A Structure Theorem in One Dimensional Dynamics

Annals of Mathematics A Structure Theorem in One Dimensional Dynamics Author(s): W. de Melo and S. van Strien Source: Annals of Mathematics, Second Series, Vol. 129, No. 3 (May, 1989), pp. 519-546 Published by: Annals of Mathematics Stable URL: http://www.jstor.org/stable/1971516 . Accessed: 23/11/2014 23:39 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. . Annals of Mathematics is collaborating with JSTOR to digitize, preserve and extend access to Annals of Mathematics. http://www.jstor.org This content downloaded from 205.133.226.104 on Sun, 23 Nov 2014 23:39:21 PM All use subject to JSTOR Terms and Conditions http://www.jstor.org/action/showPublisher?publisherCode=annals http://www.jstor.org/stable/1971516?origin=JSTOR-pdf http://www.jstor.org/page/info/about/policies/terms.jsp http://www.jstor.org/page/info/about/policies/terms.jsp Annals of Mathematics, 129 (1989), 519-546 A structure theorem in one dimensional dynamics By W. DE MELO and S. VAN STRIEN Introduction The study of the dynamics of one dimensional maps commenced with Poincare in the 1880s. We shall briefly describe his contribution. For each homeomorphism f of the circle S' without periodic points there exists a unique rigid rotation RA such that for each integer n and each x E S1, the points { x, f-l(x),.. ., f-n(x)} and {x, R['(x),..., R- n(x)} are ordered in the same way in S1. From this, one can easily show that there exists a continuous surjective and monotone map h: S1 51 which sends orbits of f into orbits of R,, i.e., hf = Ryh. In his celebrated paper of 1932, Denjoy observed that even for C' diffeomorphisms this semiconjugacy h need not be a conjugacy. He constructed a C' diffeomorphism f of S', without periodic points, having a wandering interval, i.e., an interval L c S' such that L, f(L), f2(L),... are disjoint. Since such an interval cannot exist for a rigid rotation, h must collapse L to a point. However, he showed that for C2 diffeomorphisms such wandering intervals cannot exist. Therefore, any C2 diffeomorphism of S1 either has a periodic point or is conjugate to a rigid rotation. In this way Denjoy completed the topological classification of C2 diffeomorphisms of S'. The next step in the analysis of one dimensional dynamics is to consider smooth non-invertible maps of the interval. Consider the class -W of C' maps f: [0, 1] -. [0,1] such that f(O) = f(l) = 0 and f has a unique critical point c e (0, 1). In the 1910s, Fatou and Julia, [F1], [F2], [J], showed that these maps can have very rich dynamics. In particular, they may have infinitely many periodic points. However, Fatou showed that if f is a polynomial map then only a finite number of these periodic points can be attractors. Here we say that a periodic point is an attractor if its basin of attraction, i.e., the set of points forward asymptotic to its orbit, has a nonempty interior. In particular, each map of the family f.: [0,1] -* [0, 1], f,(x) = px(l - x), p E (0,4), has at most one attractor. Following the programme of Poincare, W. Parry [Pa] and more completely J. Milnor and W. Thurston, [M-T], have shown that the one parameter family This content downloaded from 205.133.226.104 on Sun, 23 Nov 2014 23:39:21 PM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp 520 W. DE MELO AND S. VAN STRIEN f,(x) = ,px(l - x), 0 < j < 4, plays the same role in the class -? as the rigid rotation R, inside the class of diffeomorphisms of the circle. More precisely, given a map f E _V there exists a parameter value t Ee (0, 4] such that for each n the points {c} U (c)},..., f f (c)} and {1} u f-1(1) u *..* U ( are in the same order in [0, 1]. From this we can construct an order-preserving map h: U0=of-k(c) -*U0 = f1,'(2) such that hf = fth. J. Guckenheimer showed in [G] that UcL*fk(2) is either dense in [0, 1] or in the complement of the basin of the attractor of f,,. Hence h extends to a semiconjugacy, i.e., to a monotone surjective map h: [0,1] -* [0,1] with hf = fh. This shows that the analogue of Poincare's theory also holds for maps in ?sz/. In particular if L = h-'(x) is a non-trivial interval then int L n U, of- k(c) = 0 and we have two possibilities: (i) L is a periodic interval, or (ii) L is a wandering interval. Here we say that L is a periodic interval if there exists a positive integer n such that fk?n (L) = fnf(L) for some positive integer k. L is said to be a wandering interval if fk(L) n fk+l(L) = 0 for all positive integers k, 1, and if moreover no point of L is contained in the basin of a periodic attractor. Our main result shows that the analogue of Denjoy's theory also holds for a map f in -/ if the critical point c is non-flat; i.e., some derivative of f at c is non-zero. MAIN THEOREM. If the critical point of f e .d is non-flat then f has no wandering interval. More precisely, we show: THEOREM. Let f: [0,1] -* [0,1] be C3 with f(O) = f(l) = 0 and with a unique critical point c E (0, 1) such that the Schwarzian derivative Sf = D3f/Df - 3 { D2f/Df } 2 is negative in some small neighbourhood of c. (This last statement is true if f is Ck+ 1 near c and Dkf(c) # 0, for some k ? 2.) Then f has no wandering intervals. Using an example of Hall, [Ha], one can show that the Main Theorem is false if one drops the non-flatness condition. There are Co maps f: [0, 1] -* [0, 1] with a unique (flat) critical point c E (0, 1) which have wandering intervals; see [S-I], [Me]. This content downloaded from 205.133.226.104 on Sun, 23 Nov 2014 23:39:21 PM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp A STRUCTURE THEOREM 521 In a forthcoming paper we intend to show that if the critical point of f is non-flat then the (minimal) period of periodic intervals of f is uniformly bounded. Added in proof: This statement and the multinodal generalisation of the main theorem have recently been proved in [M-M-S]. It follows from our main theorem together with Milnor-Thurston's theory that if f E -/ has no attractor then f is conjugate to a quadratic map f,, for some value of the parameter ,p. If