Decay of Geometry for Fibonacci Critical Covering Maps of the Circle

(1)

www.elsevier.com/locate/anihpc

Decay of geometry for Fibonacci critical covering maps of the circle

^✩

Eduardo Colli

^a

, Marcio L. do Nascimento

^b,1

, Edson Vargas

^a,^∗^,2

aInstituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, Cep 05508-090, São Paulo, SP, Brazil bInstituto de Ciências Exatas e Naturais, Universidade Federal do Pará, Rua Augusto Corrêa 01, Cep 66075-110, Belém, PA, Brazil

Received 19 February 2008; received in revised form 9 March 2009; accepted 16 March 2009 Available online 7 April 2009

Abstract

We study the growth ofDfⁿ(f (c))whenf is a Fibonacci critical covering map of the circle with negative Schwarzian derivative, degreed2 and critical pointcof order >1. As an application we prove thatf exhibits exponential decay of geometry if and only if2, and in this case it has an absolutely continuous invariant probability measure, although not satisfying the so-called Collet–Eckmann condition.

Résumé

Nous étudions la croissance deDfⁿ(f (c))lorsquef est un revêtement critique de Fibonacci du cercle avec dérivée Schwar- zienne négative, degréd2 et point critiquecd’ordre >1. Comme application nous démontrons quef exhibe une décroissance exponentielle de géométrie si et seulement si2, et dans ce casf a une mesure de probabilité invariante absolument continue, sans satisfaire la condition de Collet–Eckmann.

MSC:37E10; 37C40; 37L40

Keywords:Circle maps; Covering maps; Fibonacci combinatorics; Decay of geometry; Invariant measures

1. Introduction

Acritical covering mapof the circleS¹=R/Zis aC^r (r1) covering mapf :S¹→S¹with just one critical point, sayc, which must be of inflection type. In the present work, we will consider critical covering maps of degree d2. This kind of map has been considered before, for example in [20–22,30]. Under the point of view of dynamical systems such critical coverings with neither wandering intervals nor periodic attractors are all topologically conju-

✩ This work is part of the “Projeto Temático – Fapesp # 2006/03829-2”.

* Corresponding author. Tel.: +55 (11) 30916156; fax: +55 (11) 30916183.

E-mail addresses:colli@ime.usp.br (E. Colli), marcion@ufpa.br (M.L. do Nascimento), vargas@ime.usp.br (E. Vargas).

1 Partially supported by CNPq, grant # 154594/2006-7.

2 Partially supported by CNPq, grant # 304517/2005-4.

doi:10.1016/j.anihpc.2009.03.001

(2)

gate to the map of the circle induced by the map x→dx of the real lineR. This implies in particular that the set B(f ):= {x∈S¹: ω(x)=S¹}is a countable intersection of open and dense subsets ofS¹. But metric properties of the underlying dynamics as the Lebesgue measure ofB(f )or the growth ofDfⁿ(f (c)), asn→ ∞, depend on the order and combinatorial behavior of the forward critical orbit. Here we will consider the case of critical points of order >1 with theFibonacci combinatorics, which will be defined below.

The Fibonacci combinatorics appeared before in the context of unimodal maps related to a question posed by J. Milnor [28] about the classification of the measure-theoretical attractors in dynamical systems on compact spaces.

Among the quadratic real polynomialsQ_α(x)=αx(1−x)there is one whose turning point has this combinatorics and, as it implies a strong recurrence of the critical point, it was considered in [14] as a candidate to exhibit awild attractor. A wild attractor forQ_α (also calledabsorbing Cantor attractorin [13]) is a compact invariant Cantor set whose basin of attraction is a meager subset of [0,1]with full Lebesgue measure [28]. It was proved in [24] that a quadratic real polynomial with the Fibonacci combinatorics has no wild attractor. This was generalized later for any combinatorics [23] (see also [7,10–13,16,26]). On the other hand, in [1] it was proved that if a unimodal real polynomial of degree big enough has the Fibonacci combinatorics then it has a wild attractor.

