof.□4. The Markov-PropertyIn this section we will study a second distortion result, called the Markov-Property.This property is much stronger than the Weak-Markov-Property and serves forstudying more complicated ergodic properties.Definition 4.1.

An S−unimodal map is said to satisfy the Markov-Property ifthere exists a set G ⊂[0, 1] with full Lebesgue measure and y ∈N such that forevery x ∈G there exist a sequence of TI-pairs (Tn, In) for the transfer range (Vy, Uy)with x ∈In and |In| →0.In particular, if Uy ̸= Vy, the mapsIn →Uyare diffeomorphisms with uniform bounded distortion.We can reformulate the Markov-Property in terms of Markov-maps:Lemma 4.2. An unimodal map f has the Markov-Property iffthere exists x ∈Nsuch that the Markov map Mx : Ex →Ux has full domain:|Ex| = 1.Proposition 4.3.

Let f be an S−unimodal map without periodic attractor. If thelimit set of the critical point is not minimal then f has the Markov-property.Let us prove this proposition for the S−unimodal map f satisfying the two condi-tions of the proposition.

In particular f is not infinitely renormalisable.

DISTORTION RESULTS AND INVARIANT CANTOR SETS OF UNIMODAL MAPS 19Let us first deal with the case when f is Misiurewicz, c /∈ω(c). Take x ∈N suchthat orb(c1) ∩Vx = ∅.

Then c1 /∈Cx and ψ(x) = x. In this situation it is easy toshow thatEx = (Cx −Vx) ∪(f −1(Cx) ∩Vx).This implies, by using lemma 2.5, that Mx has full domain: |Ex| = 1.In the sequel we will assume that the critical point c is recurrent.Lemma 4.4.

There exists x ∈N and a sequence of intervals Kn, n ≥1 such thatfor n ≥11) There are no renormalisations possible in Vx. In particular ψ(x) > x;2) ∂Kn ⊂Λx and Kn ∩Vx = ∅;3) Kn ∩ω(c) ̸= ∅;4) |Kn| →0.proof.

Let q ∈ω(c) be such that c /∈ω(q). This is possible because ω(c) is notminimal.

Choose x ∈N such thatorb(q) ∩Vx = ∅.and such that there are no renormalisations possible in Vx. Then q ∈Λx and q isaccumulated from both sides by Λx (see lemma 2.5).

Hence we can take a sequenceof intervals q ∈Kn with |Kn| →0 and ∂Kn ⊂Λx accumulating at q ∈ω(c).□Let x ∈N and the intervals Kn be given by lemma 4.4. Because Kn ∩ω(c) ̸= ∅there exists a sequence tn →∞with cj /∈Kn for j < tn and ctn ∈Kn.

We claimthat for every n ≥1 there exists an interval K′n containing c1 such thatf tn−1 : K′n →Kn

20MARCO MARTENSis monotone and onto.Indeed let K′n be the maximal interval containing c1with f tn−1|K′n is monotone and f tn−1(K′n) ⊂Kn. Assume by contradiction thatf tn−1(L′n) ⊂Kn where L′n is a component of K′n −{c1}.

Then the maximality ofK′n implies the existence of j < tn −1 such that c ∈∂f j(L′n). So f tn−1−j(c) ∈Kn.Contradicting the fact that ctn is the first critical value in Kn.

This proves theclaim.We are going to use the components of Cψ(x), which belongs to TI-pairs, in Knto show that the domain of the Markov map Mx is has positive upper density inc. This is done by pulling back the TI-pairs into K′n, close to c1.

And then onestep more to get them close to c. However we have to be careful. The whole TI-pairs have to be pulled back.

The collectionˆMn defined below will contain thecomponents for which this is impossible.The statements in the lemma below follow directly from respectively proposition2.6, the definition of Cx and lemma 2.5. We will leave the the proofs to the reader.Lemma 4.5.