f has periodic attractors then one has to be more specific. A periodic point p of f is said to be an attractor if the component Bf(p), which contains p, of the set { x; fn(x) -* { p, f(p), . . ., } as n -x co} has a non-empty interior. Let Ait, A, AX be the set of periodic points p of f such that Bf(p) is respectively a left-sided, right-sided or a two-sided neighbourhood of p. Furthermore let Az? be the set of periodic points p of f such that for some n > 0, fn(p) = p and for some sequence xi -* p, fn(xi) = xi. Finally let A =Al UA'fu XfUAz and =(cf) =U fn(Cf) where Cf is the critical point of f. Using the kneading coordinates introduced in [M-T] our main theorem easily implies that, up to conjugacy, a map f E - is determined by the position of the periodic attractors and the orbit of the critical point. In the special case that f E -V and Sf = D3f/Df - 3 { D2f/Df}2 < 0, this corollary was proved before by J. Guckenheimer; see [G]. COROLLARY. Let f, g E -V. Furthermore, assume that there exists a bijec- tion h: Af U O(Cf) -* Ag U O(cg) which is order preserving (with respect to the induced interval ordering) and such that h o f(x) = g o h(x) forall x E Af U O(cf), h(Atf) = Atg, for i = 1, r, t, z and h(cf) =cg. Then f and g are conjugate; one can extend h to a homeomorphism h: [0, 1] -* [0, 1] such that hof= goh. Proof. Let Bf be the union of the collection of open intervals I such that for some n > 0, fnl I is a diffeomorphism, fn(I) c I. Similarly define Bg. Since h maps attracting periodic points of f to corresponding attracting periodic points in a strictly order-preserving way, one can extend h to a homeomorphism k: Bf Bg such that k o f = g o k for all x E Bf. Furthermore, h: O(cf) --*O(cg), h o f = g o h and h is order-preserving. Therefore the kneading invariants of f and g are the same. Let Of(X), Og(X) E { - 1, 0, 1} be the kneading coordinates of x; see [M-T] and also [C-El], [Str2]. Note that x -- Of(X), x -- Og(x) are monotone (with respect to the lexicographi- cal ordering) and Of([O, 1]) = Og([O, 1]). This content downloaded from 205.133.226.104 on Sun, 23 Nov 2014 23:39:21 PM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp 522 W. DE MELO AND S. VAN STRIEN Now the Main Theorem implies that f has no wandering intervals. There- fore, if Of is constant on some interval I then fn(I) c Bf for some n ? 0. It follows that 0, is strictly monotone on [0, 1] \Un> ?f-n (Bf). Similarly for g. Hence there exists a unique homeomorphism h: [0, 1] \ Un 2 of n(Bf) -* [0, 1] \ Un> ofn (Bg) so that Of(X) = Og(h(x)) (this implies h o f = g o h). On compo- nents of f-n ( B) extend h continuously so that gn o h = k o fn. Then h maps components of Un > 0f- n( Bf) homeomorphically to components of Un Of- n( Bg). In this way one extends h to a homeomorphism h: [0, 1] -- [0, 1] such that hof= goh. Our main theorem is the analogue in the real case of a well known result about iterations of holomorphic maps of the Riemann sphere: Sullivan proved in [Su] that there are no wandering components in the complement of the Julia set. The main ingredient in Sullivan's proof is the theory of quasiconformal deforma- tions. The principal new tool introduced in this paper can be regarded as an extension of a technique of A. Schwartz; see [Sc]. Let N = [0, 1] or S'. This result of Schwartz states that if f: N -* N is a C2 map without critical points and J an interval such that J,..., fn- (J) are disjoint then fn has bounded non-linearity on an interval slightly larger than J. More precisely, there exist 8 > 0 and C < x with the following properties. For any interval J such that Yi = , n1- 1fi(J)I < 1 and any interval T D J with ITI < (1 + 8) IJI one has IDfn(x) I/ IDfn(y) I < C, for all x, y E T. In particular this means that the ratio of the length of two intervals in T is not distorted too much by fn. Our tools also apply in situations where one does have critical points, at least provided they are all non-flat. But instead of considering how fn distorts the ratio of the length of certain intervals, we estimate the distortion of the cross-ratio of intervals. We think that the technique proposed in this paper will be useful in many problems related to interval maps with critical points. Indeed, R. Manei, [Ma], used the techniques of Schwartz extensively and proved an extremely general hyperbolicity result for C2 maps without critical points. Combining these ideas of Mahei with our tools (rather than those of Schwartz) one can extend Ma-ne's results considerably. It turns out to be possible to prove the results on invariant measures from [Mi] and [C-E2] without the assumption that Sf < 0; see [Strl] and [N-S]. Let us make a few remarks on the history of the main theorem and its proof which has two aspects: a topological one and an analytical one. Denjoy's analytical tool was that if log I Df I has bounded variation, then the derivatives of iterates of f, Df , have bounded distortion on an interval J whose iterates, J, f(J),..., fn(J),... are disjoint. His topological tool was the semicon- This content downloaded from 205.133.226.104 on Sun, 23 Nov 2014 23:39:21 PM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp A STRUCTURE THEOREM 523 jugacy with the rigid rotation and a detailed understanding of the dynamics of rotations of the circle; see [D]. Schwartz [Sc] gives another proof of this theorem which is almost com- pletely analytic. His assumption is that log I Df is Lipschitz. Then he gets bounded distortion for the derivatives of iterates on an interval L if the sum of the lengths of its iterates is finite. This theorem is the main tool of Manie who considers C2 maps of an interval (or a circle); see [Ma]. He considers points whose forward orbit stays away from the set of critical points C(f). Using disjointness properties and a C2 theorem of Schwartz he proves that any compact set K not containing any critical points or non-hyperbolic periodic points is hyperbolic. The main problem in the study of orbits which pass close to critical points is that one gets a lot of non-linearity: the