Dimension of weakly expanding points for quadratic maps

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131(3), 2003, p. 399–420

DIMENSION OF WEAKLY EXPANDING POINTS FOR QUADRATIC MAPS

by Samuel Senti

Abstract. — For the real quadratic map Pa(x) = x²+a and a given > 0 a point x hasgood expansion propertiesif any interval containing x also contains a neighborhood J of x with P_aⁿ|J univalent, with bounded distortion and B(0, ) ⊆ P_aⁿ(J) for somen∈N. The-weakly expanding set is the set of points which do not have good expansion properties. Letαdenote the negative fixed point andMthe first return time of the critical orbit to [α,−α]. We show there is a setRof parameters with positive Lebesgue measure for which the Hausdorff dimension of the -weakly expanding set is bounded above and below by log₂M /M+O(log₂log₂M /M) forclose to|α|. For arbitrary≤ |α|the dimension is of the order ofO(log₂|log₂|/|log₂|).

Constants depend only onM. The Folklore Theorem then impliesthe existence of an absolutely continuous invariant probability measure for Pa witha∈ R(Jakobson’s Theorem).

Texte re¸cu le 11 février 2002, révisé le 15 juillet 2002, accepté le 16 septembre 2002 Samuel Senti, Department of Mathematics, Penn State University, University Park, PA, 16802 (USA) • E-mail :senti@math.psu.edu • Url :http://www.math.psu.edu/senti 2000 MathematicsSubject Classification. — 37E05, 37D25, 37D45, 37C45.

Key words and phrases. — Quadratic map, Jakobson’s theorem, Hausdorﬀ dimension, Markov partition, Bernoulli map, induced expansion, absolutely continuous invariant probability measure.

BULLETIN DE LA SOCI ÉT É MATH ÉMATIQUE DE FRANCE 0037-9484/2003/399/$ 5.00 c Société Mathématique de France

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Résumé(Dimension des points faiblement dilatants pour l’application quadratique) Pour l’application quadratique réellePa(x) =x²+aet un >0 donné, un pointxa de bonnespropriétésde dilatation si tout intervale contenantxcontient également un voisinageJdexavecP_aⁿ|Junivalent, avec distortion bornée etB(0, )⊆P_aⁿ(J) pour unn∈N. L’ensemble-faiblement dilatant est l’ensemble despointsqui n’ont pasde bonnespropriétesde dilatation. Notonsαle point fixe négatif etMle tempsde premier retour de l’orbite critique dans[α,−α]. Nousprouvonsl’existence d’un ensembleRde paramètres de mesure de Lebesgue positive pour lesquels la dimension de Hausdorff de l’ensemble -faiblement dilatant est born´ee supérieurement et inférieurement par log₂M /M +O(log2log₂M /M) s i est proche de |α|. Pour ≤ |α| quelconque la dimension est de l’ordre deO(log2|log2|/|log₂|).Lesconstantesne dependent que deM. Le théorème du Folklore implique alorsl’existence d’une mesure de probabilité absolument continue et invariante parPapoura∈ R(théorème de Jakobson).

1. Introduction

The 1-parameter family of real quadratic maps Pa(x) =x²+ais a simple model of nonlinear dynamics exhibiting surprisingly rich dynamical structure and giving rise to interesting and diﬃcult questions. For instance, the existence of a Pa-invariant probability measure which is absolutely continuous with respect to the Lebesgue measureLeb, a so-called a.c.i.p. , for a set of parameters of positive Lebesgue measure was ﬁrst proved by Jakobson in [9] (see also [4], [14], [16], [10], [17], [15]).

Much is known about the real quadratic family, including a positive answer for the real Fatou conjecture (see [8], [11]) as well as for its generalization to real analytic families of maps with one non-ﬂat critical point and negative Schwarzian derivative, so-called S-unimodal families [1]. Avila and Moreira have also shown that almost every non-regular parameter (no periodic hyper- bolic attractor) satisﬁes the Collet-Eckmann condition for the quadratic map [3]

as well as for a residual set of analyticalS-unimodal families with a quadratic critical point [2]. However, many questions remain forS-unimodal maps, H´enon maps (see [5]) and complex quadratic maps. The real quadratic map enjoys a particular status as it serves as a model to help understand these problems.

