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Author(s): WEIXIAO SHEN

Article Title: On the measurable dynamics of real rational functions

Year of publication: 2003

Link to published

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DOI: 10.1017/S0143385702001311 Printed in the United Kingdom

On the measurable dynamics of real

rational functions

WEIXIAO SHEN

Department of Mathematics, University of Warwick, Coventry CV4 7AL, UK (e-mail: wxshen@maths.warwick.ac.uk)

(Received 13 December 2000 and accepted in revised form 26 June 2002)

Abstract. Letf be a real rational function with all critical points on the extended real axis and of even order. Then:

(1) f carries no invariant line field on the Julia set unless it is doubly covered by an integral torus endomorphism (a Latt´es example); and

(2) f|J (f )has only finitely many ergodic components.

1. Introduction

In this paper, we study the measurable dynamics of a real rational functionf, which satisfies the following two conditions:

(C1) any critical pointcoff is contained inS1= ˆR=R∪ {∞}; (C2) any critical pointcoff is of even order.

Let F denote the set of all real rational functions satisfying these two conditions. Our main result is the following theorem.

THEOREM1. AnyfFcarries no invariant line field on the Julia setJ (f )unless it is a Latt´es example.

THEOREM2. LetfF.f|J (f )has only finitely many ergodic components.

It is conjectured that these theorems hold for all rational functions, see [16, 18, 22]. The question in the first main theorem is closely related to the well-known Fatou conjecture: in the space Ratd of all rational maps (and in the space Polyd of all

poly-nomials) of degreed ≥2, hyperbolic ones form an open and dense subset. Our first main theorem implies the following corollary.

COROLLARY3. Letf be a real rational map of degreed2 for which all critical points

are on the real axis. Suppose thatf is structurally stable in the space Ratd. Thenf is

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Proof of Corollary 3. Since f is structurally stable, the Teichm¨uller space of f has dimension 2d −2. In particular, all the critical points are non-degenerate which implies that condition (C2) holds. ThusfF. It follows from Theorem 1 thatf carries no invariant line field on the Julia set. By Theorem 9.4 of [22],f is hyperbolic. Some progress in the direction of this work had been made by several authors, mostly on the case that f is a quadratic polynomial. For a non-renormalizable polynomial, the absence of an invariant line field follows from the work of Yoccoz (see [9]), while McMullen [20] proved this for infinitely renormalizable quadratic polynomials which are ‘robust’, including all real ones. McMullen’s method has been refined by Levin and van Strien [12] who generalized the result to all real unimodal polynomials. For polynomials with more critical points, the only result, to my knowledge, is in [7, 8], where the authors prove the absence of invariant line fields for a real polynomial with only real critical points and such thatf is infinitely renormalizable and of bounded type. The ergodic decomposition was proved by Lyubich [14] for a smooth one-dimensional dynamics with non-flat critical points and in the category of holomorphic dynamics, by Prado [25] for a real unimodal polynomial. In the proof of all these results, some kind of ‘bound’ plays a crucial role.

In this paper, Theorems 1 and 2 will be proved using the same method and the real bounds developed in [26] play a crucial role in our proof.

The main difficulty is to study the measurable dynamics on the partJc = {zJ (f ) :

ω(z)=ω(c)}for a recurrent critical pointcwith a minimalω-limit set. It will be done in the same outline as in McMullen [20]. If a large real bound exists, then we shall transfer it to a ‘complex bound’ for a quasi-polynomial-like mapping, as in [12]. In the converse case, our argument will depend heavily on the essentially bounded geometry of the Cantor setω(c). Comparing this work with that in [20], a new ingredient is the fact that the absence of a large real bound implies thatcis ‘uniformly persistently recurrent’ in some sense, which is also proved in [26].

Let us briefly outline the structure of this paper. Section 2 contains some technical preliminaries, including some distortion lemmas. In §3, we recall the definition and give some sufficient condition for the absence of invariant line fields. In §4, the notions of real box mappings and quasi-polynomial-like mappings are recalled. Section 5 is devoted to the study of the measurable dynamics on the subset of the Julia set consisting of points accumulating on a reluctantly recurrent critical point. It turns out that we will only need to consider the partJcfor a persistently recurrent critical point. In §6, we study the dynamics

off|Jcwhen a large bound exists forc. In §7, we shall study the converse case and then

complete the proof of the main theorems.

2. Preliminaries

2.1. Distortion lemmas. We state some well-known distortion lemmas. Let (r) =

{z∈C: |z|< r}and let=(1).

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If h : UV is a proper map of degree N between two hyperbolic topological disksU, V, then for anyz0 ∈ U, we haveh({zU : dU(z, z0) < }) ⊃ {wV :

dV(h(z0), w) < δ}, wheredUanddV are the hyperbolic metric ofUandV respectively.

Moreover,0 asδ0 for a fixedN.

LEMMA2.2. For any 0< r <1 andN ∈ N, there is a constantC =C(r, N ) >1 with

the following property.

IfU ⊂Cis a topological disk andh:Uis a proper map of degreeN, then for any componentU0ofh−1((r)), we have diam(U0)2/area(U0)C, where the diameter and area are measured in the Euclidean metric ofU.

LEMMA2.3. For anyδ >0 andN ∈ N, there is a constant =(N, δ) > 0 with the

following property.

LetU be a simply connected subset ofandh : Ube a proper map of degreeN. LetXbe a measurable set withm((1/2)X)/m((1/2))≤1−δ, then for any componentU0ofh−1((1/2)), we havem(U0∩h−1(X))/m(U0)≤1−, wherem denotes the Lebesgue measure on.

All these lemmas follow easily from the following fact:

{h:;his holomorphic and proper of degree ≤d, h(0)=0}

is a compact set in the topology of uniform convergence on compact sets. We leave the (easy) proofs of this fact and the lemmas to the readers.

2.2. Real rational functions. Let f be a real rational function satisfying the condition (C1).

We shall useS1to denote the extended real lineRˆ =R∪ {∞}and use Par(f ),C(f )to denote the set of parabolic periodic points and the set of critical points off, respectively. All the metrics in this section are the spherical metric onCˆ. LetP (f )denote the post-critical set off, that is,

P (f )=

cC(f )

n=1 {fn(c)}.

The mapf|S1 : S1 → S1can be considered as a one-dimensional dynamical system, which has no wandering interval, due to [19].

PROPOSITION2.1. LetyJ (f )S1−Par(f ). For any > 0, there is aδ >0 such

that for any intervalIS1and anyn∈N, iffn(I )B(y, δ), then|I| ≤.

Proof. Otherwise, there is a positive integer0>0, and for anyk∈N, there is an interval

IkS1with|Ik| ≥0, and a positive integernksuch that

fnk(I

k)B(y,1/k).