bounded non-linearity tools of Schwartz completely break down. In [Si], Singer introduced a completely different analytic tool: the Schwarzian derivative, Sf = D3f/Df - 3f{D2f/Df } 2. He pointed out that if Sf is negative then Sfn is also negative for every n. Using the observation that if Sf < 0 then IDf I cannot have a positive local minimum he proved that, for maps with negative Schwarzian derivative, the basin of any periodic attractor must contain a critical point. Misiurewicz and Guckenheimer, [Mi], [G], showed that the Schwarzian derivative was an important analytic tool for the study of several dynamical properties. In particular, Guckenheimer proved in [G] the non-existence of wandering intervals for maps which everywhere have negative Schwarzian derivative. However this additional condition is extremely strong (it implies that I Df I cannot have any positive local minima). Also this condition is not very natural (for example, it is not coordinate invariant). Yoccoz in [Y] proved the non-existence of wandering intervals for Co homeomorphisms of the circle having only non-flat critical points. He combines techniques of Denjoy away from the critical points with some analytical esti- mates near the critical points related to the Schwarzian derivative. Our topological tools are the same as Guckenheimer's. The analytical aspect of our proof is related to the distortion of the cross-ratios under iterates of f. This part of the proof has two ingredients. The first one is the use of the Schwarzian derivative near the critical point to estimate the distortion of the cross-ratio. The other is the use of Lipschitz conditions on D2f away from the critical points. The way we combine these two ingredients resembles the proof of Schwartz. In Section 5 we prove also the non-existence of wandering intervals for a Co map f with a finite number of critical points, all of them non-flat, provided the forward orbit of the critical points does not accumulate at critical points. (This last condition is usually called the Misiurewicz condition.) This content downloaded from 205.133.226.104 on Sun, 23 Nov 2014 23:39:21 PM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp 524 W. DE MELO AND S. VAN STRIEN During the preparation of this paper, de Melo visited TH-Delft and IHES; van Strien visited IMPA. We thank these institutions for their hospitality. We are grateful to D. Sullivan for many stimulating conversations. 1. How does a map distort the cross-ratio ? Let N be either the circle S' or the interval [0, 1] and f: N -* N be a smooth map with non-flat critical points. Later in this paper we need to compare the size of intervals ff(J) and f'(T) for intervals J and T such that J c T and ff I T is a diffeomorphism. However, if f has critical points then by varying the intervals J and T the ratio of Ifn(T)I/Ifn(j)I and TI/IlJI can be made arbitrarily large (here I WI denotes the length of an interval W). So instead of considering ratios of intervals we will consider cross-ratios of pairs of intervals. In this section we will study how a map f distorts the cross-ratio of a pair of intervals. In the next section we will obtain estimates of the distortion of this cross-ratio under high iterates of f. More precisely let T c N be an open interval, Clos(T) = [a, b], and g: N -* N be a C3 map with gIClos(T) a diffeomorphism onto its image. 1.1. Definition. Let J c T be an open and bounded interval in R such that T - J has two connected components L and R. Define two cross-ratios of intervals as (1.1) ~~~~~C(T, J) = Ill I TI IL U ll IJ U R I (1.2) D(T, J) = I ll TI ILl IRI' where IJI denotes the length of J. If g is monotone on T: (1.3) A(g, T, ) - C(g(T), g(J)) c(T,J) (1.4) B(g, T, J) = D(g(T),g(J)) D(T,J) The next result is the main result in this section and shows that a smooth map f: N -* N with non-flat critical points does not contract these cross-ratios too much. Here we say that f is non-flat at a critical point c if there exists k ? 2 such that f is Ckl near c and Dkf(c) # 0. This content downloaded from 205.133.226.104 on Sun, 23 Nov 2014 23:39:21 PM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp A STRUCTURE THEOREM 525 1.2 THEOREM. Let f: N -* N be a C3 map whose critical points are non-flat. Then there exists a constant CO = CO( f ) such that if T D J are intervals (such that T - J consists of two connected components L and R) and Df( x) # 0 for all x e T then (1.5) log A(f, T, J) - CoILI IR I, A(f, T, J) ?1 - COILI IRI, (1.6) log B(f, T, I) - CoIT2, B(f, T, I) ? 1 - CoIT2. Remark. It is essential in the rest of the paper that one have estimates as in (1.5) and (1.6) rather than estimates of the type log A(f, T, I) 2 - CoITI, and log B(f, T, I) ? - CoITI (which were obtained in [Y]). In the remainder of this section we will prove this theorem. First we will show that the above operators A and B are connected with the Schwarzian derivative. 1.3. Definition. Let g: R -* R be a C3 map. If Dg(x) # 0 the Schwarzian derivative of g at x is D3g(x) 3 (D2g(x) 2 Sg~ - Dg (x) 2 Dg (x) Remark. If N = S' we fix once and for all a covering map 7: R -* S1. If T c S1 is an interval and g: S1 -* S1 is a C3 map which is monotone on T we can lift T to an interval T' c R and g to a map g': R -- R and define A(g, T, I) = A(g', T', I'), B(g, T, I) = B(g', T', I') and Sg(x) = S(g'(x')), 7(x') = x. This is clearly independent of the lift. 1.4. Elementary facts about Schwarzian derivatives and cross-ratios. 1) If p is a critical point of g which is not flat then there is a neighbour- hood W of p such that Sg(x) < 0 for x E W - { p}. 2) If g is monotone on an interval T c R and Sg < 0 on T then I Dg I does not have a positive local minimum on T. In particular, if Dg # 0 on T and x < y < z are in T then I Dg(y) I > min{ Dg(x) $,Dg(z)$} 3) If g1, g2: N -* N are Co functions then S(g1 ? g2) = (S(g1) ? g2)(Dg2)2 + Sg2. In particular if Sg < 0 then Sgn < 0 for all n > 0. This content downloaded from 205.133.226.104 on Sun, 23 Nov 2014 23:39:21 PM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp 526 W. DE MELO AND S. VAN STRIEN 4) Let M be the group of real Moebius transformations; i.e., E c M if and only if ?