Under mild hypotheses a critical covering mapf as above is ergodic with respect to the Lebesgue measure [35], that is, every totally invariant set has Lebesgue measure zero or one. Then, if it has an absolutely continuous invariant probability measure it has no wild attractor and the topological conjugacy betweenf and a map of the circle induced byx→dxis absolutely continuous.

In order to prove that there is an absolutely continuous invariant probability measure one examines the derivative on the orbit of the critical value and tries to show that it undergoes a relative expansion, for example by showing that the map satisfies the so-calledCollet–Eckmann condition, stating that the derivative on the critical orbit grows exponentially, or weaker conditions as the summability condition in [2,31] and also [3]. This may be done either by directly examining the derivative for some specially chosen moments of the orbit (at closest returns, for example) or by studying the geometric aspects of sequences of inductively defined return maps around the critical point. In the latter case, if some metric relations between successive stages decay exponentially fast then one says thatf exhibits exponential decay of geometry(see below for a precise definition), the opposite situation beingbounded geometry, where scales do not decay at all. Exponential decay of geometry and topological additional hypotheses (as finiteness of central returns) are related to the expansion of the derivative on the critical orbit and have also been a way of proving existence of absolutely continuous invariant probability measures (see [27] for the unimodal case). We also mention that in the proof of denseness of hyperbolicity in the logistic familyQα (see [8,9,25]), a central problem in one-dimensional dynamics, the exponential decay of geometry played an important role.

Several questions about dynamics on the real line were treated with tools from complex analysis. But these tools can be applied just to a somewhat narrow set of cases which do not include situations where the order of the critical point is a general real number greater than one. It is also a natural aim that real analysis is enough to solve the questions from real dynamics. In this line of thought the non-existence of wild attractors for negative Schwarzian derivative Fibonacci unimodal maps with critical point of order 1< 2 was proved by G. Keller and T. Nowicki in [17] with just real analysis (in fact the result extends a little above=2). Moreover, the exponential decay of geometry was proved by W. Shen in [33] for non-renormalizable unimodal maps with any combinatorics with critical point of order 1< 2. Shen strongly uses the natural symmetry around the critical point that arises in the unimodal case, contrary to what happens with the inflection critical point of critical covering maps.

Here the decay of geometry for a map with an inflection critical point is considered for the first time. We prove that Fibonacci critical covering maps of the circle of degree d2 having critical point of order 1< 2 display decay of geometry. Moreover, our result is sharp since in [22] it was proved that Fibonacci critical coverings have bounded geometry for every >2. This is similar to the unimodal case where bounded geometry for the Fibonacci combinatorics follows for every >2 from a result in [17].

We also mention that the study of the growth of |Dfⁿ(f (c))| connected to the question of the existence of an absolutely continuous invariant probability measure and the regularity of topological conjugacies appeared at least three decades ago, see for example [5,15,18,19,29,32,34] and others references in the books [4] and [6].

(3)

2. Basic concepts and main results

We look at the circleS¹as the quotient spaceR/Zwith the orientation and metric induced from the real lineR.

Given aC¹critical covering map f :S¹→S¹ of degreed 2 we choose one of its fixed points, sayp, and set I1:=S¹\ {p}. The distance between 2 pointsx andyinI1will be denoted by|x−y|and the length of an interval I⊂I1will be denoted by|I|.

Note that f has d branches mapping onto I1 (below we will identify I1 with(0,1)⊂R in order to consider homeomorphic extensions of these branches and prove a technical but essential lemma). From now on we will assume that the critical pointcoff is recurrent and define the sequenceI₁⊃I₂⊃I₃⊃ · · · {c}such that, forn1, the interval I_n₊₁is a component of the domain of the first return mapφ_ntoI_n. Ifφ_n(c)∈I_n₊₁, for somen1, we say thatn is acentral return moment. Thecritical return times_n is defined byφ_n(c)=f^sⁿ(c). If the sequences₁, s₂, s₃, s₄, . . . coincides with the Fibonacci sequence 1,2,3,5, . . .the critical covering mapf is called aFibonacci critical covering map. It is not difficult to show that Fibonacci critical covering maps have no central returns.