After passing to a subsequence, we may assume that there is a non-degenerate interval

IIk for allk. Then, lim infn|fn(I )| = 0. Thus, there is an attracting or parabolic

periodic pointpoff, such that supxId(fn(x), fn(p))→0 asn→ ∞.

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For any critical pointc, there is a unique involutionτcdefined byf (τc(x))=f (x)on

a neighbourhoodUc inS1. We shall fix for any critical point such a neighbourhoodUc

such that they are pairwise disjoint. An intervalIS1is called symmetric if it contains a critical point, saycand ifIUcandτc(I )=I.

For any open intervalIS1, letCIˆ denote the hyperbolic surface Cˆ −(S1−I ). LetD(I )denote the disk such thatD(I )S1=I and the boundary∂D(I )intersectsS1

orthogonally. The following is an observation of Sullivan, see [29].

LEMMA2.4. There is a universal constantθ > 0 such thatD(I ) = {z ∈ ˆCI : d(z, I ) < θ}, whered denotes the hyperbolic distance inCIˆ .

PROPOSITION2.2. LetfF. Then either of the following holds:

(i) J (f )= ˆCandf is ergodic; or

(ii) for a.e. zJ (f ), and anyyω(z)−Par(f ), there is a recurrent critical pointc

such thatω(z)candω(c)y.

Proof. LetEJ (f )be a measurable set of positive measure such thatf (E)E (mod 0)

andm(Cˆ −E) >0. LetzEbe an arbitrary Lebesgue density point ofE. Letyω(z)

be an arbitrary point which is not a parabolic periodic point. We claim that there is some recurrent critical pointcoff such thatcω(z)andyω(c).

First of all, we choose a small positive number >0 so that

(1) for anycC(f )which is not recurrent, we havefn(c)B(c, )for anyn ∈ N; and

(2) for anycC(f )ω(z), we havefn(z)B(c, )for anyn∈N.

Letδ >0 be a small positive number. LetD0 =B(y, δ/2)D =B(y, δ)be small

disks centred aty. There exists a sequence of positive integersn1 < n2 <· · · such that

fnk(z)y as k → ∞. LetA

k (A0k, respectively) denote the component offnk(D)

(fnk(D

0), respectively) containingz.

Iffnk :A

kDhas a uniformly bounded degree, then sincezis a Lebesgue density

point ofE, by Lemmas 2.2 and 2.3,m(D0−E) = 0. SinceEJ (f ), we then have

D0 ⊂ J (f ). Sofn(D0) = ˆCfor somen ∈ Nand henceE = ˆC(mod 0), which is a

contradiction. So after passing to a subsequence we may assume thatfnk|A

khas a degree

which tends to infinity ask→ ∞. For the same reason, we may assume thatfnk|A0

k also

has a large degree.

Thus, there are 0≤ pk < qk < nk such thatfpk(Ak)andfqk(Ak)contain the same

critical pointckandfnk−pk|fpk(Ak)has a uniformly bounded degree. By Lemma 2.1, the

hyperbolic diameter offpk(A0

k)infpk(Ak)is uniformly bounded from above. By passing

to a further subsequence, we may assume thatck =cfor allk.

Ifδ was chosen small, then it follows from Proposition 2.1 that max{|fn(Ak)S1| :

pknnk}< . Since the forward orbit ofcentersfqk(Ak)S1, which is contained in

B(c, ), it follows from (1) thatcis a recurrent critical point. Sincefpk(A0

k)S1= ∅, and

fpk(A

k)⊂ ˆCfpk(Ak)S1, and since the hyperbolic diameter off

pk(A0

k)in the hyperbolic

surfacefpk(A

k)is bounded from above, we have that the spherical diameter offpk(A0k)

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Sinceδcan be taken arbitrarily small, the claim follows. The proposition easily follows

from the claim.

LetCr(f )C(f )denote the set of non-periodic recurrent critical points. For any

c, dCr(f ), we saycd ifω(c)d andω(d) c. An order is defined on the set of

equivalence classes as follows:[c] ≺ [d]ifω(c)dbutω(d)c. Let

Jc= {zJ (f ):ω(z)cbutω(z)dfor any[d] ≺ [c]}.

It follows from Proposition 2.2 that eitherJ (f )= ˆCandf is ergodic or

J (f )

cCr(f )

Jc(mod 0).

3. Invariant line fields

Recall that a line fieldµis a measurable Beltrami differential onCˆ such that|µ| =1 on a setEof positive measure andµvanishes elsewhere. The setEis the support ofµ. A line field is holomorphic on an open setU ifµ = ¯φ/|φ|a.e. forφa holomorphic quadratic differential onU. A line fieldµis called (f-)invariant iff(µ) = µa.e. We say thatf

carries an invariant line field on the Julia set if there is anf-invariant line fieldµwith a support contained in its Julia set (up to a set of measure zero).

A rational mapf is called a Latt´es example if it is doubly covered by an integral torus endomorphism. Such a rational map carries an invariant line field on its Julia set, which isCˆ. It is a conjecture that Latt´es examples are the only rational maps which carry an invariant line field on the Julia set, see [18, 22].

It is useful to remember the following, which is Lemma 3.16 in [20].

LEMMA3.1. Letµbe anf-invariant line field which is holomorphic on a non-empty open set contained in the Julia set. Thenf is a Latt´es example.

PROPOSITION3.1. LetfF. Assume that f carries an invariant line field µ with supportEJ (f ). Then either

(i) for a.e. zEand for anyyω(z)which is not a parabolic periodic point, there existscCr(f )such thatcω(z)andω(c)y; or

(ii) f is a Latt´es example.

Proof. Take a Lebesgue density pointzofEand let yω(z)−Par(f ). If there is no

cCr(f )ω(z)such thatω(c) y, then by the same argument as that in the proof

of Proposition 2.2, we will obtain a sequence of proper mappingsfnk : A

kB(y, δ)

for someδ > 0 such that xAk and fnk|Ak has a uniformly bounded degree, and

fnk(x)yask→ ∞. It then easily follows thatµis holomorphic on some non-empty open set and hencef is doubly covered by an integral torus endomorphism. We shall give more criteria for the non-existence of invariant line fields. The idea is taken from [20] and [21]: Uniformly nonlinearity implies the absence of invariant line

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An invariant line fieldµis called almost continuous at a pointxif for any >0,

m({zB(x, r): |µ(z)µ(x)| ≥})

m(B(x, r)) →0

asr → 0, whereµ = µ(z) dz/dz¯ on a conformal coordinate aroundx. Note that the definition does not depend on the choice of the coordinate.

Given a rational functionf, consider the collection H(f ) of all holomorphic maps

h:UV, whereU, V are open sets such that there existi, j ∈Nsuch thatfih=fj

onU. Obviously for any elementh:UV inH(f ),h|V )=µ|U, i.e.

µ(z)=µ(h(z))h(z)/h(z)a.e. onU.