(t) = (at + /3)/(yt + 8) with a, /, y, 8 real numbers. If ? c M then S4o O(in fact So O iff c E M). 5) There is a similar relationship between Moebius transformations and the above operators. In fact, if O E M is monotone on T C R and J C T then, A(O, T, J) = I and B(O, T, J) = 1. 6) From (3) and (4) it follows that if Sg is negative on W and A, 4 c M then S(O o g o 4) is negative on {-'(W). 7) The operators A and B are multiplicative: A(g o f, T, J) = A(g, f(T), f(J)) x A(f, T, J) and B(g o f, T, I) = B(g, f(T), f(J)) x B(f, T, I) if f is monotone on T and g is monotone on f(T). 1.5 LEMMA. Letf: [a, b] -* R be a C3 function such that Df(x) # 0 and Sf(x) < 0 for all x E [a, b]. Then for Clos(J) C Int(T) C [a, b]: (1.7) A(f, T, J) > 1, (1.8) B(f, T, J) > 1. Proof Given three points x < y < z in [-so, so] there is a unique Moebius transformation 4 such that ?(x) = 0, ?(y) = 1 and ?(z) = so. Hence we can find 4 e M mapping T diffeomorphically onto [0, 1]. For every X < 1, the Moebius transformation ox(x) = x/(Xx + 1 - X) maps the interval [0, 1] diffeomorphically onto itself. If z E (0, 1), ox(z) --*1 as X -A 1 and fO(z) -* 0 as X -oo. Hence, given a, a e (0, 1) there exists X such that OX(') = a. Since composition with Moebius transformations preserves cross-ratio and the sign of the Schwarzian derivative, we may assume that T = [0, 1], f maps T onto T preserving orientation and maps L onto L. So we let L = [0, a] and J = [a, /3]. We have that f(O) = 0, f(a) = a and f(l) = 1. Since f has negative Schwarzian derivative on [0, 1], f has no other fixed point. We claim that f(x) > x for all x E (a, 1). Suppose, by contradiction that this is not the case. Since f has no fixed point in (a, 1) we have then that f(x) < x for all x E (a, 1). This implies Df(l) ? 1 and Df(a) < 1. If f(x) > x for all x E (0, a) then Df(O) ? 1 and Df(a) is not strictly bigger than min(Df(O), Df(l)). This contradicts Sf < 0. If f(x) < x for all x E (0, a) then Df(a) = 1 and there is an x E (a, 1) such that Df(x) < 1. So Df(x) is not strictly bigger than This content downloaded from 205.133.226.104 on Sun, 23 Nov 2014 23:39:21 PM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp A STRUCTURE THEOREM 527 min(Df(a), Df(l)) again contradicting Sf < 0. Since f has no fixed point in (0, a) these are the only two possibilities, proving the claim by contradiction. Let us show that A(f, T, 1) > 1. From the claim, it follows that f(1) > A. Now, f(3) - a 1 /3-a 1 A(gTJ)= f() 1-a > 3 1-a since f(/) > > a. Similarly, B(f, T, 1) > 1; see Figure 1. f (a) =X a FIGURE 1 1.6. LEMMA. Let f: N -- N be a C3 map and V be a neighbourhood of the critical points C(f). There exists a constant CO" = Co'(f, V) such that if T c N - V then A(f, T, J) ?2 - Coj'Lj IRI, B(f,T,J) ?1 - ColT12. Proof Let us prove only the first inequality. The second one is similar and, in fact, easier. Suppose, by contradiction that the first inequality is not satisfied. This content downloaded from 205.133.226.104 on Sun, 23 Nov 2014 23:39:21 PM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp 528 W. DE MELO AND S. VAN STRIEN Then there is a sequence of intervals Jn C Tn C N - V such that A A Tn, Jn) - I - 0 jLLn jIRnI as n - oo. By taking subsequences we may assume that an -- a, bn -*b, Cn- c, d n-- d where Tn = [a n, dn], Jn = (bnI Cn). There are two possibilities: b -a nd n- c -ac b -a Let us assume we are in the first case. The other is similar (interchange the roles of Ln and Kn). Since f is C3 and a E N - V we have (1.9) f(an + x) = f(an) + XnX + ? n(x)x2 where IXnI > min{jDf(x)j, x E N - V} > 0, P n is differentiable and max{ ( nJM(x)jI, jDPn(x)j; x E N - V} < oo. Let us calculate A n: f(cn) - f(bn) f(dn) - f(aj) f(c.) - f(an) f(dn) - f(bn) (I. 0)c C- b dn -an c -an dn -b n ACJ ) -fan ) fdn) -f&n) ( a)d-n cn -a d b (b - - C) Since we are in the case sup(bn - an)/(dn -Cn) < oo and Tn C N - V, then An - o ximplies that bn - an 0 (i.e. a = b) and f(cn) - f(bn) f(dn) - f(a.) f(cn) - f(a.) f(dn) - f(b.) ( 1.11) Bn = n (n n - an ) n - an) dn - bn (bn - an)(dn - Cn) 0--oo. Let us calculate Bn using (1.9). We get, using the notation bn = bn- an, Cn- = C(n-)a din d = dn-a B - (d c-)( ,|n ? nb)?xb ~~~~~~~~~~~~~n . Bn (C)C - Xn + (Xn 11 ('(t+b)+ln ) yn(dn)dx b This content downloaded from 205.133.226.104 on Sun, 23 Nov 2014 23:39:21 PM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp A STRUCTURE THEOREM 529 and we have B A - )M(d2))b, ? c - ) ? O(bn2) n dbn(dn-c) Since we have assumed sup bn/(dn - cn) 530 W. DE MELO AND S. VAN STRIEN Case 1. cn -- c # a. In this case, An -- -o implies Bn - - oo (compare equations (1.10) and (1.11)). But now, from c # a, d # a, b = a, (1.14) f(cn) -f(bn) f(cn) -f(an) (1.15) f(dn) - f(bn) = f(dn) - f(an) + 02 f(cn) - f(an) - ?~~~+02?0E)3, c - a where 01 = (bn - an), e)2 = O(bn - an), 03 = O(dn - Cn) More precisely, 1 1 (cn - bn)(Cn - an) 2 (dn-bn)(dnn - Aan x(fd nC )(n n- bn) b--f(bn)) ( bn -an), Substituting (1.14) and (1.15) in the expression (1.11) we get (bB- )(d- ( (f(c -Aan) ? )(f(cn)fban) ? 0 3) -2(f(cn) f(Un))(f(C)f(Un) ?02?+3)) (dn~~ - a c(d - an (b= - f(d - (f(c) f(n) (an - 02) ? 13} Since 1/[(d - bn)(d - an)] = [1 ? O(d - cn)]/[(c - bn)(c - an)] we have that (81- - an) = O(dn - ca). Thus Bb does not tend to -x. Case 2. ctn 1 c = a. Let WO be the component of W containing a. Choose dn e WO and a constant a > 0 such that [f(dn) - f(an)]/(dn - an) = a* [f(dn) --f(aa)]/(dc--nan)2 Then [f(dn) -f(bn)]/(dn -bn) = a [f(dn) - f(bn)]/(dn - bn) o(b - an) This content downloaded from 205.133.226.104 on Sun, 23 Nov 2014 23:39:21 PM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp A STRUCTURE THEOREM 531 Substituting this into (1.10) one gets: di - ( o(b -a )(-b ) -' n n x { expression (1.10) with d replaced by d } o(bn - an) f(cn) - f(an) C - a ? f(cn) -f(a ) f(dJ) -f(bn) cn - a d - bn (bn - an)(dn -n c,~ -a d- The first term is 2 0 because the interval [a, d'] C W (see Lemma (1.5)). The last term is bounded because dn *d a, an, cn, bn -*a. Proof of Theorem (1.2). Clearly Theorem (1.2) follows from Proposition (1.7). 