If I is a component of Cψ(x) with I ∩Kn ̸= ∅then there exists aninterval T containing I such that1) (T, I) is a TI-pair for (Vx, Ux);2) T ⊂Kn.Furthermore|Cψ(x) ∩Kn| = |Kn|.The map f tn−1 : K′n →Kn is monotone and differentiable hence we can pull backthe TI-pairs in Kn into K′n and form the pairwise disjoint collection:In = {I ⊂K′n|f tn−1(I) is a component of Cψ(x) ∩Kn}.

DISTORTION RESULTS AND INVARIANT CANTOR SETS OF UNIMODAL MAPS 21From the previous lemma we get for every I ∈In an interval TI ⊂K′n such that(TI, I) is a TI-pair for (Vx, Ux). Furthermore | ∪In| = |K′n|.LetˆMn consists of I ∈In with1) I ∩[0, c1] ̸= ∅;2) c1 ∈TI.From lemma 2.7 we know that the collection {TI|I ∈ˆMn} consists of nestedintervals.

In particular we can count ˆMn = { ˆMi|i ≥0} such that T ˆMi+1 ⊂T ˆMi−ˆMi.Denote by ˆLi the component of K′n −∪i≥1 ˆMi next to ˆMi such that c1 /∈[0, ˆLi].Lemma 4.6. If I ∩ˆLi ̸= ∅for some i ≥1 and I ∈In then TI ⊂ˆLi.Furthermore there exists ˆρ > 0 such that |ˆLi| ≥ˆρ| ˆMi| for all i ≥0.proof.

The first statement follows directly from lemma 2.7. The Koebe-Lemmagives ˆρ > 0 such that |TI| contains a ˆρ−scaled neighborhood of I where (TI, I) isa TI-pair for (Vx, Ux).

Again the statement follows from lemma 2.7: ˆLi contains acomponent of T ˆMi −ˆMi.□Let Tn be the symmetric interval f −1(K′n). Observe that ∪i≥0 ˆLi ⊂f(Tn).

SodefineEn = f −1(∪i≥0 ˆLi).from the previous lemma we getLemma 4.7. Every y ∈En has a TI-pair for (Vx, Ux).

Furthermore this TI-pairis inside Tn.The next step is to show that En ⊂Tn has some universal metrical properties.Denote the components of Tn −En by Mn = f −1( ˆMn). Again Mn is countable,

22MARCO MARTENSsay Mn = {Mi|i ≥0}. If c ∈∪Mn then the corresponding component is M0.

Ifnot we define M0 = ∅.For i ≥1 let Li be the component of Tn −∪i≥0Mi next to Mi but not between Miand c. Furthermore let Ri be the component of Tn −Mi containing c. If M0 ̸= ∅define R0 and L0 to be the components of Tn −∪i≥0Mi next to M0.Lemma 4.8. There exists ρ > 0 such that for i ≥0|Li| ≥ρ|Mi| and |Ri| ≥ρ|Mi|.This bound is independent of n ≥1.proof.

For i ≥1 Ri contains one component of Tn −{c}. From the non-flatnessof the critical and the fact that Tn is a symmetric interval we get ρ1 > 0 with|Ri| ≥ρ1|Mi|.Now Li ∪Mi is mapped monotonically onto ˆMj ∪ˆLj for some j ≥1.

The spacegiven by lemma 4.5 can be pulled back without being distorted to much becausethe non-flatness of the critical point. Hence we get a constant ρ2 > 0, dependingon the behavior of the critical point such that|Li| ≥ρ2|Mi|.For i = 0 the situation is even better: L0 and R0 are mapped onto some ˆLj and M0into the corresponding ˆMj.

Again the non-flatness allow us to pull back space.□proof of proposition 4.3. Let R be the set of points whose orbit accumulates onto c.In section 2 we saw |R| = 1.

We are going to show that the domain of the Markovmap Mx has a positive upper density in every point y ∈R. From |R| = 1 it followsthat |Ex| = 1.