PROPOSITION3.2. Letf be a rational function of degree2 andxbe a point inJ (f ). If there is a positive constantC > 1 and a positive integer N2 and a sequence

hn:UnVnof elements ofH(f )with the following properties:

(i) Un, Vnare topological disks and

diam(Un)→0, diam(Vn)→0

asn→ ∞;

(ii) hnis a proper map with degree between 2 and N;

(iii) for someuUnsuch thathn(u)=0 and forv =hn(u)we have

max

z∂Un

d(z, u)Cd(u, ∂Un)

and

max

z∂Vn

d(z, v)Cd(v, ∂Vn);

(iv)

d(Un, x)Cdiam(Un), d(Vn, x)Cdiam(Vn),

where diam and, d denote the diameter and the distance in the spherical metric respectively.

Then for anyf-invariant line fieldµ,x∈supp(µ)orµis not almost continuous atx. Proof. By changing the coordinates using a M¨obius transformation, we can assume that

x ∈C. The conditions in (i), (iii) and (iv) also hold for the Euclidean metric by changing the constantC. Here, all distances, diameters and areas are measured in the Euclidean metric onCunless otherwise stated.

Fix anyunUnsuch thathn(un)=0 and letvn =hn(un). Denote byαnandβnthe

linear transformation ofCsuch thatαn>0, βn >0 and

αn(un)=βn(vn)=0, diam(αnUn)=diam(βnVn)=1.

DenoteXn =αnUnandYn =βnVn. And denote byHnthe functionβnhnαn−1 :

XnYn. By condition (iii), we have

min

z∂Xn

|z| ≥ max

z∂Xn

|z|/C≥1/(2C),

min

z∂Yn

|z| ≥ max

z∂Yn

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ThusXn, YnB(0,1/(2C)). So after passing to a subsequence we may assume that

(Xn,0)(X,0)and(Yn,0)(Y,0)asn→ ∞in the Caratheodory topology for some

topological diskX, Y and there is a holomorphic functionH :(X,0)(Y,0)such that for any compact setKinX,Hn|KH|Kuniformly asn→ ∞.

CLAIM. The functionHis not a constant function.

Indeed, sinceXnB(0,1), for someδ =δ(C) >0, in the hyperbolic surfaceXn, the

hyperbolic ballBXn(0, δ)is contained in the Euclidean ballB(0,1/(4C)). By Lemma 2.1, there is a constant > 0 which depends only onC andN, such thatHn(B(0,1/(4C)))

contains the hyperbolic ball BYn(0, ) in the hyperbolic surface Yn. Since Yn

B(0,1/(2C)),BYn(0, )B(0, )for some>0. For anyn∈N, letznB(0,1/(4C)) be a point such that|Hn(zn)| =. After passing to a further subsequence, we may assume

thatznzasn→ ∞. Thus|H (z)| =. NoteH (0)=0, and hence the claim follows.

Consequently,H is a proper map of degree between 2 andN.

Assume now that|µ(x)| = 1 and thatµis almost continuous atx. Without loss of generality we can assume thatµ(x)=1. Let us prove that for anyδ >0, we have

m({zUn: |µ(z)−1| ≥δ})

m(Un)

→0, (1)

asn→ ∞.

Letrn=maxzUnd(x, z)andsn =d(un, ∂Un). Thenrn→0 asn→ ∞. Thus, by the definition of almost continuity, we have

m({zB(x, rn): |µ(z)−1| ≥δ})

m(B(x, rn))

0.

Note that by (iii) and (iv), we have

m(B(x, rn))rn2sn2m(Un).

Thus, statement (1) holds.

Similarly, we can prove that for anyδ >0,

m({zVn: |µ(z)−1| ≥δ})

m(Vn)

→0, (2)

asn→ ∞.

Letµn =(αn−1)|Un)be a Beltrami differential onXn andνn = (βn−1)|Vn)be

a Beltrami differential onYn. ObviouslyHnνn =µn. It follows from (1) and (2) that for

anyδ >0

m({zXn: |µn(z)−1| ≥δ})→0, (3)

and

m({zYn: |νn(z)−1| ≥δ})→0, (4)

asn→ ∞.

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SinceHn|LH|Luniformly, we have that fornsufficiently large,Hn|Lis univalent and

also|Hn(z)| ≥ηonL.

By (3) and (4), for anyδ >0, we have

m{zL: |µn(z)−1| ≥δ} →0

and

m{wHn(L): |νn(z)−1| ≥δ} →0

asn→ ∞. Note that

m{zL: |νn(Hn(z))−1| ≥δ} ≤η−2m{wHn(L): |νn(w)−1| ≥δ} →0

asn→ ∞.

For anyδ(0,1), let

An(δ)=

zL:Hn(z) Hn(z)−1

δ

asn→ ∞. For a.e.zAn(δ), we haveµn(z)=νn(Hn(z))Hn(z)/Hn(z). Thus, for such

az, either|νn(Hn(z))−1|> δ/10, or|νn(z)−1|> δ/10. Thus, we have

m(An(δ))→0

asn→ ∞. Consequently, we have that, for anyδ >0,

m

zL:H

(z)

H(z)−1

> δ

=0.

Sinceδcan be taken to be arbitrarily small, we know that for a.e.zL,H(z)=H(z). But thenHis a constant function, and hence it is equal to 0, i.e.His a constant function,

which is a contradiction.

To construct the family{hn}we usually first construct a sequence of restrictions of

forward iterates off, then try to pull them back to the neighbourhood ofx.

For any annulusAwith a finite modulus, we define the core curve ofAto be the unique simple geodesic inA. In other words, ifφ :A→ {z∈C:1 <|z| < R}is a conformal map, then the core curve isφ−1({z ∈ C: |z| =√R}). For any topological diskU, and any compact setKU, we use mod(K, U )to denote the supremum of the modulus of an annulus which is contained inUand surroundsK.

COROLLARY3.3. Letf be a rational function of degreed2 andx be a point in the Julia setJ (f ). If there exists a positive constantδ, a positive integerN2, sequences

{sn},{pn},{qn}of positive integers, sequences{An},{Bn} of topological disks with the

following properties:

(i) fsn:A

nBnis a proper map whose degree is between 2 andN;

(ii) diam(Bn)0 asn→ ∞;

(iii) fpn(x)A

n, fqn(x)Bnand

mod({fpn(u):uA

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(iv) for anyn∈N, there existsgnA:An→CandgnB:Bn→Csuch that

fpngA

n =idAn, g

A n(f

p

n(x))=x

and

fqngB

n =idBn, g

B

n(fqn(x))=x;

(v) sn→ ∞, pn→ ∞, qn→ ∞asn→ ∞;

then for anyf-invariant line fieldµ,µ(x)=0 orµis not almost continuous atx. Proof. By condition (iii) there is an annulus;nBn surrounding{fpn(u) : uAn,

(fpn)(u)=0} ∪ {fpn+sn(x), fqn(x)}such that mod(;

n)=δ. Letγnbe the core curve

of;nandBn be the topological disk bounded by this curve. There is a constantC(δ) >1

such that for anyb∈ {fsn(u):(fsn)(u)=0} ∪ {fpn+sn(x), fqn(x)}, max

zγn

d(z, b)C(δ)d(b, γn).