A common technique to build an a.c.i.p. for an interval map f is to look for expansion for some iterate of the original map on a properly restricted domain Ji. The map T|J_i := fⁿⁱ|J_i is called the induced map. The Folklore Theorem (see for instance [12], [6], [13]) asserts the existence of a T-invariant a.c.i.p. ν provided there exists >0 and a T-invariant interval A containing the critical point satisfying the following three conditions: a)Ais the countable union, modulo sets of zero Lebesgue measure, of intervals Ji with disjoint interiors, b)T restricted to eachJi is a diﬀeomorphism with uniformly bounded distortion and c) Leb(T(Ji)) ≥ . If additionally the summability condition

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niν(Ji)<∞holds, then there exists an a.c.i.p.µ which is invariant by the original map f.

Let B(0, ) denote the ball of radius centered at 0. For the quadratic family Pa and a given > 0, we will say that a point xhas good expansion properties if any interval containing x also contains a neighborhood J of x with P_aⁿ|J univalent with bounded distortion and B(0, ) ⊆ P_aⁿ(J) for some iterate n∈N. We call points which do not have good expansion properties - weakly expanding. For such points there are only a ﬁnite number of iterates for which one can ﬁnd a neighborhood J ofxwithP_aⁿ|J univalent with bounded distortion and B(0, )⊆P_aⁿ(J)

In this work, we estimate the Hausdorﬀ dimension of the -weakly expanding set for the quadratic map for a set of parameters of positive Lebesgue measure.

More precisely if αis the negative ﬁxed point and M := min

n∈N;|P_aⁿ(0)|<|α|

is the ﬁrst return time of the critical orbit to [α,−α], we prove the following:

Main Theorem. — For a set of parameters Rwith lim

δ→0

Leb(R ∩[−2,−2 +δ))

δ = 1

the Hausdorﬀ dimension of the -weakly expanding setE is dim_H(E ) =log₂M

M

1 +Olog₂log₂M log₂M

for c2⁻^M < ≤ |α| and a constantc >0. For0< ≤ |α| arbitrary, we have dimH(E) =Olog₂|log₂ |

|log₂ |

.

Constants depend only on the return time M.

For parameters in Rclose to−2 the return timeM is large, so points with good expansion properties have full Lebesgue measure. For =|α|the Folklore Theorem applies and Jakobson’s Theorem follows as a Corollary once we prove the summability condition.

Corollary (Jakobson [9]). — For the real quadratic map Pa, there exists a probability measure which is invariant by Pa and absolutely continuous with respect to the Lebesgue measure for a set of parameters a ∈ R of positive Lebesgue measure.

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Structure of the paper. — We consider the neighborhoodA:= [α,−α] of the critical point in Section 2. We build a partition of A consisting of those pre- images ofAon which the induced mapT can be extended as a diﬀeomorphism to a given larger domain containing A. Such extendibility insures uniform bounded distortion by Koebe’s distortion property and the image by T of the partition elements obviously contain the ball of radius <|α|centered at 0.

In Section 3 we identify the sets that contain -weakly expanding points with =α.

In Section 4 we introduce Yoccoz’s strongly regular parameter conditions [17]

and show that for these parameters theα-weakly expanding set is the attractor of an iterated function system (IFS) with countably many generators.

In Section 5, we obtain derivative estimates for the generators of the IFS by viewing the quadratic map as a perturbation of the Chebyshev polynomialx²−2 for parameters close to −2 (large M). We also prove that strongly regular parameters satisfy the Collet-Eckmann condition.

In Section 6 we bound the Hausdorff dimension of the α-weakly expanding set. The lower bound is obtained by approximating this set by the attractor of an IFS with finitely many generators and estimating the dimension of the approximation (which obviously contains the α-weakly expanding set). If the number of generators of the approximation is large enough, the difference be- tween both dimensions lies within the error bounds produced by the derivative estimates giving us an upper bound. The complement of the partition of A coincides with the -weakly expanding set withc2⁻^M < ≤ |α|for some constantc >0. We prove a measure estimate which implies both the summability condition and the full Lebesgue measure of the partition of A. The Folklore Theorem together with the positive Lebesgue measure of strongly regular parameters (see [17], [15]) then implies Jakobson’s Theorem [9].