One of the main geometric aspects of one-dimensional dynamics with critical points is the evolution of thescaling factorμ_n= |I_n₊₁|/|I_n|, particularly in the subsequence(n_i)_i of non-central return moments (which is the sequence itself in the case of Fibonacci critical covering maps). Ifμ_n_i →0 exponentially fast asi→ ∞we say thatf has exponential decay of geometry.

LetCddenote the set ofC¹critical coveringsf of degreed2 which also have the following properties:

• The critical pointcis recurrent and not periodic.

• There exist aC¹mapψ:I₁→I₁satisfying limx→cψ (x)=0 and real constantsϑ >0 and >1 such that f (x)=f (c)+ϑsgn(x−c)|x−c|

1+ψ (x)

(1) for everyxin a neighborhood ofc.

• Restricted toS¹\ {c}, the mapf isC³and has negative Schwarzian derivative.

Remark that a mapf ∈Cdcannot have neither a wandering interval nor a periodic attractor, see [6] and also [35].

Our main result claims the exponential growth ofDf^sⁿ(f (c))asn→ ∞for Fibonacci critical coverings.

Theorem 1.There are constantsρ >1and C >0 such that iff ∈Cd has the Fibonacci combinatorics and order ∈(1,2]thenDf^sⁿ(f (c))Cρⁿ, for alln1.

Three consequences are the following.

Theorem 2.If f ∈Cd has the Fibonacci combinatorics and order∈(1,2]thenf has an absolutely continuous invariant probability measure.

Theorem 3.If f ∈Cd has the Fibonacci combinatorics and order ∈(1,2]thenf exhibits exponential decay of geometry.

Theorem 4.Iff ∈Cd has the Fibonacci combinatorics and order∈(1,2]then theω-limit setω(c)of its critical point is a minimal invariant Cantor set with zero Hausdorff dimension.

In order to prove Theorem 1 we find a recursive difference equation (12) involvingDf^sⁿ(f (c))and the parameter λ_n= |f^sⁿ(c)−c|/|f^sⁿ⁺²(c)−c|. Then we proceed as in [17] to get the exponential growth ofDf^sⁿ(f (c))withn.

Using a classical argument, we show that the sequence of derivativesDf^k(f (c))goes to infinity ask→ ∞and this is exactly the criterion in [3] assuring the existence of an absolutely continuous invariant probability measure, as stated in Theorem 2. Finally, Theorem 1 together with distortion control imply the third and the fourth theorems.

A technical difference with respect to the work of Keller and Nowicki [17] is that we do not get ana priorinumeric lower bound forλ_n, and in fact we do not need it for the proof. It suffices to use thatλ_nis uniformly bounded away from one, which is a consequence of the real bounds (see Theorem 9 below). Another difference is a cross-ratio argument where an asymmetric huge extendibility allowed by topological properties plays a central role. For this

(4)

argument, the assumption of negative Schwarzian derivative is essential, since the range of extendibility goes much beyond the neighborhood of the critical point.

We also derive some more precise statements about the geometry and the derivative growth in the Fibonacci combinatorics, as a consequence of these theorems and a technical result proved in [22] for any >1. This result states, in particular, that the two components ofI_n\I_n₊₁are comparable withI_n. HenceI_n₊₁is not only exponentially small with respect toI_nbut also with respect to these two adjacent components.

The derivative growth presents three distinct pictures, depending on the value of, as the following theorems tell.

Theorem 5.Letf ∈Cdwith the Fibonacci combinatorics and∈(1,2). Then

nlim→∞

1

nlog logDf^sⁿ f (c)

=μ, where

μ=1+√ 1+4

2 .

In particular, the geometry decays super exponentially.

This theorem says thatDf^sⁿ(f (c))grows super exponentially withn, for <2. But ass_ngrows withωⁿ, where ω=¹⁺₂^√⁵is the golden number, andμ=μ()decreases fromω, when=1, to 1, when=2, then we can conclude thatDf^sⁿ(f (c))does not grow exponentially withs_n. Therefore, as a consequence,f is not Collet–Eckmann. But we can extend this conclusion to all iterates, by the following theorem.