LetAnbe the component offsn(B

n)contained inAn. LetUn=gnAAn, Vn=gnBBn and

hn=gnBfsngnA:UnVn.

Then{hn:UnVn}is a sequence satisfying the hypothesis of Proposition 3.2.

4. Quasi-polynomial-like mappings

In this section, we recall two notions: one is ‘a real box mapping’ which we used in [26]; and the other is ‘a quasi-polynomial-like mapping’, used by Levin and van Strien in [12], which turns out to be the complex counterpart of the first notion.

Fix anfF. A (real) maximal chain is a sequence of open intervals{Gi}ni=0such that

f (Gi)Gi+1andGi is the maximal interval with this property, for any 0≤in−1.

The order is the number ofGis containing a critical point. If{Gi}ni=0is a maximal chain

we shall say thatG0is a pull-back ofGn. IfGi does not contain a critical point for any

0≤in1, then the chain is called monotone andG0is called a monotone pull-back of

Gn. IfGi does not contain a critical point for any 0< i < nbutG0contains exactly one,

we shall say that the chain is unimodal, andG0is a unimodal pull-back ofGn.

Recall that an open interval T is called nice if fn(∂T )T = ∅ for any n ∈ N. Clearly, ifGandGare two pull-backs ofT then either they are disjoint or one is contained in the other. We call an intervalT properly periodic if there is a positive integers ≥ 1, such that the interiors ofT , f (T ), . . . , fs−1(T ) are pairwise disjoint andfs(T )T,

fs(∂T )∂T. In this casesis called the period ofT andfs :TT will be called a

renormalization off. Let us say thatf is renormalizable atcif there is a properly periodic interval containingcand thatf is infinitely renormalizable atcif there are a sequence of properly periodic intervals{Tn}containingcwith periodssn→ ∞asn→ ∞. Iff is not

infinitely renormalizable atc, we shall say thatf is only finitely renormalizable atc. LetcCr(f )and letc1, c2, . . . , cbbe the critical points in[c]. LetIi ci be small

nice intervals,i = 1,2, . . . , b. Let Jij, 0 ≤ j < ri +1 be pairwise disjoint intervals

contained inIi such thatJi0ci, whereri ∈N∪ {0,∞}. A mapping

B:

b

i=1

ri

j=0

Jij

b

i=1

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is called a (real) box mapping (induced byf) corresponding tocif for each 1≤iband each 0≤j < ri+1 there is ak(i, j )∈ {1,2, . . . , b}andp(i, j )∈Nwith the following

properties:

(B1) B|Jij =fpi,j|Jj

i ;

(B2) there is a maximal chain{Gk} pi,j

k=0 withGpi,j = Iki,j andG0 = J

j

i , moreover the

chain is unimodal ifj =0 and monotone otherwise; (B3) Jij ∩orb([c])= ∅;

(B4) orb([c])Ii

ri

j=0J

j i ;

(B5) for anyx∈orb([c])Jij, we havef (x), f2(x), . . . , fpi,j−1(x)b

i=1Ii.

Remark 4.1. In the language of [26], (5) is a box mapping of type II which is the first return

map to its image.

The following is a natural way to construct box mappings. For any small nice interval

I cand anyc ∈ [c],c = c, letI (c)be the component containingc of the domain of the first return map (off) toI. Denote these intervalsI (c)byI2, I3, . . . , Iband put

I1=I. For any 2≤ibletri =0 and letJi0=Ii. For anyx ∈orb([c])I, letJ (x)

denote the component of the domain of the first return map toIcontainingx. Letr1+1 be

the number of these intervalsJ (x)and letJ10c, J11, . . . be these intervals. Finally define

BI :

b i=1

ri

j=0J

j

i

b

i=1Ii to be the first return map off to

b

i=1Ii. We shall call

BIthe box mapping associated toI. Note that ifω(c)is a minimal set, thenr1is finite.

Let Vi, be topological disks containing ci, i = 1,2, . . . , b, such that Vi1 ∩Vi2 ∩ orb([c]) = ∅ for any 1 ≤ i1 < i2 ≤ b. For each 1 ≤ ib, letUij be topological

disks contained inVi, 0 ≤ j < ri +1, for some ri ∈ N∪ {0,∞}such that for any

0≤j1< j2< ri+1, we haveUij1∩Uij2∩orb([c])= ∅.

A collectionFi,j : Uij

b

i=1Vi1 ≤ ib,0 ≤ j < ri +1

is called a

quasi-polynomial-like mapping (induced byf) corresponding tocif the following holds. For any 1≤iband any 0≤j < ri +1, we have 1≤ki,jbandpi,j ∈N, such that:

(H1) Fi,j =fpi,j|Uij;

(H2) Fi,j(Uij)=Vki,j and the mapFi,j is a branched covering with a unique critical point

ci forj =0 and is conformal otherwise;

(H3) for any 1≤iband 0≤j < ri+1, orb([c])Uij = ∅;

(H4) for any 1≤ib, we have orb([c])Vi ⊂orb([c])rji=0Uij ;

(H5) for any 1≤iband 0≤j < ri+1 and anyx∈orb([c])Uij, ifFi,j =fp, then

f (x), f2(x), . . . , fp−1(x)bi=1Vi.

Remark. Quasi-polynomial-like mappings are not mappings in the classical sense.

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A quasi-polynomial-like mapping F= {Fi,j}determines a map

F :

i,j

Uij

(i,j)=(i,j )

Uij

i

Vi

in a natural way. LetK(F)be the set of pointszsuch thatFn(z)is well defined for any

n∈N. We shall callK(F)the Julia set of{Fi,j}, which is always a measurable set. By the

definition, orb([c])iVi belongs to the setK(F).

For anyxK(F), and anyn ≥ 0, there is a unique(in, jn)such thatFn(x)Uijnn.

So for any topological disk Fn(x)AVin, we can consider the pull-back of A

under F along the orbitx, F (x), . . . , Fn(x), which is defined to be the topological diskP

containingx, such thatFim−1,jm−1 ◦Fim−2,jm−2◦ · · · ◦Fi0,j0(P )U

jm

im for all 0≤m < n andFin−1,jn−1◦Fin−2,jn−2◦ · · ·◦Fi0,j0(P )=A. Note that such a topological diskP always exists. We shall writeP =Compx(F, n, A)and sometimes we will useFn :PAto denote the mapFin−1,jn−1 ◦ · · · ◦Fi0,j0|P. Similarly, we shall useV (F, x) to denote the topological diskVi containingx, andU (F, x)the topological diskUij containingx. We

shall sometimes omit the quasi-polynomial-like mapping F in these notations, provided that it is clear from the context which F is referred to.