In Section 7 we consider the -weakly expanding set for general . We gen- eralize the construction introduced in the ﬁrst part, showing that the -weakly expanding set is the attractor of an IFS with countably many generators. In this case only very crude derivative estimates are necessary. We bound the number of contractions for which the same estimates are used and proceed similarly as in the ﬁrst part to estimate the dimension of the -weakly expanding set.

Acknowledgments. — The author would like to thank J.-C. Yoccoz for his un- counted time, support and inspiration. Thanks also to M. Jakobson for his interest and support, to the Universit´e d’Orsay, where this research was con- ducted, and to the Penn State University, where this article was written.

2. Deﬁnitions For the real quadratic map

Pa(x) =x²+a

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with parameters −₄³ > a∈ Rdenote the ﬁxed points by β > 0 and α < 0.

Forn∈N, setα⁰:=αand deﬁne inductivelyαⁿ as the unique negative point for which

Pa(αⁿ) =−αⁿ⁻¹.

Ifa < αⁿ⁻¹deﬁne inductivelyαⁿ as the unique negative point for which P_a(αⁿ) =αⁿ⁻¹.

ChooseM ∈Nlarge and consider parameters for whichPa(0)∈(α^M⁻¹, α^M⁻²).

Let

A:= [α,−α]⊆[α¹,−α¹] =:A.

SetC₁⁺= [α¹, α] and

C_n⁺:= [αⁿ⁻¹,αⁿ] for 2≤n≤M −2.

Also writeC_n⁻=−C_n⁺ for the interval symmetric to C_n⁺.

Definition 2.1. — An intervalJ ⊆[−β, β] is regular of ordern≥0 if there exists a neighborhood Jof J such that the restriction of P_aⁿ to Jis a diﬀeo- morphism onto Aand P_aⁿ(J) =A. We denote the order of a regular interval byord(J). Denote

J :=

J A regular ;∀J regular with int(J)∩int(J)=∅⇒J⊆J . Elements ofJ are maximal with respect to inclusion.

Proposition 2.2. — a)Regular intervals are nested or have disjoint interiors.

b) [αⁿ, αⁿ⁻¹] is regular and maximal for all n ≥ 1. C_n^± ∈ J are regular and maximal for 2 ≤ n ≤ M −2. As limn→αⁿ = −β, regular intervals cover[−β, α]. As the multiplier ofαis negative, there are no maximal regular intervals of order1in[α,−α] ([α¹, α]and−[α¹, α]are the only regular intervals of order 1).

c) The image Pa(J) of a regular interval J of order n≥1 is regular of or- dern−1andPa(J) =Pa(J)⊆(a, β). Conversely, ifJ is(maximal)regular of ordern≥0, and ifJ⊆(a, β), then both components ofP_a⁻¹(J)are(maximal) regular of order n+ 1.

d) For each regular interval J there are two adjacent regular intervals J_i, i = 1,2, with int(J_i)∩int(J) =∅ and ord(J_i) = ord(J) + 3. If J ∈ J there are two adjacent maximal regular intervals J_i with int(J_i)∩int(J) = ∅ and ordJ_i−ordJ ={±1,±3}.

Definition 2.3. — For a regular intervalJ denote byGJ :A→Jthe local inverse toPa^ord(J)|J. Also

Wn:=

J∈J 2≤ord(J)≤n

int(J), W =

n≥2

Wn

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θk A(k) θk+1 A(k+ 1)

0

L B(k) B(k+ 1)

P^M

A ξk J(k) GB(k)

Figure 1.

Forx∈int(J), J ∈ J, thereturn time is the functionN :W →Ngiven by N(x) :=ord(J)

and theinduced map is the functionT :W →int(A) given by T(x) :=P_a^ord(J)(x).

Accept the abuse of notation T(0) :=P^M(0).