2. Distortion of the cross-ratio C under iteration Let f: N -* N be a C3 map. If f has negative Schwarzian derivative, then S(f') is negative for all m. Therefore, in this case, fm expands the cross-ratio. In general this is not true. However, Theorem (2.1) below tells us that if M - 1 jfi(J)I is not too big then A(fm, T, J) is bounded away from 0. This is the analogue of a result of A. Schwartz, [Sc], for C2 diffeomorphisms f: N -- N without critical points which says that there exist C > 0 and 8 > 0 with the following property: For each interval T D J, and each integer m such that M-l Ifi(J) I < 1 and I Tl < (1 + 8) IJI one has Df mx) < Dfm(y) - for each x, y E T. We will prove and use this result of A. Schwartz in Lemma (4.1). 2.1. THEOREM. Let f: N -* N be a C3 map whose critical points are non-flat. There exist constants 8 > 0 and 1 > - > 0 such that if T D J are intervals satisfying: i) fm is a diffeomorphism on Clos(T); ii) jM_ 01 Ifk(J) I < ; iii) ILjI IRI < IjI2, where L and R are the components of T - J. then A(fTm Tw J) s ne-e 8 I aLemm To prove this theorem we first need a lemma. This content downloaded from 205.133.226.104 on Sun, 23 Nov 2014 23:39:21 PM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp 532 W. DE MELO AND S. VAN STRIEN 2.2. LEMMA. If g is monotone on an interval T D J and C1 > 0 is such that (2.1) A(g, T, J) ? 1 - CljLI IRI and ILI IRj(C1 + 1/(jJ U LI IJ U RlI)) < 1 then (2.2) jL j R j 0. Proof Notice that (2.4) C(T, J) ILI iUR IL UJj jJUlRI' Equation (2.1) implies: Ig(L) g(R)| g(L) u g(J) |g(J) U g(R)j I g(J) g(T) I g(L) U g(J)| g(J) U g(R) - - C(g(T), g(J)) - 1 > (1-CtLLI IRl) * C(T, J)- 1 - (1- C1lLI iit)(i _J UjL jU l )-1 (I-l 1j +RC I - LI RI>O VIJUIIJUIJ }IUR This implies (2.2). Let us now prove (2.3). We have: Ig(L U 1)I Ig(J U R)I = Ig(J)I Ig(T)I + Ig(L)I Ig(R)j. Hence, from (2.2), jg(LIj jg(RIj< C21g(J) I |g(T) I + C21 g(L) I Ig(R) 1. ILI IRI - This content downloaded from 205.133.226.104 on Sun, 23 Nov 2014 23:39:21 PM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp A STRUCTURE THEOREM 533 Taking the last term to the left-hand side, we get jg(L)j jg(R)j C (1 - C21Lj IRI) ?L IRI < C2 g(L)l g(T)|. Since 1 - C21LII RlI > 0, this proves the lemma. Proof of Theorem 2.1. Choose 1 > j > 0 so that (2.5(a)) IxI < | Ilog(l - x) < 2Ix, (2.5(b)) Ilog(y) I < q =* jy - Ij < 21log y. Choose 8 > 0 so that (2.6) 36COS < 1, 2 < O < < where CO = Co(f) is as in Theorem 1.2. Choose e > O so that - < j, e < 1/18Co, E < 1 . Let L = A(fn, T, J). Write Ln = 1- KnjLII R. Let us prove by induction that Kn < 8/ IJI2. From Theorem (1.2), (2.7) Li = A(f, T, J) ? 1 - CoILI IRj. From assumption (ii) of the theorem and inequalities (2.6), Ij2< (E lfk(j)l| < 82 < - < . - ~ k=0 CO Co Using this in (2.7) we get: 8 K1 < Co < Suppose, by induction, that 8 < Jr for j= 1,.,n< m. Notice that 8 L = 1- KjILI IRI > I1- -JrIlLI IRI > 1- 8e > O. Taking in Lemma (2.2) C1 = 8/ IJI2 we get C2 = 1/IJI2 + 8/ IJI2 = 9/IJI2. Hence C21L IRli < 9/jIj2ILII RlI < 9 < 2 < 1. Then, by Lemma (2.2), This content downloaded from 205.133.226.104 on Sun, 23 Nov 2014 23:39:21 PM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp 534 W. DE MELO AND S. VAN STRIEN Since C2' = C2/(1 - C21L IRli) and C21LII RlI < 2 as above, we have C2' < 2C2. Hence Ifi(L)I Ifi(R)I 18 Ll 0 gRI A STRUCTURE THEOREM 535 2.4. COROLLARY. Under the conditions of Theorem (2.1): (2.12) Ifmr(L) Ifm(R)I 18 ILl IRI I I12fm(1)hfm(T)L Proof From Theorem (2.1) we have 8 A(f am, T. J) 2 I1- IJ 2IL I IR I Now, taking C1 = 8/I12, C1 + 1/I12 = 9/ I12, we have that (C1 + 1/Ij12)ILI IRI < 9e < 1. So we can use Lemma (2.2) and we get jfm(L)j fm(R)) < C2'1 fm( I I fm(T) 1 ILI IRI ?c f() f)I where C2 18 C2 1 - C2ILI IRI 2 IjI 2 because C2 = C1 + 1/IjI2 = 9/IjI2. 3. Distortion of the cross-ratio D under iteration In the last section we have shown that ffn cannot contract the cross-ratio C too much on T :DJ provided XT 01fk(J)l and (ILl IRI)/Ij12 are small. In this section we will show that ffl cannot contract the cross-ratio D too much either, but now we need a stronger condition: En=0Ifk(T) must be not too big. However, from this cross-ratio D we can get some infinitesimal information whereas C describes more global properties of f on T. 3.1. THEOREM. Let f: N -- N be a C3 map whose critical points are non-flat. If T D J are intervals satisfying: i) fm is a diffeomorphism on Clos(T); ii) yM2= If?(T)I = S < 3; then: (3.1) log B(fm T. J) 2- Co( E Ifi(T) 12) 2-3C0 * S i =o where CO is the constant from Theorem 1.2. This content downloaded from 205.133.226.104 on Sun, 23 Nov 2014 23:39:21 PM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp 536 W. DE MELO AND S. VAN STRIEN Proof: From the multiplicative property of the operator B and Theorem 1.2 we get: m-1 m-1 log B(fm, T, J) =log H B(f, fi(T), fi(J)) = E log B(f, fi(T), fi(J)) i=O i=O m-1 2 E (-Co ICf'(T) 2) The lemma below describes the infinitesimal information which can be obtained by the operator B. 3.2. LEMMA. Let g: T -- R be a C3 diffeomorphism with T = [a, b]. Let x E (a, b). If for every J C T* c T (3.2) B(g, T*, J) 2 C2 > 0 then one has either (3.3) IDg(x)l 2 C231Dg(a)I or IDg(x)l 2 C231Dg(b)I or both. Proof. Let us consider the following two operators: Bo(g , T*) = I T* 12 D I T* 12 Dg(a*) IDg(b*) 1 Dg(x) gIT Bj(g, T, x) = I | ILI IRI where T* = [a*, b*] C T and L and R are the connected components of T - x }. Observe that Bo(g T*) = lim B(g, T*, J), Bl(g, T. x) = lim B(g, T. J). Hence BO(g, L), BO(g, R), Bj(g, T, x) ? C2> 0. Let |g(L) I| |X Ig(R) an | g(T)| 2\1= LI | 2\2 RI and X\= - ILl g(R) I T Since BO(g, L), BO(g, R) ? C2 we have: (3.4) X, ? C21Dg(a)| IDg(x) , (3.5) ?22 C21Dg(x)l IDg(b) . This content downloaded from 205.133.226.104 on Sun, 23 Nov 2014 23:39:21 PM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp A STRUCTURE THEOREM 537 Since B1(g, T, x) ? C2 > 0 we have: (3.6) | Dg(X)IX ? C2X1X2. It is clear that if X 1 < X then X 2 2 X and if X 2 < X then X 1 2 X. So, suppose X 1< X. Then: IDg(x )1(I6)2 2 C2gC31Dg(a) I IDg(x) 1. ? C2 > C Hence I Dg( x) I 2 C23 I Dg( a) I and similarly for X 2 < X. Since it is impossible that both X 1 and X2 are bigger than X, the lemma follows. Remark. For a diffeomorphism g: T -- R satisfying Sg < 0, one has IDg(x)I > min{IDg(a)i, IDg(b)I}. So the above lemma gives a slightly weaker version of this inequality. 