LEMMA4.1. If 0m < nare two integers andxK(F), then we have

Compx(n, V (Fn(x)))⊂Compx(m, V (Fm(x))).

Given a real box mapping as in (5), we can construct a quasi-polynomial-like mapping in the following way. For any 1≤ib, letVi = ˆCIi. For any 0≤j < ri +1, letU

j i be

the component offpi,j(V

i)containingJij, wherepi,j is as in (B1). LetFi,j =fpi,j|Uij.

Then{Fi,j}is a quasi-polynomial-like mapping. This construction can always be done but

the extended map may not possess any geometric property.

5. The measurable dynamics off|Jcfor a recurrent critical pointcwith a non-minimal

ω-limit set

The purpose of this section is to consider the dynamics off|Jcfor a recurrent critical point

csuch thatω(c)is not minimal. (Jcis defined at the end of §2.) The result is as follows.

THEOREM5.1. LetfF, andcCr(f )be such thatω(c)is not minimal. Then

(i) m(Jc)=0 orJ (f )= ˆCandf is ergodic; and

(ii) f carries no invariant line field onJc.

We shall use the real bound developed in [26] to construct a quasi-polynomial-like mapping corresponding toc. Theorem 5.1 will be shown by a combinatorial argument.

The following is Proposition 6.2 in [26].

PROPOSITION5.2. For any ρ > 0 and any > 0, there is a real box mapping

B : bi=1ri

j=0J

j

i

b

i=1Ii corresponding to csuch that maxib=1|Ii| ≤ ; and for

any 1ib, there exists a symmetric intervalTi containing theρ-neighbourhood ofIi,

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(i) ifB|Jij =fpandB(Jij)Ik, then the maximal chain{Gn}pn=1withGp=Tkand

G1⊃f (Jij)is a monotone chain; and

(ii) ifG0is the component off−1(G1)containingJij, thenJijTi; moreover,

(iii) ifJij =Ii, thenG0⊂Ii.

COROLLARY5.3. There is a quasi-polynomial-like mapping F = Fij : Uij

b

i=1Vi1 ≤ ib,0 ≤ j < ri +1

corresponding toc, such that bi=1ri = ∞

and for someC >0 andN ∈N, we have:

(i) for each 1ib, there is a topological diskWi ⊃⊃Vi, such that mod(WiVi)C;

(ii) for any 1iband 0j < ri+1, ifFi,j =fpi,j andFi,j(Uij)=Vk, then there

is a topological diskXijUijsuch thatfpi,j :Xj

iWkis a proper map of degreeN. Moreover, ifUij =Vi, thenXjiVi.

Proof. Let B : bi=1ri

j=0J

j

i

b

i=1Ii be a real box mapping satisfying the

requirements in Proposition 5.2 for a largeρ and small. Sinceω(c) is not minimal,

iri = ∞ if is sufficiently small. We shall extend this box mapping to a

quasi-polynomial-like mapping. To fix the notation, we assume that Ji0 = Ii if and only if

b+1≤ibfor some 1≤ib. By possibly interchanging the indices, we can always assume this to be so. For each 1 ≤ ib, letSiIi be the symmetric interval such

thatTi is theρ/2-neighbourhood ofSi and letWi =D(Si). For any 1≤ ib, define

Vi =D(Ii).

For anyb+1≤ib, lets=sibe the minimal positive integer such thatBsi(Ii)Ii

for some 1≤ ib. Such a positive integer always exists and is no more thanbb. IfBsi|I

i =fki|Ii, then letVibe the component offki(D(Ii))containingIi. LetV˜iVi

be a component offki(D(T

i)), thenfki : ˜ViDTiis a proper map of degree uniformly bounded from above. SinceVihas a small diameter in the hyperbolic Riemann surfaceWi,

Vi has a small diameter in the hyperbolic Riemann surface V˜i. ThusVi is compactly

contained inWiand the annulusWiVi has a large modulus.

For any 1≤iband 0≤j < ri+1, letm=m(i, j )be such thatB(Jij)Imand

letUij (Xij,Yij, respectively) be the component offpi,j(V

m)(fpi,j(Wm),fpi,j(CTˆ m), respectively) containingJij, wherepi,j ∈ Nis such thatB|Jij =fpi,j|Jij. Thenfpi,j :

Yij → ˆCTm is a proper map of a degree uniformly bounded from above. SinceWm has

a small diameter in the hyperbolic Riemann surfaceCTˆ m, by Lemma 2.1,X

j

i has a small

diameter in the hyperbolic Riemann surfaceYij. ThusXjiVi if 1≤ib, since in that

case, the real trace ofYij is contained inIi.

DefineFi,j|Uij =fpi,j|Uij. Then{Fi,j}becomes a quasi-polynomial-like mapping as

desired.

PROPOSITION5.4. Assume thatω(c)is not minimal. Then for someN ∈N, and for any

xJc, there are somed ∈ [c]and some neighbourhoodsU⊂⊂Uofdwith the following

property: for infinitely many positive integersn, there is a neighbourhoodP ofxsuch that

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Proof. Let F = Fi,j : Uij

b

i=1Vi1 ≤ ib,0 ≤ j < ri +1

be the quasi-polynomial-like mapping as in the previous proposition. The real trace of F is a real box mapping. Letpi,j, ki,j ∈ Nbe such thatFi,j = fpi,j|Uij, Fi,j(Uij)Vki,j. We may assume thatr1= ∞and that for some 1≤bb,Ui0=Vi if and only ifb+1≤ib.

LetF be the map determined by F as described at the top of page 967. For any 1≤ib, letti be the minimal non-negative number such that

Fti(c

i)b

i=1

Vi,

and let si be the non-negative integer such that Fti = fsi on a neighbourhood of ci.

(In particular,ti =si =0 for 1≤ib.)

For any 0 ≤ h < ∞, and any c ∈ [c], let ah(c) be the minimal non-negative

integer such that fah(c)(c)Uh

1 and let Ph(c) be the component of fah(c

)

(U1h)

containingc. Thenfah(c):P

h(c)U1his a proper map of a uniformly bounded degree.

Writeqh = p1,h+sk1,h. Then for some 1≤ ihb,f

ah(c)+qh :P

h(c)Vih is also a proper map of a uniformly bounded degree. For eachc ∈ [c], iffah(c)+qh(c)Ujh

ih, then letQh(c)be the component off(ah(c

)+q h)(Ujh

ih)which containsc .

LEMMA5.1. For any xJc and any large h, if nh is the minimal non-negative

integer such thatfnh(x)

d∈[c]Qh(d), and ifc = ch is the critical point such that

fnh(x)Q

h(ch), and ifAh(x)is the component offnh(Ph(c))containingx, then the

proper mapfnh:A

h(x)Ph(c)has a degree uniformly bounded from above.