Definition 2.4. — A gap of order n is a connected component of A\Wn. The set of all gaps of order n is denoted by ε(n). The primary (α-)weakly expanding set

E:=

n≥1

A\Wn=

n∈N

ε(n)

is the set of non isolated points ofA\W. The (α-)weakly expanding set is E=

k≥0

P_a⁻^k(E).

We have dimH(E) = dimH(E).

For 2 ≤n≤M −2,ε(n) = (αⁿ,−αⁿ) sinceC_n^±∈ J. As [α^M⁻²,α^M⁻¹] is not regular,J has no elements of order eitherM −1 orM so

ε(M −1) =ε(M) = (α^M⁻²,−α^M⁻²).

Hence ε(n)⊆(α^M⁻²,−α^M⁻²) forn≥M−2.

In the following assume: T(0)∈

n≥1Dom(Tⁿ).

Definition 2.5. — F ork≥1, letJ(k)∈ J withT^k(0)∈int(J(k)). Set n0:= 0, nk:=N

T^k(0) ,

N0:=M, Nk=Nk−1+nk−1=M +

0≤i≤k−1

ni.

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Deﬁne a nested sequence of regular intervalsB(k) containingP^M(0) by B(1) :=A, B(k) =GB(k)(A) :=GJ(1)◦ · · · ◦ GJ(k−1)(A)

(see Fig. 1). Let ζ ∈ [α^M⁻²,α^M⁻¹] be the point with P_a^M⁻¹(ζ) = 0 and let Q:=P_a^M|(ζ,−ζ). Then deﬁne

A(k) :=Q⁻¹ B(k)

.

Let L be the maximal (with respect to inclusion) regular interval adjacent to B(k) and contained inP_a^M((ζ,−ζ))∩(B(k) \intB(k)). Set

A(k) := [θ k,−θk] :=Q⁻¹

B(k)∪L .

IfP_a^M((ζ,−ζ))∩(B(k) \intB(k)) is regular andA(k)=A(k+1), then by Propo- sition 2.2A(k) \int(A(k)) is maximal and regular of orderN_k =Nk+ 1. Other- wise,θkis the left endpoint of a maximal regular interval of orderN_k =Nk+ 3.

Also setξ₀=α¹ andξ_k =P_a^N^k(θ_k+1).

Let us make the following orientation assumption:

P^N^k(x)≤P^N^k(0) for all x∈A(k).

The reversed orientation case is similar. Write Kk:=

I regular ;P_a^N^k(0)∈IandI⊆[ξk, β]

.

Proposition 2.6. — For all I, I ∈ Kk we have int(I)∩int(I) =∅, except for the two regular intervals adjacent toJ(k)of order ord(I) :=p=nk−3and ord(I) =nk−1. Also ord(I)=ord(I) andJ(k)⊂I.

If nk≤M−2thenKk ={C_n⁺

k−1}if0∈P_a^N^k(A(k))andKk =∅otherwise.

If nk≥M + 1then M−2≤p < nk

Proof. — Assume ﬁrst ξ_k ∈ int(I). Regular intervals are nested or disjoint.

IfI⊂Ithenp :=ord(I)≥p+2. LetIdenote both regular intervals of order p+ 3 adjacent to but not contained inI. ThenI ⊆I∪I or else there would be a point x∈int(I\I) withP_a^p+3(x) =−α. As J(k)∈ J and ξk ∈int(I) we have J(k)⊆I so I∩J(k) =∅andI ∈ Kk. SoI, I ∈ Kk have disjoint interiors and forIbetweenIandJ(k) we haveI ⊆I\IasI∩int(J(k))=∅ so p+ 3≤p.

Now assume ξk ∈int(I). The component ofJ(k)\int(J(k)) containing ξk

is regular, or else, by Deﬁnition 2.5 and Proposition 2.2, ξk is the endpoint of a maximal regular interval. This contradicts ξk ∈ int(I) with I regular.

Regular intervals are nested or have disjoint interiors soIandJ(k) are adjacent andJ(k)∈ J impliesp=nk−1 ornk−3.