Theorem 6.Letf ∈Cdwith the Fibonacci combinatorics and∈(1,2). Then

klim→∞

log logDf^k(f (c))

logk =logμ logω<1,

whereμ=μ()is given as in Theorem5andωis the golden number. In particular,f does not satisfy the Collet–

Eckmann condition.

For=2 the exponential growth ofDf^sⁿ(f (c))withnis the best that we can hope.

Theorem 7.Letf ∈Cdwith the Fibonacci combinatorics and=2. Then the growth ofDf^sⁿ(f (c))is not more than exponential. In particular, the geometry decays with a ratio which is not more than exponential.

Finally, for >2 we conclude thatDf^sⁿ(f (c))and the geometry are bounded. This last assertion, in particular, is also proved in [22].

Theorem 8.Letf ∈Cdwith the Fibonacci combinatorics and >2. ThenDf^sⁿ(f (c))is bounded, and in particular the geometry is bounded.

This paper is organized as follows. In Section 3 we review the technical tools related to the uniform distortion control of powers off which are widely used in one-dimensional dynamics. The so-called real bounds play a central role: they state ana prioriproperty on the geometry of the induced sequence of first return maps. In Section 4 we discuss in more details the Fibonacci combinatorics, which is explicitly used in the proofs. The facts stated in this section are very similar to the Fibonacci combinatorics of unimodal maps. In Section 5 we state the extendibility properties of a first entry map from the critical value to a neighborhood of the critical point which will be essential to get lower and upper bounds for its derivative, through a cross-ratio argument. These bounds are obtained in Section 6 and used to show Theorem 1, in Section 7. In this section we also prove the existence of an absolutely continuous invariant probability measure, the exponential decay of geometry and the zero Hausdorff dimension ofω(c). Finally, in Section 8 we derive, using a result in [22], the more precise consequences of these findings, which are stated in Theorems 5, 6, 7 and 8.

(5)

3. Cross-ratio, distortion control

Koebe Principles are the most important tools to controlling distortion and finding lower bounds for derivatives that will ultimately prove Theorem 1. We list some important definitions and results that may be found, for example, in [6].

LetJ, T ⊂I₁⊂S¹be a pair of intervals such thatJ⊂T andT \J has 2 non-empty connected componentsL andR. We say thatJ isα-well insideT if min{|L|,|R|}α|J|. Thecross-ratioofJ andT is the ratio

C(T , J )=|T|

|J|

|L|

|R|.

Leth:T →S¹be a diffeomorphism onto its image and define B(h, T , J )=C(h(T ), h(J ))

C(T , J ) .

IfhisC³and has negative Schwarzian derivative thenB(h, T , J ) >1. As a consequence, ifJ⊂T is degenerated to a pointxwe have

Dh(x)|h(L)||h(R)|

|h(T )|

|T|

|L||R|. (2)

On the other hand, if we degenerateRto a pointxthen Dh(x)|h(T )||h(J )|

|h(L)|

|L|

|T||J|. (3)

There is also the Macroscopic Koebe Principle saying that ifh(J ) isα-well insideh(T ) thenJ is alsoα-well insideT. Under the same condition the distortion is controlled by

Dh(x) Dh(y)

1+α α

2

, (4)

for allx, y∈J.

In [20] and [35] one may find the following important result, that will be used everywhere throughout this work.

Theorem 9(Real bounds). There isα=α() >0 (which can be uniform for₀, for every₀>1), such that if f ∈Cd has orderthenI_n₊₁isα-well insideI_nfor every non-central return momentnafter two non-central return moments. In particular, iff has the Fibonacci combinatorics, this is true for everyn3.

Again we considerφ_n, the first return map to the intervalI_nassociated to the critical coveringf ∈Cd. All branches ofφ_n map their domains diffeomorphically ontoI_nexcept by thecritical branchφ_n|I_n+1, which is aC¹homeomorphism ontoI_n. In the case of the critical branch, even if the distortion cannot be bounded because of the presence of the critical point, the corollary of the next lemma guarantees that it can be written as the composition off with a map (a power off) having uniformly bounded distortion.