Proof. LetEhk =fk(Ah(x))for 0≤ knh. We need to count the number of times a

critical point appears in someEkh. LetTkh =EkhS1. Then{Tkh}nh

k=0is a (real) maximal

chain. (We admit the situation thatTkh = ∅for somek.) Since all critical points off

are contained inS1, it suffices to show that the maximal chain{Tkh}nh

k=0 has a uniformly

bounded order. Note that forhsufficiently large maxnkh−=01|Tkh|is small sincef|S1has no wandering interval. So eachEkhcontains, at most, one critical point. For the same reason, anydC(f )Cr(f )appears in, at most, one of the domainsEkh.

Assume that the maximal chain has order larger than 2#C(f ). Then there exists some

dCr(f )such thatdTkh1, Tkh2 for some 0 ≤ k1 < k2 ≤ nh and the maximal chain {Tkh}nh

k=k1 has order≤2#C(f ). Sincef

nh−k1 :Eh

k1 →Ph(c

)has a degree bounded from above andQh(c)has a small diameter in the hyperbolic Riemann surfacePh(c), and since |Tkh

1|has a small diameter,f

k1(x)is close tod. If such adappears for infinitely manyhs, then it is contained inω(x). SincexJc, we must haved ∈ [c]. So there are positive

integersm1 > m2such thatfnh−k1 = Fm1,fnh−k2 = Fm2 holds in a neighbourhood

ofd. It is then easy to see thatEkh

2 ⊂Ph(d)andE

h

k1 ⊂Qh(d). Sincef

k1(x)Eh

k1, this contradicts the minimality ofnh. Thus the maximal chain{Tkh}knh=0has order≤2#C(f ).

The proof of the lemma is then completed. We continue the proof of Proposition 5.4. By now, we have, for eachh large, a proper map fnh+ah(ch)+qh : A

h(x)Vih of a uniformly bounded degree such that

fnh+aj(ch)+qh(x)is contained inUjh

ih for some 1≤ihb

and 0j

h< rih+1. By the previous proposition, for eachh, there is a topological diskXjh

ihVih such thatf

pih,jh :

Xjh

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Proof of Theorem 5.1. LetXJc be a measurable set of positive measure such that

f (X)X (mod 0). LetxXbe a Lebesgue density point. Letd ∈ [c], andU ⊂⊂ U

be neighbourhoods of d as in the previous proposition. There exist n1 < n2 < · · ·

neighbourhoodsAj ofx, such thatfnj : AjU has a uniformly bounded degree

andfnj(x)U. Hence by Lemma 2.2 and 2.3, m(UX) = 0. Since fn(U ) = ˆC

(mod 0)for somen ∈ N, we have thatX = ˆC(mod 0). ThereforeJ (f ) = ˆCandf is ergodic.

Iff carries an invariant line fieldµonJc, letx be an almost continuous point ofµ.

One can construct a sequence of proper mappingsfnj :A

jUas in the previous proof,

which then implies thatµis holomorphic on a non-empty open set. Thus f is doubly covered by an integral torus endomorphism. In particular,f has no recurrent critical point.

Contradiction!

Reduction. From now on, we shall assume that for anycCr(f )withω(c)non-minimal,

the set Jc has measure zero. It follows from a similar argument as in the proof of

Proposition 2.2 that for a.e. zJ (f ),ω(z) ⊂ Par(f )cC

r(f ),ω(c)minimalω(c) . The last set is a disjoint union of finitely many minimal sets. Since the set{zJ (f ) : ω(z)⊂Par(f )}is countable, we may assume that:

(1) J (f )=cC

r(f ),ω(c)minimalJc(mod 0); and

(2) Jc = {zJ (f ) : ω(z) = ω(c)}(mod 0)for any cCr(f )such thatω(c) is

minimal.

6. Large real bound implies the non-existence of an invariant line field

We begin the study of the measurable dynamics off|Jc in the case thatω(c) cis a

minimal set. We shall assume the following throughout this section.

Assumption. There exists a sequence of symmetric nice intervalsI (n)c,n=1,2, . . ., such that|I (n)| →0 andd(ω(c)I (n), ω(c)In)/diam(ω(c)I (n))→ ∞asn→ ∞.

Letb=#[c], andc1=c, c2, . . . , cbbe the critical points contained in[c].

PROPOSITION6.1. There is a constantρ0such that for anyρρ0the following holds. LetI c be a symmetric nice interval interval. Suppose that|I|is very small and satisfiesd(ω(c)I, ω(c)I )ρdiam(ω(c)I ),letBI : rj=0Jj

b

i=2Ii

b

i=1Ii be the real box mapping associated toI. Then there is a quasi-polynomial-like

mapping F=F :Uijbi=1Vi1≤ib,0≤j < ri+1

corresponding tocsuch that:

(i) r1=r,r2=r3= · · · =rb=0 andUi0=Vifor any 2ib;

(ii) Viω(c) = Iiω(c), U1jω(c) = J1jω(c), for any 1ib and any

0≤jr;

(iii) F|(ViIi)=BI|(ViIi),F|(UjJj)=BI|(UjJj), for any 2iband

any 0jr;

(iv) there is topological diskDcontainingV1such thatDω(c)=V1∩ω(c), and

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Proof. We first assume thatr =0.

Sinced(ω(c)I, ω(c)I )/diam(ω(c)I )is very large, there is a round annulusX

inCˆ−ω(c)centred atcseparatingω(c)I fromω(c)Iwith a large modulus. Take the core curveγ ofX. It is a round circle centred atc. LetE1 denote the topological disk

bounded byγ. Letkj (0 ≤ jr),mi (2 ≤ ib)be the positive integers such that

BI|Ii = fmi andBI|Jj = fkj. Letmi(2 ≤ ib) andkj(0 ≤ jr)be the first

return time ofIi,Jj toI, i.e. the minimal positive integers such that fm

i(Ii)I and

fkj(Jj)I. LetE

i be the component offm

i(E1)containingci, 2≤ib, whereci is the critical point off which is contained inIi. For 0≤jr, letDj be the component of

fkj(E

1)containingω(c)Jj. It is easy to see that ifBI(Ii)Ii, thenfmi(Ei)=Ei

and that ifBI(Jj)Ii, thenfkj(Dj)=Ei.

We claim thatDjE1for any 0≤jr.

Indeed, sinceXE1is an annulus with a large modulus (=mod(X)/2) which is disjoint

fromω(c), it is easy to show by Koebe’s distortion theorem that any;∈ {Ei, Dj : 1≤

ib,0≤jr}is an ‘almost round’ disk as seen from anyx;ω(c), that is max

z∂;d(z, x)(1+)d(x, ∂;),

where > 0 can be taken close to 0 ifρ is large. Sincer ≥ 1, we have diam(Dj)

2(1+)diam(ω(c)I ). SinceE1∩S1is an interval containing a large neighbourhood of

ω(c)I, it follows thatDjE1.