3.3. THEOREM. Let f: N -- N be a C3 map which is non-flat at all of its critical points, and let CO = Co(f) be the constant from Theorem (1.2). If T = [a, b] c N is such that: i) fm is a diffeomorphism on Clos(T), ii) 2M - 1 If?(T)I2 = S < 3, then either (3.7) Dfm(x) ? 2 exp(-gCoS) |Dfm(a)|, or (3.8) lDfm(x) I 2 exp(-gCOS) Dfm(b) , or both. Proof. This follows immediately from Theorem (3.1) and Lemma (3.2). 4. Non-existence of wandering intervals for unimodal maps Assume N = [0,1] and f: N -- N is a C3 unimodal map, i.e., a map with just one critical point. In this section we show that if the critical point is non-flat then f cannot have a wandering interval. We start with a more general result which follows from the proof of Schwartz's theorem (see for example [C-E1, pp. 111] or [Ma]). Versions of this result are proved in several papers but in order to be complete we include the proof here. 4.1. LEMMA. Let N be either [0, 1] or S' and f: N -- N be a C2 map. If J is a wandering interval off then there exists a sequence n(i) -x oc such that fn(i)(J) -C Cf ) where Cf ) is the critical set of f. This content downloaded from 205.133.226.104 on Sun, 23 Nov 2014 23:39:21 PM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp 538 W. DE MELO AND S. VAN STRIEN Proof: For simplicity assume that diam(S') < 1. Let J be a wandering interval of f. We may assume that J is a maximal wandering interval, i.e., that J is not strictly contained in some bigger wandering interval. Suppose, by contra- diction, that the forward iterates of J do not accumulate at the critical set of f. Then, there is a neighbourhood V of C(f) such that fl(J) n V = 0 for all n ? 0. Let W be a neighbourhood of C(f) such that Clos(W) C V. Since f is C2, there exists a constant K > 0 such that IDlogIDf(x)I I < K for every x E N - W. Let T be an interval such that fi(T) nl W = 0 for every i < n. Then, for x, y E T we have, from the chain-rule and the mean value theorem, that Iog I-r~ I - E (log Df(f'(x)) - log Df(f (y)) I n-1 < E K Ifi(T). i=O Hence, (* exp(-K E I f (T))< Dfyl< exp K E I f (T)) for all x, y E T. From this and the mean value theorem we get (**a) eDfp(x) < K E ?f'(T) X I 1 and (**b) | Dfn>x) | 2 exp -K E I f (T) l) x IfT)I for every x E T. In particular, since the intervals fi(J) are disjoint, we get from (**a) that Dfn(x)I < exp(K) Ij) 'I' Let T D J be an interval such that L # 0 and R # 0 are the connected components of T - J. Choose 8 > 0 so small that 2 . 8 exp(2K) is less than the length of each of the connected components of V - W. We claim that if ILl, JRI < 8 * Il, then: i) =01fi(T)l < 2, ii) exp(-2K)Ifn~'(I)I/II ? IDfn+1(x)I < exp(2K)Ifn+'(j)1I/III and iii) fn+'(T) fl W = 0 for every n > 0 and every x E T. We prove the claim by induction. Suppose i), ii) and iii) hold This content downloaded from 205.133.226.104 on Sun, 23 Nov 2014 23:39:21 PM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp A STRUCTURE THEOREM 539 for every n = 1, ...,m - 1. From ii) we have IDf+ 1(x) I < exp(2K)/IJI for i < m. Hence If" i(T) I < Ifi+ '(J) I + 28 * IJI exp(2K) Ifi+ '(J) I/IJI for i < m and X'' 0If (T) I < (1 + 28 * exp(2K)) (Y'= 0If'(J) 1) < 2, since E2Ifi(J) I < 1 and 8 < 2 exp(- 2K). This shows that i) holds for m. Using this, (**a) and (**b) we get that ii) holds for m. From this we have ifm+1(T)l I-fm+'(J)I < 28 . exp(2K)Ifm+l(J)I < 28 * exp(2K). By the choice of 8 this is less than the minimum of the lengths of the connected components of V - W. Hence, since fm+1(J) l V = 0, we conclude that iii) also holds for m. This proves the claim. Let T R J be an interval such that i), ii) and iii) are satisfied. From ii) it follows that ffn I T is a diffeomorphism for every n > 0. Since T D J and T # J it follows from the maximality of J that T is not a wandering interval. So either a) T contains points which are contained in the basin of some periodic attractor or, b) there exist integers n and k such that fn+k(T) n fnf(T) # 0. First assume we are in case a). Since J does not contain points which are in the basin of some periodic attractor, T contains a point in the boundary of the (immediate) basin of the periodic attractor. Hence Ifk(T) I does not tend to zero, contradicting Ifn(T)I < (1 + 28 exp(2K)) * Ifn(j)I -O 0. In case b) it follows that, for every integer j, fn+(j+l)k(T) n fni+ jk(T) # 0. Thus, I = Uoo ofn+ijk(T) is an inter- val and fn maps I homeomorphically into itself. Hence all points in I are asymptotic to periodic points. This is a contradiction because I contains fn(J) and J does not contain points which are in the basin of periodic attractors. So in both cases we get a contradiction, and it follows that f cannot have a wandering interval. 4.2. COROLLARY. If f is a C2 unimodal map and the critical point of f is in the basin of an attractor then f has no wandering interval. From now on f: N -- N will be a C3 unimodal map with a non-flat critical point c. Assume by contradiction that f has a wandering interval J. By replacing J with another wandering interval containing some iterate of J we may assume that: (4.1) fn(Clos(J)) n C(f) = 0, for all n > 0 and 00 (4.2) E fk(J)I < 80, k=O where 8S is the constant from Theorem 2.1. In fact (4.1) holds for any wander- ing interval of unimodal maps because of Lemma (4.1). Let Km be the max- This content downloaded from 205.133.226.104 on Sun, 23 Nov 2014 23:39:21 PM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp 540 W. DE MELO AND S. VAN STRIEN imal interval containing J such that fmlKm is a homeomorphism. From (4.1) Km - Clos(J) is a disjoint union of two non-empty intervals Lm and Rm. To be definite choose Rm so that d(fm(Rm), c) < d(fm(J), c). 