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Fornk≤M−2, if 0∈P_a^N^k(A(k)) thenJ(k) =C_n⁺

k andI∩int(J(k)) =∅for all regularI⊆[ξk, β] exceptI=C_n⁺_k₋₁ for whichJ(k)⊆I. If 0 ∈P_a^N^k(A(k)) thenJ(k) =C_n⁻

k andI∩int(J(k)) =∅for all regularI⊆[ξk, β].

For nk ≥ M + 1, no interval of order < M −2 intersects int(J(k)). As- sume p≥nk. Thenξk ∈int(I) and I∩int(J(k))=∅so one of the endpoints of J(k) is contained in I\I and p+ 3 ≤ nk contradicting the assumption.

SoM−2≤p < nk. Also ifJ(k)⊆I, then int(J (k)) would contain an endpoint ofIasI∩int(J(k))=∅. But then p+ 1≥n_k+ 2 contradictingp < n_k.

3. Classiﬁcation of the gaps

ForM+ 3≤n≤N+1, let$be the largest integer for whichN

≤nwithN from Deﬁnition 2.5. Deﬁne

ε(n,0) =

J∈ε(n) ;J ⊆A\int(A(1))

, εc(n) =

J∈ε(n) ;J ⊆A($) and for 1≤k < $

ε(n, k) =

J ∈ε(n) ;J ⊆A(k) \int(A(k + 1)) . Proposition 3.1. — A gap ofε(n, k)is one of both pre-images

• either byQ⁻¹◦ GB(k) of a gap of ε(n−N_k)∩[α, ξ_k];

• or by Q⁻¹◦ GB(k)◦ GI of a gap of ε(n−N_k−ord(I)) whereI∈ Kk. Proof. — LetJ ∈ J withJ ⊆A(k) \intA(k+1). By Proposition 2.2 P_a^N^k(J)⊆ [α¹, ξ_k] is a regular interval of order ord(P_a^N^k(J)) =ord(J)−N_k. IfP_a^N^k(J) is not maximal there existsI ∈ Kk with P_a^N^k(J)⊆I. ThenPa^N^k^+ord(I)(J)∈ J, unless I is the regular interval adjacent to J(k) with ord(I) = nk −3 and P_a^N^k(J)I forI∈ Kk withord(I) =nk−1. ThenPa^N^k^+ord(I⁾(J)∈ J.

Conversely, for any maximal regular intervalI⊆[α¹, ξ_k], we haveI⊆(a, β) and by Proposition 2.2 both components of P_a⁻¹(I) are regular and maximal with respect to inclusion. If I ∈ Kk, the same argument yields that both pre-images Q⁻¹◦ GB(k)(I) contained inA(k) are maximal regular intervals of order ord(I) +N_k. If I ∈ Kk consider the images GI(J) of maximal regular intervalsJ. Passing to the complementary set yields the proposition.

Corollary 3.2. — A gap of ε(n,0) is contained in either [α^M⁻¹,α^M⁻²] or its symmetric and is a pre-image byPa⁻^(M⁻¹⁾of a gap of ε(n−M+ 1).

Definition 3.3. — SetD0:= [α^M⁻²,−α^M⁻²] andDk=D0∩[α, ξk]. Denote:

• G0^± the restriction to D0 of the positive/negative branch of the inverse map to P_a^M⁻¹|[α^M−2,α^M−1].

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• G_k^±the restriction toDk=∅of the positive/negative branch of the inverse map to Q⁻¹◦ GB(k).

• G_k,I^± the restriction to D0 of the positive/negative branch of the inverse map to Q⁻¹◦ GB(k)◦ GI forI∈ Kk=∅.

No mapping is to be deﬁned ifDk =∅orKk =∅. DenoteG0,I^± :=G0^± and Γ =

Gk; =±, k≥0, Dk∩E =∅

∪

Gk,I ; =±, k≥0, I∈ Kk =∅ .

4. Strongly regular parameters

Fix a largeM ∈Nwith log^κM < M < ²₃M for someκ >1 andρ∈Rwith M2⁻^M ρ1.