DISTORTION RESULTS AND INVARIANT CANTOR SETS OF UNIMODAL MAPS 23Take y ∈R and n ≥1. The first time the orbit of y hits Tn is denoted by sn.

LetHn be the maximal interval containing y such that f sn maps Hn monotonically intoTn. Before we saw that there are no renormalisations possible inside Ux.

Hence theLemma from the appendix givesf sn(Hn) = Tn.Now let An ⊂Hn be the preimage of En. By using corollary 2.2 we get a universalconstant ρ > 0 such that|An| ≥ρ|Hn|.The last observation to be made is An ⊂Ex.

The TI-pairs in En are part of Tnhence they can be pulled back into Hn. So every point in An has a TI-pair henceis in Ex.Thus the upper density of Ex in y is bigger than ρ.□Let us finish this section with the fundamental distortion property which holds forMarkov maps.Fix x ∈N and consider the Markov map Mx : Ex →Ux.

Let the collection B0consists of the branches of Mx. That is it consists of the components of Ex.

Defineinductively the collections of branches of M nx as follows. An interval I ⊂Ux is inBn+1 if Mx(I) ∈Bn.proposition 4.9.

If f satisfies the Markov-property, say Mx : Ex →Ux has fulldomain, then there exists K > 0 such that1K ≤|DM nx (y1)||DM nx (y2)| ≤Kfor y1, y2 ∈I ∈Bn and n ≥1.

24MARCO MARTENSProof. Suppose that f also satisfies the Misiurewicz property, that is, the criticalpoint is not recurrent.

In this case we may assume that the critical orbit does notintersect Vx. Then it is easy to see that the branches of M nx have essentially biggermonotone extensions.

Using the Koebe-Lemma the proposition can be proved.In the other case, the critical orbit is recurrent, we also find easily monotone ex-tensions. Indeed, using lemma 2.7 it is easy to see that every branch of M nx has amonotone extension to Vx.

Again the Koebe-Lemma finishes the proof.□Corollary 4.10. If f satisfies the Markov-property then every 0-dimensional closedinvariant set has Lebesgue measure zero.Proof.

Let X ⊂[0, 1] be a 0-dimensional closed invariant set and let y ∈N be suchthat the Markov-Property holds for the transfer range (Vy, Uy).Suppose |X| > 0. Then there exists a density point x of X with x ∈X ∩G.

Let{(Tn, In)}n>0 be the TI-pairs for (Vy, Uy) given by the Markov-Property. As in theproof of lemma 3.9 we showlimn→∞|Xc ∩Ux||Ux|≤limn→∞K |Xc ∩In||In|= 0.Hence, because X is closed, Ux ⊂X.

Contradiction.□5. The Ergodic Properties of Unimodal mapsIn this section the distortion results from the previous sections will be used togive a measure theoretical description of S−unimodal dynamics.First we willuse the developed technics to prove two fundamental theorems which were alreadyproved by Blokh and Lyubich in [BL1]: the ergodicity and the Ergodic ClassificationTheorem.

However we will include the relation with the Markov-Property.

DISTORTION RESULTS AND INVARIANT CANTOR SETS OF UNIMODAL MAPS 25Theorem 5.1. Every S−unimodal map which doesn’t have periodic attractors isergodic.The ergodicity follows directly from lemma 3.9.We will fix in this section an S−unimodal map f without periodic attractor.

Itscritical point is denoted by c.Let n ≥0 and x ∈[0, 1].The maximal interval containing x on which f n ismonotone is denoted by Tn(x). Furthermore the components of Tn(x) −{x} aredenoted by respectively Ln(x) and Rn(x).

Define rn : [0, 1] →R byrn(x) = min{|Ln(x)|, |Rn(x)|}.This functions turn out to be fundamental for understanding the ergodic theory ofunimodal maps.Furthermore definer(x) = lim supn→∞rn(x)for every x ∈[0, 1].The ergodicity of f implies the existence of a number r ≥0 such thatr(x) = r for a.e. x ∈[0, 1].Proposition 5.2.