Lemma 10.Iff ∈Cd andx∈I₁\I_n₊₁is a point such that there exists the smallestk1 such thatf^k(x)∈I_n₊₁ then there is an intervalT containingxwhich is mapped byf^kdiffeomorphically ontoI_n.

Proof. The statement follows from the fact thatInis a nice interval in the sense of Martens [26]. 2

Lemma 11.There isK=K() >0such that iff ∈Cd has the Fibonacci combinatorics and order >1 then for everyx∈I₁\I_n₊₁,n3, andkthe first positive integer such thatf^k(x)∈I_n₊₁there is an intervalJ to whichx belongs such thatf^k(J )=I_n₊₁and

Df^j(y) Df^j(z)K,

(6)

for ally, z∈J andj=0,1,2, . . . , k. The constantKis given by K=

1+α α

2

,

whereα=α() >0is the constant given in Theorem9.

Proof. By Lemma 10 there is an intervalT such thatx∈T andf^k|T is a diffeomorphism ontoI_n. LetJ be the pre- image ofI_n₊₁underf^k|T. By Theorem 9,I_n₊₁isα-well insideI_n. Now we apply the Macroscopic Koebe Principle toh=f^k⁻^j|^j_f(T )and conclude thatf^j(J )isα-well insidef^j(T ), for everyj=0,1, . . . , k. Finally the distortion control of (4) applied toh=f^j|T implies the lemma. 2

4. The Fibonacci combinatorics

The domain ofφ_nhas a connected componentM_n₊₁, calledpost critical domain, which containsφ_n(c). Following the terminology in [9], a branch ofφnwhich is a restriction of a branch ofφn−1is called animmediate branch.

Lemma 12.Iff∈Cdhas the Fibonacci combinatorics then the following properties hold:

(1) The post critical domain ofφ_nis an immediate branch. In particularφ_n₊₁(c)=φ_n₋₁◦φ_n(c), for alln2.

(2) Two consecutive post critical domains M_n and M_n₊₁ (and in particular φ_n₋₁(c) and φ_n(c)), n1, are on opposite sides ofcinsideI1.

Proof. Due to the fact that the critical branch ofφnis a homeomorphism fromIn+1ontoIn, for alln1, φn has exactly one immediate branch, for everyn2. Asd2 the first return mapφ1has exactlyd branches and all these branches are restrictions of f. Now φ₂ has its critical branch with return time equal to 2 (since by the hypothesis s₂=2), its immediate branch with return time equal to 1 and any of its branches which is not an immediate branch with return time at least equal to 2. Then, as s₃=3,φ₃(c)=φ₁◦φ₂(c). By induction, assume, forn2, that the immediate branch of φ_n has return times_n₋₁ and that the return time of its others branches (including the critical branch) are at least s_n. So, ass_n₊₁=s_n₋₁+s_n, the post critical branch ofφ_n must be its immediate branch. Also, except for its immediate and critical branches, all the other branches ofφ_n₊₁will have return time greater or equal than 2sn, which is greater thans_n₊₁, and the induction follows.

The second statement follows from the fact that the branches ofφn preserve orientation and the fact that its post critical branch is an immediate branch. 2

The following lemma illustrates the strong recurrence property of Fibonacci critical coverings.

Proposition 13.Ifg∈Cd has a sequence of return timess1, s2, s3, s4, . . .smaller(with respect to the lexicographic order)than the Fibonacci sequence1,2,3,5, . . .thenghas central returns.

Proof. We argue that ifg∈Cd has a sequence of return times s₁, s₂, s₃, s₄, . . .smaller (with respect to the lexicographic order) than the Fibonacci sequence 1,2,3,5, . . .thenghas central returns. Among the branches of the first return map of a critical covering the immediate branch has the smallest possible return time. So, if a critical covering has no central returns its sequence of critical returns is at least as big as the Fibonacci sequence 1,2,3,5, . . .. 2

The following lemma is one of the applications of Lemma 11 in the Fibonacci case.

Lemma 14.LetK=K() >0be the constant given in Lemma11. Iff ∈Cd has the Fibonacci combinatorics and order >1then

K⁻¹<Df^j(f^sⁿ⁺¹(f (c))) Df^j(f (c)) < K, for alln4and for allj=0,1, . . . , sn−1.