Definition 4.1 (see [17]). — A parameteraisstrongly regular if a) T(0)∈

n≥1Dom(Tⁿ),

b)

1≤i≤k,

N(Tⁱ(0))≥M

N(Tⁱ(0))< ρk for all k∈N.

Proposition 4.2. — For a strongly regular parameter and q = [ρ⁻¹M] the integer part ofρ⁻¹M,

N T^k(0)

< M for k≤q, N T^k(0)

< ρk for q < k.

Corollary 4.3. — LetF be the attractor of the iterated function system gen- erated by the Gk,Gk,I ∈Γ with k≤q. For strongly regular parameters F ⊆E andEis invariant by allGk,Gk,I ∈Γ with k≤q.

Proof. — Follows from Deﬁnition 2.4 and from Proposition 3.1 asE ⊂ D0=Dk

for strongly regular parameters.

5. Derivative estimates

Proposition 5.1. — Let x∈Dk. For strongly regular parameters, there ex- ists a constant c >0 with

1) For0≤k≤q:

c⁻¹2⁻¹²^(M^+N^k⁻ⁿ^k⁾≤DGk^±(x)≤c2⁻¹²^(M+N^k⁻ⁿ^k⁾ 2) Fork > q andx∈Dk:

c⁻¹2⁻¹²^(M+N^k⁾⁻^ρk≤DG_k^±(x)≤c2⁻¹²^(M^+N^k⁾⁺ⁿ^k^+ρk. 3) For0≤k≤q:

c⁻¹2⁻¹²^(M+N^k+1⁾≤DG_k,I^± (x)≤c2⁻¹²^(M+N^k+1⁾

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4) Fork > q:

c⁻¹2⁻¹²^(M+N^k^+ord(I))⁻^2ρk ≤DGk,I^± (x)≤c2⁻¹²^(M+N^k+ord(I))+2ρk

Lemma 5.2. — For parameters with Pa(0) ∈ (α^M⁻¹, α^M⁻²), there exists a constant c >0 with

c⁻¹4⁻ⁿ ≤αⁿ−Pa(0)≤c4⁻ⁿ for 0≤n < M−2, c⁻¹2⁻ⁿ ≤ −αⁿ≤c2⁻ⁿ for 0< n≤M−2, c⁻¹4⁻^M ≤β+Pa(0)≤c4⁻^M.

Proof. — For these parameters, considerPaas a perturbation of the Chebyshev polynomial. The estimates follow. See [15]

Lemma 5.3. — Writeh(x) := (β²−x²)⁻¹² for |x|< β. Then DP_aⁿ(x)= 2ⁿ h(x)

h(P_aⁿ(x))

n−1 i=0

1 + β+a (P_aⁱ(x))²

₋¹₂ .

Proof. — Follows by induction usingβ²+a=β.

Lemma 5.4. — For strongly regular parameters, there is a constantc >0 with 1) log₂DP_aⁿ(x)^h(P

n a(x)) h(x)

−n≤cn4⁻^M for 1≤nandx∈[αⁿ, αⁿ⁻¹].

2) log₂DG_C^±_n(x)^h(^G_h(x)^Cⁿ^±^(x))+n≤c4ⁿ⁻^M for 1≤n≤M −2 andx∈A.

3) 4⁻^ord^J ≤DGJ(x)≤c for all regularJ andx∈A.

4) log₂DGB(k)(x)^h(^G^B(k)_h(x)^(x))+ (Nk−M)≤2ρkfor 1≤k andx∈A.

Proof. — 1) and 2) follow from Lemma 5.3 using estimates from Lemma 5.2.

3) Since|DPa| ≤4 on [−β, β]⊆[−2,2] andGJ sendsAintoJwith bounded distortion.

4) Rewrite 3) as |log₂|DGJ()(x)h(GJ()(x))/h(x)|+n| ≤ n+c.Use the chain rule with 2) forn≤M and 3) else wise. For strongly regular parameters 4^M⁻^M ρand

k−1

=1 n≥M

(n+c)< ρk+c

k−1

=1 n ≥M

n

M <(1 +cM⁻¹)ρk.

Thus (1 +cM⁻¹)ρk+ck4^M⁻^M <2ρk.