Let f be an S−unimodal map without periodic attractor. Themap f has the Markov-Property if and only if r > 0.Proof.

The Markov-Property easily implies r > 0.Suppose we have r > 0 for a map f not satisfying the Markov-Property. Using theContraction Principle we get that the function δ : [0, 1] →[0, 1]δ(ǫ) = sup{|I||f n maps the interval I monotonically onto T with |T | < ǫ}

26MARCO MARTENStends to zero as ǫ →0.Take x ∈N close enough to the critical point c to get δ( 12|Vx|) < 12r.Because f doesn’t satisfy the Markov-Property, we get |Ex| < 1, which implies, byusing the ergodicity, that for almost every point y ∈[0, 1] the exists ny ≥0 suchthat for every n ≥ny with f n(y) ∈Uxrn(y) < 12|Vx|.Now take such a point y and n ≥ny. Let s ≥0 be the smallest number suchthat f n+s(y) ∈Ux.

Suppose that rn+s(y) is determined by the piece Rn+s(y) ofTn+s(y) (see definition rn(y)). Now we claim that one piece, say Rn(y) of Tn(y) ismapped monotonically onto Rn+s(y).

In fact this follows easily from the fact thatRn+s(y) ⊂Vx and that orb(∂Ux) ∩Vx = ∅.Then we conclude that rn(y) < δ( 12|Vx|) < 12r for all n ≥ny. Hence lim sup rn < 12r.Contradiction.□Using the number r ≥0 we can give The Ergodic Classification of S−unimodalmaps.P = {f ∈U|f has a periodic attractor };C = {f ∈U|r = 0};I = {f ∈U|r > 0}.Let us remember the definition given in [Mr] of an attractor (the ergodicity of ourmaps makes the definition a bit simpler).Definition 5.3.

A closed invariant set A ⊂[0, 1] is called an attractor if for almostevery x ∈[0, 1]ω(x) = A.

DISTORTION RESULTS AND INVARIANT CANTOR SETS OF UNIMODAL MAPS 27Theorem 5.4 (The Ergodic Classification Theorem). Every S−unimodalmap has an attractor A.

It can be of three different types:1) if f ∈P then A is a periodic orbit;2) if f ∈C then A = ω(c) which is a minimal Cantor set;3) if f ∈I then A is the orbit of a periodic interval.In particular f ∈I if and only if f has the Markov-Property.Proof. Consider an S−unimodal map f. If f ∈P we are finished.If f ∈C then f doesn’t have the Markov-Property.

Hence using Proposition 4.3we get that ω(c) is a minimal Cantor set. Now r = 0 implies that almost all orbitsaccumulates on ω(c).

Again using the minimality of ω(c) the attractor turns outto be ω(c).Take f ∈I. By lemma 5.2 we get the Markov-Property for f, say |Ex| = 1 forsome x ∈N.

It easily follows from Proposition 4.9 that for almost every x ∈[0, 1]Ux ⊂ω(x)which implies [c2, c1] ⊂ω(x). Because ω(x) ⊂[c2, c1] for every x ∈[0, 1] we getA = [c2, c1] = ω(x) for almost every x ∈[0, 1].□Theorem 5.5.

If f is a S−unimodal map whose limit set ω(c) of the critical pointis zero-dimensional then|ω(c)| = 0.In particular if f ∈C ∪P we have |A| = 0.Proof. Suppose that the critical point is not recurrent.

Then there exists x ∈Nwith ω(c) ∩Vx = ∅or ω(c) ⊂Λx. From lemma 2.5 we know |Λx| = 0 which impliesthe theorem.

28MARCO MARTENSThe second case deals with the situation when c ∈ω(c) but the limit set is notminimal. In this case the map has the Markov-property.