(7)

Proof. Asφ_n|I_n+1 hascas its only critical point andcis not periodic, then the intervalJ=f (I_n₊₁)is diffeomorphically mapped ontoIn byf^sⁿ⁻¹, it is disjoint fromIn, it is the first entry map intoIn and it contains the pointsf (c) andf (f^sⁿ⁺¹(c)), since candf^sⁿ⁺¹(c)are inIn+1. Therefore the lemma follows directly from Lemma 11 (applied toIn). 2

5. Extendibility around the critical value

From now on it will be always assumed in the statements that we are considering a functionf ∈Cd with the Fibonacci combinatorics.

We definef^m(c)=cm, form1, the points of the critical orbit. Among them there are the pointscs_n, that belong toM_n₊₁⊂I_nand are in alternate sides ofc, accordingly to the second statement of Lemma 12. We also define, for eachn1, the pointz_n=(φ_n|I_n+1)⁻¹(c). The pointsz_nwill help us to find the right cross-ratios in order to evaluate Df^sⁿ⁻¹(f (c)).

Lemma 15 (Definition and placement ofz_n). Let z_n=(φ_n|I_n+1)⁻¹(c), for every n1. Then z_n is between c_s_n−1 andc_s_n+1.

Proof. Asφ_n(c)=c_s_n∈M_n₊₁it is easy to see thatz_nandc_s_n are in opposite sides ofc. Thatz_nis betweencand c_s_n₋₁ follows from the fact thatc_s_n₋₁ ∈M_n andz_n∈I_n, and that they are both in the same side ofc. Thatc_s_n₊₁ is betweenz_n andcfollows from the monotonicity ofφ_n|In+1, that send the pointsz_n,c_s_n₊₁,ctoc,c_s_n₊₂,c_s_n, in this order, andcs_n₊₂ is betweencandcs_n. 2

We now identifyI₁with the interval(0,1)in the real line, in order to consider extendibility properties of the return mapφ_n|In+1 not restricted by the circle topology. Regarding as an interval map,f hasd branches in(0,1), two of them beingφ₁|M2 andφ₁|I2.

This identification allows thatall results will be valid also in the context of interval maps withd branches. These maps are in correspondence with circle maps with (in general)d points of discontinuity of the derivative. But these discontinuity points will make no difference in the arguments.

Letg0 andg1 be the lifts off that extend φ1|M₂ andφ1|I₂, respectively, to the whole line. Their difference is constant and equal to some integerτ with absolute value smaller thand. More precisely, defining the unitary vector en=_|^c_c^sn_sn⁻₋^c_c_| then

g1=g0+τ e1, (5)

for someτ∈ {1, . . . , d−1}.

Therefore if we define by inductiong_n=g_n₋₂◦g_n₋₁, forn2, theng_nis the homeomorphic extension ofφ_n|In+1

to the whole line. We claim that

g_n◦g_n₋₁=g_n₋₁◦g_n+(d−1)τ en+1. (6)

Forn=1 the assertion follows after explicitly developing the equality g₁◦g₀=(g₀+τ e₁)◦(g₁−τ e₁).

If it is true fornthen

gn+1◦gn=gn−1◦gn◦gn=

gn◦gn−1−(d−1)τ en+1

◦gn=gn◦gn+1+(d−1)τ en+2. Another auxiliary point isx_n, a pre-image of one of the boundary points of(0,1).

Lemma 16(Definition and placement ofxn). Forn1let∂nbe the boundary point of(0,1)which is opposite tocs_n

with respect tocand letxn=g_n⁻¹(∂n). Then (1) x_nis betweenc_s_n−1 andz_n, for everyn2;

(2) x_n=g⁻_n₋¹₁(x_n₋₂), for everyn3.

(8)

Proof. (1) It is enough to prove thatg_n(x_n)is betweeng_n(c_s_n−₁)andg_n(z_n)=c. But by (6) g_n(c_s_n₋₁)=c_s_n₊₁−(d−1)τ en,

that is,g_n(c_s_n−₁)is outside(0,1)and in the same side of∂_nwith respect toc. The first claim follows.