F1,j :Uj

iVi, Fk :Vk

iVi0≤jr,2≤kb

is a quasi-polynomial-like mapping satisfying the required conditions.

We now turn to the case thatr=0.

In this case there is a permutation σ : {1,2, . . . , b} → {1,2, . . . , b} such that

BI(Ii)Iσ (i), whereBIis the real box mapping associated toI, for otherwise the forward

orbit ofcwill not enter some of the intervalsIi, which contradicts the hypothesis thatω(c)

contains all critical points inbi=1Ii. By possibly changing the subscripti, we may assume

thatσ (i)=i+1 for any 1≤ib−1 andσ (b)=1. Letki be the positive integer such

thatBI|Ii =fki. Letm=k1+k2+ · · · +kb.

ConstructX,E1as in the caser=0. LetD0be the pre-image offm(E1)containingc.

G = fm : D0 → E1 is a proper map. For any uD0 such that G(u) = 0,

Gn(u) never escapes the domain D0. Write P = {Gn(u):G(u)=0, nN}, then

mod(P , E1) ≥ mod(ω(c)I, E1) ≥ mod(X)/2 is large. By Theorem 5.12 in [20],

there is a domainUD0such thatfm :UV=fm(U)is a polynomial-like map with the same degree asGand mod(VU)is large sincef has no attracting cycle inD0.

DefineU0 =U,V1=V andVs =fk1+k2+···+ks−1(U0). Then{fk1 : U0 →V1, fki :

ViVi+1, fkb : VbV1 | 2 ≤ ib−1}is a quasi-polynomial-like mapping as

required.

Forδ > 0 let be the collection of quasi-polynomial-like mappingsFi,j : Uij

b

i=1Vi1 ≤ ib,0 ≤ jri

corresponding to c induced by f such that

ri =0, Ui0 =Vi for all 2≤iband such that there is a topological diskDcontaining

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For each quasi-polynomial-like mapping F in, and each 2≤i0≤b, we can associate

to it another quasi-polynomial-like mappingF in the following way. Let˜ c1∈U0, ciVi

be the critical points off. DenoteV˜1 = Vi0. For each 1 ≤ ib,i = i0, letk(i)be the minimal positive integer such thatFk(i)(ci)∈ ˜V1and letV˜i =Compci(F, k (i),V˜1).

For eachxω(c)∩ ˜V1, letk(x)be the minimal positive integer such thatFk(x)(x)∈ ˜Vi

for some 1≤ib, and letU (x)˜ =Compx(F, k(x),V˜i). Then define ˜

F=

Fk(x): ˜U (x))

b

i=1 ˜

Vi, Fk(i

)

: ˜Vib

i=1 ˜

Vi

xω(c)∩ ˜V1,1≤ib, i=i

.

Note that there is a constantδ >˜ 0 depending only onδ (andf) such thatF˜ ∈ ˜. Moreover, we can takeδ˜arbitrarily large ifδis sufficiently large.

THEOREM6.2. If there is a sequence of symmetric nice intervalsIncsuch that

(1) |In| →0 asn→ ∞; and

(2) d(ω(c)In, ω(c)In)/diam(ω(c)In)→ ∞asn→ ∞, then

(i) ifm(Jc) >0, thenf|Jcis ergodic; and

(ii) Jccarries nof-invariant line field.

Proof. By Proposition 6.1, there is a sequence of quasi-polynomial-like mappings

F (n)=

F (n)1,j :Uj(n)b

i=1

Vi(n);F (n)k,0:Vk(n)

b

i=1

Vi(n)

0≤jr(n),1≤kb

corresponding toc(induced byf) which are contained in the class Eδn forδ

n → ∞as

n→ ∞such that diam(V1(n))→0 asn→ ∞. LetW (n)V1(n)be a topological disk

such thatW (n)ω(c)=V1(n)ω(c)and mod(W (n)V1(n))δn.

(i) LetEJc be a measurable set of positive measure withf (E)E (mod 0)and

letxbe a Lebesgue density point ofEsuch thatω(x)=ω(c)andfn(x)C(f )for any

n≥0.

For anyn ∈ N, let k(n)be the minimal non-negative integer such that fk(n)(x)U0(n)ni=2Vi(n) and let 1 ≤ i(n)b be such that fk(n)Vi(n)(n).

Then k(n) → ∞ as n → ∞ since maxbi=1 diam(Vi(n)) → 0. Let A(n) be the

component offk(n)(Vi(n)(n))containingx. Let p(n)be the positive integer such that

fp(n)(Vi(n)(n)) = V1(n) if i(n) = 1 and let p(n) = 0 otherwise. Let X(n) be

the component offp(n)(W (n))containingVi(n)(n) and letA(n)be the component of

fk(n)(X(n)) containing A(n). Then fp(n) : X(n)W (n) is a proper map of a uniformly bounded degree, andX(n)ω(c) = Vi(n)(n)ω(c). We claim that forn

sufficiently large,fkn :A(n)X(n)is also a conformal map.

Assume that the claim fails. Letk < k(n)be the maximal number such thatfk(A(n))

contains a critical pointdoff. First let us show thatdω(c). Sincef (d)fk+1(A(n)),

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Since diam(fk+1(A(n)))and|fk+1(A(n))S1|are small, we have thatfk+1(x)is close tof (d), and hencefk(x)is close to d. So we have dω(x) = ω(c)if n is large. SinceX(n)ω(c)= Vi(n)(n)ω(c),dfk(A(n)). Sincefk(n)k(d)

b

i=1Vi(n),

there is some positive integert such thatF (n)t is well defined in a neighbourhood ofd

and coincides withfk(n)k. Then it is easy to see thatfk(A(n))U0(n)bi=2Vi(n).

Sincefk(x)fk(A(n)), this contradicts the minimality ofk(n).

Thusfk(n)+p(n) : A(n)W (n)is a proper map of a uniformly bounded degree. Sincex is a Lebesgue density point and fk(n)+p(n)(x)V (n), it follows thatm(EV (n))/m(V (n))→1 asn→ ∞.

Ifm(Jc) > 0 andf|Jc is not ergodic, then there is a measurable setEJc such

that f−1(E) = E (mod 0) and m(E) > 0, m(JcE) > 0. Then it follows that

m(EV (n))/m(Vn)→1 andm((JcE)V (n))/m(V (n))→1 asn→ ∞, which is

obviously absurd.

(ii) Assume thatf carries an invariant line fieldµwith supportEJc. We need to

show a contradiction.

LetxEbe an almost continuous point ofµsuch thatµ(x)=0 andfn(x)C(f )

for anyn≥0. By the same argument as in the previous proof, for eachnlarge, there are

k(n)∈N, 1≤i(n)bsuch thatfk(n)(x)Vi(n)(n)and there is a univalent branchh(n)

offk(n), defined onVi(n)(n), mappingfk(n)(x)tox.