Corollary 4.10 tells thatevery 0-dimensional closed invariant set has Lebesgue measure zero. In particular|ω(c)| = 0.Now assume that c ∈ω(c) and that ω(c) is minimal.

Suppose f is infinitely renor-malizable. So f|ω(c) is injective.

Together with lemma 3.9 we get |ω(c)| = 0. Hencewe may assume that f is not renormalizable.Let {(Vxn, Uxn)}n≥1 be the sequence of δ−transfer ranges given by theorem 3.4.Fix n ≥1.

The minimality of ω(c) and lemma 2.5(3) implies that every pointx ∈ω(c) is contained in some component I ⊂Cψ(xn). So we get a finite collectionof intervals TI covering ω(c).

Here the intervals TI form together with the intervalsI, I ∩ω(c) ̸= ∅, a TI-pair for (Vxn, Uxn). Using the fact that these intervals TI arenested, see lemma 2.7, and form a finite collection we can find a component In ofCψ(xn) such that1) In ∩ω(c) ̸= ∅;2) (TIn −In) ∩ω(c) = ∅;Because (TIn, In) is a TI-pair for the δ−transfer range (Vxn, Uxn) we get a ρ1 > 0,not depending on n ≥1 such that both components of TIn −In are bigger thanρ1|In|.

Observe that these components do not contain points of ω(c): we foundsome space in the Cantor set ω(c).Let tn ≥1 be minimal such that ctn ∈In. Using 2) above we get as before aninterval T ′n containing c1 such that f tn−1 : T ′n →TIn is an onto diffeomorphism.Let Tn = f −1(T ′n) and Mn ⊂Tn be the maximal interval with f tn(Mn) ⊂In.

LetLn and Rn be the components of Tn −Mn. Using the fact that the critical point isnon-flat we find ρ > 0 such that

DISTORTION RESULTS AND INVARIANT CANTOR SETS OF UNIMODAL MAPS 291) (Tn −Mn) ∩ω(c) = ∅;2) |Ln| = |Rn| ≥ρ|Mn|.Now we are able to prove that ω(c) does not have density points.For this letx ∈ω(c). Because c ∈ω(x) we can define sn ≥0 to be the smallest number withf sn(x) ∈Tn.

Then apply the Lemma from the appendix and we find for everyn ≥1 an interval Kn around x such that f sn : Kn →Tn is an onto diffeomorphism.Again the Contraction-principle assures that |Kn| →0. Using the Macroscopic-Koebe-Lemma and property 1) and 2) of Tn above we see that x is not a densitypoint of ω(c).

Hence |ω(c)| = 0.□Theorem 5.7. Let A be the topological attractor of f ∈I.

Then f|A is conserva-tive.Proof. Let D ⊂A with positive Lebesgue measure.

The map f has the Markov-Property. Hence by Lemma 4.2 there exists x ∈N with |Ex| = 1.

Because D ⊂Athere exists C ⊂Ux with positive Lebesgue measure such that f n(C) ⊂D for somen ≥0. Let D0 ⊂D be the set of points whose orbits return to D. Now by using theMarkov-Property we see that almost every point of D has a positive upper densityfor D0.

Hence |D0| = |D|.□All the results are stated for S−unimodal maps. In fact it can be shown that theorbits of the intervals considered in the proofs of section 3 satisfy the necessarydisjointness conditions needed for applying the C2−Koebe-Lemma.

Hence the re-sults in this section also hold for C2−unimodal maps. The difficulties for provingthe C2 versions of the Theorems in this paper occur when dealing with the orbitsof TI-pairs.

These orbits do not satisfy some disjointness conditions needed forapplying the C2−Koebe-Lemma. However the intervals in these orbits are nested,

30MARCO MARTENSas described by Lemma 2.7. This property makes it possible to make the necessaryestimates for applying the C2-Koebe-Lemma.