4.3. Remark. It follows that fm+1 is a homeomorphism on Lm U J and Km+ C Km. 4.4. LEMMA. The lengths of the intervals Lm and Rm go to zero as m -x 0. Proof If this were not the case then K = nfl=oKm would be an interval that strictly contains J because Km + 1 c Km. Let us show that K is a wandering interval. Since fnl 1K is monotone for all n 2 0 and no point of J C K is contained in the basin of a periodic attractor, no point of K can be in the basin of a periodic attractor. Hence it suffices to prove that fi(K), i ? 0, are disjoint. So assume by contradiction that for some n > 0 and some k ? 0, fk(K) n fk+n(K) # 0. Then fk+ijn(K) n fk+(j+l)n(K) # 0 for every j ? 0. Therefore K = U_0fk+ jn( K) is an interval and fn maps K* into itself monotonically. Hence every non-periodic point in K* is asymptotic to a periodic attractor. This is a contradiction because K D J. Hence K is a wandering interval. But this contra- dicts the maximality of J (J was chosen so that it is not contained in a strictly bigger wandering interval). Let us introduce some notation. For x near the critical point c we denote by x' the "symmetric" of x; i.e., f(x') = f(x) and x' # x. The symbol x# wijJ denote either x' or x. If U1, U2 C N we define (U1, U2) as the smallest interval containing U1 U U2. In particular (a, b) is the (closed) segment connecting a and b. Let us define, by induction, a sequence of integers k(n) that describe the closest approach of iterates of J to the critical point. More precisely, k(O) = 0 and, for n > 1, k(n) = min k; fk(j) c Kfk(n1-)(j), (fk(n-)(J)) )}. By Lemma 4.1 there exists a sequence n(i) -x oc such that fn(i)(j) -J c. Hence the sequence k(n) is well-defined and fk(n)(J) > c. 4.5. LEMMA. fk(n)(Kk(n)) contains either (fk(n-1)(I) c) or ((fk(n 1)(j))' c). Proof Since Kk(n) is maximal, there exist integers i, j < k(n) such that f (Rk(n)) and fj(Lk(n)) 0 c. Since i, j < k(n), fi(J) n (fk(n-1)(J) (fk(n-1)(J#))) = 0. This content downloaded from 205.133.226.104 on Sun, 23 Nov 2014 23:39:21 PM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp A STRUCTURE THEOREM 541 Therefore f'(J U Rk(n)) and fj(Lk(n) U I) contain either (fk(n-1)(J), c) or ((fk(n-1)(J))' c) (the ith and the jth iterates may contain different segments). Suppose, by contradiction that fk(n)(Lk(n)) does not contain fk(n-1)(J)#. From the choice of Lk(n) and Rk(n) one has d(f k(n)(Lk(n)), c) > d(f k(n)(J), c) and hence fk(n)(Lk(n) U J) is contained in f'(Lk(n) U J)# = ((f k(n1)(J))y c) and fk(n)i maps fj(Lk(n) U J)# onto f k(n)(Lk(n) U J) homeomorphically. This im- plies that fi(J) is contained in the basin of an attractor and, therefore, is not a wandering interval. This contradiction shows that fk(n)(Lk(n)) contains f k(n- 1)(J)# Let us now prove by contradiction that fk(n)(Rk(n)) contains c. If this is not the case, then fk(n)-i maps fi(J U Rk(n))# homeomorphically into itself. As before this is a contradiction. So fk(n)(Kk(n)) contains both (fk(n - 1)(J))# and c. This finishes the proof of Lemma (4.5). Define Vn ={x; fn(x) E (x, x') and fi(x) E (x, x') for i < n}. 4.6. LEMMA. Let (a, b) be a connected component of Vn. Then fn I (a, b) is a diffeomorphism, fn(a) = a# and fn(b) = b#. Proof Since b is in the boundary of Vn there is a smallest 0 < i < n such that fi(b) = b#. Let us show that fn(b) = b#. Write n = ki + I with 0 < I < i. Then fn(b) = fl(fki(b)) = fl(b#) = f'(b) Z (b, b') if I > 0, since I < i. Hence I = 0 and fn(b) = fki(b) = b#. Similarly fn(a) = a#. Now, fni(a, b) is a diffeomorphism because, otherwise, there would exist x e (a, b) and i < n such that fi(x) = c e (x, x'). 4.7. Remark. Let A = a or b. Assume f(A) = A# for some 0 < i < n. If Df'(A#) > 0 then i = n and if Df'(A#) < 0 then either i = n/2 or i = n. Proof. Let Df'(A#) > 0. Then, since n = ki we have, for x E (a, b) near A#, fi(x#) E (X#, fn(X#)) C (x', x). Hence i = n because (a#, b#) is also a component of Vn. Let Df'(A#) < 0. We distinguish between two cases. If A is the inner boundary of (a, b), i.e., (A, A') n (a, b) = 0, then i = n because fi(x#) E (A, A') C (x, x') for x near A. If A is the outer boundary then for x E (a, b) near A, x# and fn(x#) are on the same side of A#. Hence n/i is even and i < n. Since f2=(x) c (x, fn(x)) C (x, x') we have 2i = n. 4.8. LEMMA. Let (a, b) be a connected component of Vn. If fn(a) = a and fn(b) = b orfn(a) = a' and fn(b) = b' then (a, b) is contained in the basin of an attractor. This content downloaded from 205.133.226.104 on Sun, 23 Nov 2014 23:39:21 PM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp 542 W. DE MELO AND S. VAN STRIEN Proof This follows because in this case fn maps either (a, b) or (a', b') homeomorphically onto itself. In order to use the estimates from Section 3 we need some disjointness of iterates of intervals. 4.9. LEMMA. Let (a, b) be a connected component of Vn. Let I be the interval ((a, b), (a, b)') (this is int((a, a') U (b, b'))). For x E (a, b) let n(x) = inft i > 0, fi(x) E I). Then n(x) = n for all x E (a, b). Proof. To be definite let I = (b, b'). For x near b, n(x) = n. In fact assume that fi(x) E (b, b') for some 0 < i < n. If fi(b) 0 (b, b') then this is impossible. But if fi(b) E (b, b') then i = n/2 and Df'(b) < 0 (see Remark above Proposition 1.7). But then fi(x) 0 (b, b'). Hence, by contradiction, n(x) = n. We claim that n(a) = n. Suppose by contradiction that fi(a) E (b, b') for some i < n. Hence fi((a, b)) 3 b' because fi(b) 0 interior (b, b'). So let y e (a, b) such that fi(y) = b#. Thus fn(y) = fn-ifi(y) = fn-i(b#) = fn-i(b) 0 (y, y'). On the other hand fn(y) E (y, y'). This shows that n(a) = a. Summarizing, wehavethat if i A STRUCTURE THEOREM 543 ic point in the boundary of the immediate basin of a', and similarly for b. Hence Df2m(a#) ? 1 and Df2m b#) ? 1. Since Df2m( a) = Dfm(fm(a#)), Dfm(a#) ? 1 and int(a#, fm(aJ#)) is in the basin of ad, we can choose a, b E (a, b) so that IDfm(a-#)I ? 1, 1Dfm(b#)? 