(2) By definitiong_n(x_n)=∂_n=∂_n₋₂, henceg_n₋₂(g_n₋₁(x_n))=∂_n₋₂. This implies thatg_n₋₁(x_n)=x_n₋₂, proving the second claim. 2

Lemma 17.Forn3, the restriction of gn to(xn, zn−1)is a C¹homeomorphism onto (∂n, cs_n₋₂)havingcas its unique critical point.

Proof. Although the statement is forn3, we first look atg₁andg₂. LetU₁=I₂andU₂=g⁻₁¹(M₂). Theng₁|U₁ is aC¹homeomorphism onto(0,1)havingcas its unique critical point. The same is true forg₂|U2, sinceg₂=g₀◦g₁ andg0|M2 is a diffeomorphism onto(0,1). Notice thatx1∈∂U1,x2∈∂U2, andU2⊂U1.

Now we letUn=(xn, zn−1), for everyn3. We claim thatznandxn+1belong toUn, for alln1, henceUn+1⊂ Un for everyn1. This assertion immediately follows in the casen3 from the localization Lemmas 15 and 16.

Moreover,z1∈U1=I2sincez1=(φ1|I₂)⁻¹(c)andz2∈U2=g⁻₁¹(M2), sinceI3⊂g₁⁻¹(M2)andz2=(φ2|I₃)⁻¹(c).

Thatx₂∈U₁follows fromx₂∈∂U₂andU₂⊂U₁. Thatx₃∈U₂follows fromx₃=g₂⁻¹(x₁)(by the second assertion of Lemma 16),g₂(U₂)=(0,1)andx₁∈(0,1).

We claim thatg_n|Unis aC¹homeomorphism with unique critical pointcfor alln1. As forn=1,2 it is already proved, assume by induction that n3 and that the claim is valid forn−1 and n−2. FirstU_n⊂U_n₋₁ implies that g_n₋1|Un is aC¹ homeomorphism with unique critical point c. Moreover, as z_n₋1∈∂U_n andg_n₋1(z_n₋1)=c thengn−1(Un)intersects just one component of Un−2\ {c}. But in fact gn−1(Un)is exactly this component, since gn−1(xn)=xn−2∈∂Un−2, by the second assertion of Lemma 16. Thengn−2restricted togn−1(Un)is a diffeomorphism onto its image and the compositiongn=gn−2◦gn−1, restricted toUn, is aC¹homeomorphism onto its image havingcas its unique critical point.

Asg_n(x_n)=∂_nandg_n(z_n₋₁)=g_n₋₂(c)=c_s_n−2 the image ofg_n|Unis the interval given in the assertion. 2 It is the following corollary of Lemma 17 that will be explicitly used when estimating the derivative growth using cross-ratio expansion. The notation introduced in the lemma closely follows previous works on this subject, for example [1].

Lemma 18.Forn2letx_n^f =f (xn),z^f_n =f (zn),c^f_s_n=f (cs_n),c^f =f (c)(in this case,xn,zn,cs_n and care all in the closure ofI2, andf means the continuous extension of the branchφ1|I₂ toI2). Then, for everyn3,f^sⁿ⁻¹ diffeomorphically sends(x_n^f, z^f_n₋₁)onto(∂n, cs_n₋₂). Moreover, the pointsc^f andc^f_s_n₊₁ are sent tocs_nandcs_n₊₂. Proof. By Lemma 17,f^sⁿ sends(x_n, z_n₋₁)homeomorphically onto(∂_n, c_s_n−2)withcas its unique critical point. As c is a critical point for the first iterate, iff^sⁿ⁻¹restricted to (x_n^f, z^f_n₋₁)had a critical point then there would exist j < snsuch thatf^j(c)=c. This would be a contradiction, sincecis not periodic.