All results stated in this paper willturn out to be true for C2-unimodal maps.Appendix: Basic notions in real 1-dimensional dynamicsLet f : X →X be a measurable map on the borel measure space (X, λ). We willgive some basic definitions dealing with the ergodic theory of this map.0) The orbit {x, f(x), f 2(x), ...} of a point x is denoted by orb(x) and the set of alllimits of the orb(x) is denoted by ω(c).1) A borel set A ⊂X is invariant ifff(X) ⊂X.2) f is called ergodic iffX cannot be written as the union of two disjoint invariantsets both with positive measure.3) f is called conservative ifffor every set D ⊂X with λ(D) > 0 the first returnmap fD on D can be defined in almost every point of D.4) A closed invariant set D is called minimal if the orbit of every point in D isdense in D.5) acip stands for absolutely continuous invariant probability measure.

And acimstands for σ−finite absolutely continuous invariant measure.Dealing with maps on the interval we will use the following notation:6) ∂D is the boundary of the interval D;7) The Lebesgue measure will be denoted by |.|;8) A δ−scaled neighborhood T of the interval I is an interval such that bothcomponents of T −I have length δ|I|.9) Denote the maximal interval containing x ∈[0, 1] on which f s is monotone byTs(x).

DISTORTION RESULTS AND INVARIANT CANTOR SETS OF UNIMODAL MAPS 31The collection U consists of the S−unimodal maps. This are maps f : [0, 1] →[0, 1]having the following property:1) f has negative Schwarzian derivative;2) f(0) = f(1) = 0;3) there is exactly one point c ∈[0, 1] where the derivative of f vanishes.

Fur-thermore this critical point is non-flat: around c the map behaves like x →xα(α > 1). The number α > 1 is called the order of the critical point.For every S−unimodal map f the homeomorphism τ is defined to be the orderreversing map satisfying f ◦τ = f. The non-flatness of the critical point impliesthat this map is Lipschitz.

Furthermore intervals of the form (x, τ(x)) are calledsymmetric.An S−unimodal map f is called renormalisable if there exists a symmetric intervalV such that the first return map to V is of the form f n|V , for some n ≥0, and upto scaling S−unimodal. f n|V is called a renormalisation of f. It is called infinitelyrenormalisable if there are arbitrarily small symmetric intervals on which f can berenormalised.The limit set of the critical point of an infinitely renormalisable S−unimodal mapis a minimal Cantor set.

Furthermore the map acts like a homeomorphism on it.Lemma. Let f ∈U be non-renormalisable and V a symmetric interval.

If s ≥0is minimal such that f s(x) ∈V , x ∈[0, 1], then V ⊂f s(Ts(x)).Contraction Principle. Let f ∈U without periodic attractor.

Then for everyx ∈[0, 1]|Ts(x)| →0

32MARCO MARTENSwhen s →∞.The proofs of the above lemmas can be found for example in [MMS].References[BL1] A.M.Blokh, M.Ju.Lyubich, Attractors of maps of the interval, Func. Ana.

andAppl. vol.

21 (1987), 32-46. [BL2] A.M.Blokh, M.Ju.Lyubich, Measurable Dynamics of S−unimodal maps of theinterval, preprint 1990/2 at SUNY.

[G] J. Guckenheimer, Limit sets of S−unimodal maps with zero entropy, Comm.Math. Phys.

vol. 110, (1987), 133-160.

[GuJ] J.Guckenheimer, S.Johnson, Distortion of S−unimodal maps, Ann. of Math.vol.

132 no. 1 (1990), 71-131.

[M] M.Martens, Interval Dynamics, Thesis at Technical University of Delft, theNetherlands, (1990). [Mi] M.Misiurewics, Absolutely continuous measures for certain maps of the inter-val, Publ.

Math. IHES vol.

53 (1981) 17-51. [MMS] M.Martens, W.de Melo, S.van Strien, Julia-Fatou-Sullivan theory for real one-dimensional dynamics, to appear in Acta Math.

[Mr] J.Milnor, On the concept of attractors, Commun. Math.

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