Case 1. There are infinitely manynsuch thati(n)=1.

By passing to a subsequence, we may assume thati(n)=1 for alln∈N. Lets(n)be the first return time ofU0(n)toV1(n). Thenfsn:U0(n)V1(n)is a proper map whose

degree is no less than 2 and bounded uniformly from above.

Sincefk(n)(x)U0(n)V1(n), it follows from Corollary 3.3 that this is impossible. Case 2. Forn#1,i(n) >1.

Let

˜

F (n):

r(n)˜

j=0 ˜

Uj(n)

b

i=2 ˜

Vi(n)

b

i=1 ˜

Vi(n)

be the holomorphic box mapping associated to F (n) andi(n) as in the remark before Corollary 7.3. ThenF (n)˜ belongs to the class˜n withδ˜

n→ ∞. Letk(n)˜ be the minimal

non-negative integer such that

fk(n)˜ (x)∈ ˜U0(n)

b

i=2 ˜

Vi(n)

and let 1≤ ˜i(n)bbe such thatfk(n)˜ (x)∈ ˜V˜i(n)(n).

If there are infinitely manynsuch that˜i(n)=1, then we return to case 1 and the proof is completed. So let us assume that forn#1,i(n)˜ =1. So there is a univalent branch of

˜

h(n): ˜Vi(n)˜ (n)→ ˆCoff−˜k(n)such thath(n)(f˜ k(n)˜ (x))=x. Lets(n)˜ be the first return

time ofV˜˜i(n)(n)toV˜1(n). Thenfs(n)˜ : ˜Vi(n)˜ (n)→ ˜V1(n)is a proper map whose degree

is no less than 2 and bounded from above uniformly. A contradiction again follows from

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7. Bounded geometry implies the non-existence of an invariant line field on the Julia set

We fix anfF andcCr(f )withω(c)minimal. Assume that there is a constant

M >0 such that for any symmetric nice intervalI containingc, we have

d(ω(c)I, ω(c)I )

diam(ω(c)I )M. (6)

By takingM larger, we may assume that the inequality (6) holds for any small nice interval intersectingω(c).

To conclude the proof of Theorem 1 and 2, it suffices to prove the following theorem.

THEOREM7.1. In the previous situation, we have: (1) m(Jc)=0 orf|Jcis ergodic;

(2) f carries no invariant line field onJc.

Since f has a recurrent critical point, it has infinitely many periodic points on the extended real axisS1. Using conjugacy by real M¨obius transformation, we may assume that∞is a periodic point off.

Take an arbitrarily small symmetric interval T c and let BT : rj=0Sj

b

i=2Ti

b

i=1Ti be the real box mapping associated toT. As described in §4, this

real box mapping can be extended to a quasi-polynomial-like mapping F=F1,j :Uij

b

k=1Vk1≤ ib,0 ≤jri

withVi = ˆCTi andr1 =r,r2 =r3= · · · = rb =0. We shall fix F from now on. LetF be the map determined by F as described at the top of page 967.

From now on, all metrics are the Euclidean metric in C unless otherwise stated. All intervals are assumed to be contained inbi=1Ti. For any two disjoint open intervals

A1, A2, we shall use (A1, A2) to denote the maximal bounded open interval which is

disjoint fromA1∪A2, and similarly use (A1, A2] to denote the minimal open interval

which contains(A1, A2)A2.

For any δ > 0, let be the collection of intervals I satisfying the following

conditions: I is a symmetric nice interval I containing a critical point in ω(c) such that theδ-neighbourhood ofI is disjoint fromω(c)I; moreover, iff is only finitely renormalizable atc,I is aδ-nice interval and iff is infinitely renormalizable atc,I is a properly periodic interval. By Proposition 3.1 and Corollary 4.7 in [26], for someδ0>0, 0contains an arbitrarily small interval containingc. (By saying thatIis aδ-nice interval, we mean that for anyxω(c)I, ifJ is the component of the domain of the first return map toI, thenI contains theδ-neighbourhood ofJ.)

LetIbe a small interval in0and letJi, i=1, . . . , nbe the components of the domain of the first return map toI which (are contained inkTkand) intersectω(c)I and let

J0=I.

LEMMA7.1. There is a constantδ1>0 such that for any 0in,

d(Ji, ω(c)Ji)

|Ji| ≥

δ1.

Proof. For anyi =0, letr =ri be the minimal positive integer such thatfr(Ji)I.

(20)

fr(D)which containsJi. Thenfk :UDis a proper map with uniformly bounded

degree. Obviously,U∩Rcontains a definite neighbourhood ofJi, and is disjoint from

ω(c)Ji.

For any 0≤in, let 1≤kibbe such thatJiTki.

LEMMA7.2. There exists 0i, jnwithi=j,ki =kj, such that there is no point in

ω(c)betweenJi andJj, and such that

d(Ji, Jj)C1min(|Ji|,|Jj|), C1−1|Jj|<|Ji|< C1|Jj|,

whereC1is a constant depending only onδ1andf. Proof. Take 0inbe such that

|Ji| = n

min

m=0|Jm|.

First assume that both components of TkiJi contain an interval of the form Jj. Letj1 = j2be such thatJj1, Jj2 ⊂Tki and such that(Ji, Jj1)and(Ji, Jj2)are disjoint frommJm. It cannot happen that both(Ji, Jj1]and(Ji, Jj2]are much larger than|Ji, for otherwise, a large neighbourhood ofJi is disjoint fromω(c)Ji, which contradicts

assumption (6). Thus the lemma holds. Now let us consider the case that only a component, sayL, ofTkiJiintersects

mJm. Letj be such thatJjLis closest toJi. WhenI is

small,Ji is also small, and hence the other component ofTkiJi is much larger thanJi. Similarly as before, we must have(Ji, Jj]is not too large compared toJi. So the lemma

holds in this case as well.

We shall say that a topological diskP is admissible if∂P is a smooth curve disjoint fromω(c), and if one of the following holds:

(A1) Pω(c)= ∅; or

(A2) there exists a nice intervalAP such thatPω(c)=Aω(c).

LetC >1 be a constant and letP be a admissible topological disk. We say thatP is

C-bounded if the following hold:

(D1) l(∂P )2≤Carea(P );

(D2) forxω(c)Pi, and eachzPi, we have

d(x, z)Cd(x, ∂Pi);

(D3) mod(ω(c)P , P )C−1;

(D4) there is a topological diskQP, such that(QP )ω(c)= ∅, and mod(QP )C−1.

We shall often consider a couple(P1, P2)of admissible topological disks. We shall say

that the couple(P1, P2)isC-bounded if the following hold:

(T1) P1∩P2∩ω(c)= ∅;

(T2) both ofP1andP2areC-bounded;

References

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