2 1 and (a, b) - (a-, b) is in the basin of attractors. So fm(J) C (a, b). Let a = inf{IDf(x)I/IDf(x')I, x near c }). Notice that a is positive because f is non-flat at c. Then I Dfm(ad) I > a and I Dfm(b) I 2 a. Claim: Each point x E N belongs to at most three of the intervals f (a, b) for i=O,...,m-1. In fact, [a-,b]=(a-,a) U(ab)U(bb), (a-,a) isin the basin of a# and int(b, b) is in the basin of b#. Therefore, each point x belongs to at most one of the intervals fi(a, b), i = 1,..., m (because they are disjoint), to at most two of the sets fi((ad, a) U (b, b)) because the basins of two different attractors are disjoint and the period of a# and b# are either n or n/2. Hence each point belongs to at most three of the intervals fi(a, b) and the claim is proved. Let us take, in Theorem (3.3), T = (a, b). Then we have Dfm(x) I > C2 min I Dfm(a) lDfm(b) } > C2a, for all x E fk(n)(J). In particular, Ifk(n+1)(J)I > C2alfk(n)(J)I. This finishes the proof of Proposition (4.11). Let us now show that there are no wandering intervals for unimodal maps as above. Choose a subsequence ni -x oc such that I fk(ni-1)(J) I > I fk(ni)(J) | From Lemma (4.5) we have that fk(n:)(Lk(n)) D (fk(ni- 1)(J))# and fk(ni)(Rk(nl)) D (fk(ni + 1)(J))#. Hence fk(ni)(Lk(n) > a fk(ni )(J) and |( )(Rk(n,))| > oa| >kn~l() ?P4ealkni(l where, as above, a = inf{ I Df(x) I / I Df(x') I }. From Corollary (2.4) we have fk(nif)(Lk(na) fk (Rk(ni)) ILk(nf)I IRk(ni)I _ Lfkn 1( fk(fl)Lk(ni) ) ? | fk(ni)(j )I ? f k(ni)( R( ) )A This content downloaded from 205.133.226.104 on Sun, 23 Nov 2014 23:39:21 PM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp 544 W. DE MELO AND S. VAN STRIEN By shrinking Kni we get Ifk'nj)(ji)j | fk(ni)(1fJ) I Ifk(ni)(j J I I fk(ni) (Aq ) I Ik(ni)l I k(ni)l 1 ~ lli 18 < 12 1fk(ni)(j)I X (I fk(ni)(L + I fk(ni)(j) + fk(ni)( i)| for every interval K2 = U J U R c K . Choose Li and RA so that Ifk(ni)(Li) I = min{Ifk(ni)(j) ,al fk(ni-)(I)I}, I fk(ni)(.) | = min I fk(ni)(j) I, eal fk(ni)(J)I. Then If (Li) | f (Ri) ] < 3 x 18 Ifk(ni)() 12 k(ni)I IRk(ni)I IJIj2 Ifk(ni)()I I 1 Ifk(ni)()I 1\ fk(n q 5i)( s tmxt1 ea Jt fk(ni)( Li) I < maxt 1,-a because Ifk(ni- 1)(J) I > Ifk(ni)(j) I Hence 1 18 ( 1 \ ( 1\ This contradicts Lemma (4.4) and proves the theorem. 5. Non-existence of wandering intervals for some other maps In this section we consider maps, with an arbitrary number of critical points, which satisfy the so-called Misiurewicz condition. 5.1. THEOREM. Let f: N -- N be a C3 endomorphism, where N is either the interval [0, 1] or the circle S'. Suppose f satisfies the following properties: a) every critical point is non-flat; b) the forward orbit of each of the critical points does not accumulate to the critical set C( f). Then f has no wandering intervals. Proof Suppose, by contradiction, that f has a wandering interval J. We may assume that Z00 ?=Ifk(j) < where 8 is the constant in Theorem 2.1. For each integer n, let Tn be the maximal interval containing J so that fn is monotone on Tn. Let Ln and Rn be the connected components of Tn - J. As in Lemma 4.4 we have that the lengths of Ln and Rn tend to zero as n -x oc. In This content downloaded from 205.133.226.104 on Sun, 23 Nov 2014 23:39:21 PM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp A STRUCTURE THEOREM 545 particular, if n is big enough, we have ILn1 IRnI < 611j2, where ? is as in Theorem 2.1. Therefore, from this theorem and Lemma 2.2 we get: (I n I 1C2 I fn (Ln U j) I I fn(j U R)I where C2 = 1jl2/(ljl4 + 8). On the other hand, by maximality of Tn, (5.2) Clos(Tn) - Tn C P = U fk(C(f)) k=1 By Lemma 4.1, there exists a sequence n(i) -x oc such that fn(i)(J) converges to C(f). Since, by hypothesis, Clos(P) l CQf ) = 0 we get that Ifn(i)(Ln(0) I and jfn(i)(Rn(i,)J do not tend to zero. This and (5.1) imply that ILn(0)J IRn(0) does not tend to zero. This contradiction proves the theorem. IMPA, RIO DE JANEIRO, BRAZIL TECHNICAL UNIvESrry, DELFT, THE NETHERLANDS REFERENCES [C-El] P. COLLET and J. EcKmANN, Iterated Maps of the Interval as Dynamical Systems, Birkhauser (1980). 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[M-M-S] M. MARTENS, W. DE MELo and S. VAN STRIEN, Julia-Fatou-Sullivan theory for real one-dimensional dynamics, preprint, Delft University. [M-T] J. MILNOR and W. THURSTON, On iterated maps of the interval: I, II, preprint, Princeton University (1977). [Mi] M. MisiuREwicz, Absolutely continuous measures for certain maps of an interval, Publ. Math. IHES 53 (1981), 17-51. [N-S] T. NoWICKI and S. VAN STRIEN, Absolutely continuous measures for Collet-Eckmann maps without Schwarzian derivative conditions, Invent. Math. 93 (1988), 619-635. This content downloaded from 205.133.226.104 on Sun, 23 Nov 2014 23:39:21 PM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp 546 W. DE MELO AND S. VAN STRIEN [Pa] W. PARRY, Symbolic dynamics and transformations of the unit interval, Trans. A.M.S. 122 (1966), 368-378. [Si] D. SINGER, Stable orbits and bifurcations of maps of the interval, SIAM J. Appl. Math. 35 (1978), 260-267. [Sc] A. 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(Received September 18, 1986) This content downloaded from 205.133.226.104 on Sun, 23 Nov 2014 23:39:21 PM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp Article Contents p. [519] p. 520 p. 521 p. 522 p. 523 p. 524 p. 525 p. 526 p. 527 p. 528 p. 529 p. 530 p. 531 p. 532 p. 533 p. 534 p. 535 p. 536 p. 537 p. 538 p. 539 p. 540 p. 541 p. 542 p. 543 p. 544 p. 545 p. 546 Issue Table of Contents Annals of Mathematics, Second Series, Vol. 129, No. 3 (May, 1989), pp. 427-652 Compactifying Complete Kähler-Einstein Manifolds of Finite Topological Type and Bounded Curvature [pp. 427-470] Multiplicity Estimates on Group Varieties [pp. 471-500] Algebraische Punkte auf Analytischen Untergruppen algebraischer Gruppen [pp. 501-517] A Structure Theorem in One Dimensional Dynamics [pp. 519-546] Arithmetic Theory of Arithmetic Surfaces [pp. 547-589] Excision in Cyclic Homology and in Rational Algebraic K-theory [pp. 591-639] Ondelettes et Intégrale de Cauchy sur les Courbes lipschitziennes [pp. 641-649] Correction to "Martin's Maximum, Saturated Ideals, and Non-Regular Ultrafilters. Part I" [pp. 651]