Moreover,f^sⁿ⁻¹(c^f)=f^sⁿ(c)=c_s_nand f^sⁿ⁻¹

c^fs_n₊₁

=f^sⁿ(cs_n+1)=f^sⁿ⁺²(c)=cs_n+2. 2

6. Derivative on the critical orbit

We will control the growth ofDf^sⁿ(f (c))stated in Theorem 1 using the expansion of suitably chosen (degenerated) cross-ratios, that will allow us to construct difference equations, similarly as was done in [17].

λ_n= dn

dn+2

,

(9)

that will play an important role in the proof. Sincec_s_n andc_s_n+₂ are in the same side of{c},λ_nis always greater than one. But the real bounds of Theorem 9 give more.

Lemma 19.Letα >0be as in Theorem9,f ∈Cdwith the Fibonacci combinatorics andλ_ndefined as above. Then λ_n1+α

for everyn2.

Proof. There is a componentR of I_n₊1\I_n₊2 betweenc_s_n₊₂ ∈I_n₊2 andc_s_n∈I_n\I_n₊1. Theorem 9 implies that

|R|α|In+2|. Then

d_nR+d_n₊₂α|I_n₊₂| +d_n₊₂(α+1)dn+2. 2

The local approximation given in (1) gives useful estimates involving thed_n^f’s and also the derivative off at thecs_n’s. One of the estimates also uses Lemma 19.

Lemma 20.For sufficiently highn, there are numbersθ_n⁽¹⁾,θ_n⁽²⁾andθ_n⁽³⁾going to 1 asn→ ∞, such that

d_n^f=θ_n⁽¹⁾ϑ d_n, (7)

Df (c_s_n)=θ_n⁽²⁾ϑ d_n⁻¹, (8)

d_n^f

d_n^f −d_n^f₊₂=θ_n⁽³⁾ d_n

d_n−d_n₊₂=θ_n⁽³⁾ 1

1−λ⁻_n, (9)

whereϑis the constant given in(1).

Proof. Eq. (7) follows directly from (1). From the same equation we get that Df (x)=ϑ |x−c|⁻¹

1+ψ₁(x)

(10) forx in a neighborhood of the critical point, whereψ1is a continuous function such that limx→cψ1(x)=0, and this proves (8).

Letξ(a, x)=₁¹₋⁻_ax^x , forx∈ [0, (1+α)⁻¹]anda <1+α, whereα >0 is the constant given in Lemma 19. We have

d_n^f

d_n^f −d_n^f₊₂= 1 1−^dⁿ^f⁺²

d_n^f

= 1

1−^θⁿ⁽¹⁾⁺²

θ_n⁽¹⁾λ⁻_n

=θ_n⁽³⁾ 1 1−λ⁻,

whereθ_n⁽³⁾=ξ(^θ

(1) n+2

θ_n⁽¹⁾, λ⁻), which is defined fornsufficiently high, sinceθ_n⁽¹⁾→1. Asξ(a,·)uniformly tends to 1 as a→1 thenθ_n⁽³⁾→1. 2

Now taking into account Lemma 18 we apply (2) to the diffeomorphismf^sⁿ⁻¹in the interval(x_n^f, c^f)and obtain a fundamental lower bound for its derivative.

Lemma 21.For sufficiently highn, there isK_n⁽¹⁾going to1asn→ ∞such that Df^sⁿ⁻¹

c_s^f_n₊₁

K_n⁽¹⁾d_n−d_n₊₂ d_n^f₊₁

d_n^f₋₁ d_n^f₋₁−d_n^f₊₁

.

Proof. LetT_n=(x_n^f, c^f)with subintervalsL_n=(x^f_n, c^f_s_n₊₁),R_n=(c^f_s_n₊₁, c^f)andJ_n= {c^f_s_n₊₁}(degenerated), which are sent, underf^sⁿ⁻¹, to the intervals(∂_n, c_s_n),(∂_n, c_s_n+2),(c_s_n+2, c_s_n)and{c_s_n+2}, respectively. Eq. (2) yields

Df^sⁿ⁻¹ c_s^f_n₊₁

|f^sⁿ⁻¹(L_n)|

|f^sⁿ⁻¹(Tn)|

d_n−d_n₊₂ d_n^f₊₁

x_n^f x_n^f −d_n^f₊₁

.