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Proof of Proposition 5.1. — 1) Set

y:=Gk(x) :=Pa◦ Gk^±(x) =Pa(Q⁻¹◦ GB(k))(x).

By Proposition 4.2, for strongly regular parameters GB(k) is a composition of GC_n^± with 2≤n < M. Also c⁻¹h(x)≤h(GB(k)(x))≤ch(x). As k≤q 4^M⁻^M, combining 1) and 2) of Lemma 5.4 with Lemma 5.2 yields

(1) c⁻¹2⁻^M⁻^N^k ≤DGk(x)≤c2⁻^M⁻^N^k

y = Pa((y −a)¹²) implies |D(P_a⁻¹)^±(y)| = ¹₂|y−a|⁻¹². Bounded distortion forGk yields

c⁻¹DGk(x)≤ |y−a|

|x−Pa^N^k⁻¹(0)| ≤cDGk(x). Forx∈Dk andP^N^k(0)∈J(k) withnk< M, Lemma 5.2 yields

c⁻¹2⁻ⁿ^k≤x−P_a^N^k(0)≤c2⁻ⁿ^k.

The statement now follows from the chain rule. For k = 0 note that c⁻¹ ≤

|x−P_a^M⁻¹(0)| ≤c.

2) Lemma 5.4 part 4) yieldsc⁻¹2⁻^M⁻^N^k⁻^2ρk ≤ |DGk(x)| ≤c2⁻^M⁻^N^k^+2ρk. As above the chain rule and the bounded distortion argument yield

c⁻¹2⁻¹²^(M^+N^k⁾⁻^ρk ≤DG_k^±(x)·x−P_a^N^k(0)¹² ≤c2⁻¹²^(M^+N^k^)+ρk. J(k) is regular, so by Proposition 2.2 it is adjacent to a regular intervalJ⊆J of ordern_k+ 3 contained in the image ofQ. Bounded distortion ofGJ(k) and part 3) of Lemma 5.4 imply c⁻¹4⁻ⁿ^k ≤c⁻¹|J(k)| ≤ |J| ≤ |x−P^N^k(0)| ≤ c forx∈Dk and the result follows.

3)Gk,I :=P_a◦Gk,I^± . By Proposition 4.2 and Proposition 2.6ord(I) =n_k−1.

Proceed as in 1), as estimates of Lemma 5.4 2) also hold forC₁^±. For bounded distortion of Gk,I note that c⁻¹ ≤ |x−P^N^k+1⁻¹(0)| ≤ c as P^N^k+1⁻¹(0) ∈ A andx∈D₀.

4) By Proposition 2.6 and Proposition 4.2ord(I)< nk < ρk. The same ar- gument as in Lemma 5.4 part 4) holds forGB(k)◦ GI (substituting 2ρkby 3ρk).

c⁻¹2⁻^(M^+N^k^+ord^I)⁻^3ρk ≤ |DGk,I(x)| ≤c2⁻^(M+N^k^+ord^I)+3ρk. Bounded distortion yields:

c⁻¹2⁻¹²^(M+N^k^+ord^I)⁻^2ρk ≤DGk,I^± (x)·x−P^N^k^+ord(I)(0)¹²

≤c2⁻¹²^(M+N^k^+ord^I)+2ρk. One has J(k)⊆I\int(I) by Proposition 2.6 soc⁻¹≤ |x−P^N^k^+ord(I)(0)| ≤c as x∈D₀.

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Corollary 5.5. — Strongly regular parameters satisfy the Collet-Eckmann condition:

lim inf

n→∞

1

nlnDP_aⁿ(a)>0.

Proof. — Letn, k ∈Nwith Nk ≤n < Nk+1 andq < k. For strongly regular parameters, the same reasoning as in Proposition 5.1 with y =P_a^N^k⁻¹(a)∈A yields

c⁻¹2^M^+N^k ≤DP_a^N^k⁻¹(a)=|DGk|⁻¹≤c2^M+N^k

for allk∈N. If we had|DP_a^N^k⁻¹(a)| ≤c2^M⁺¹²^N^k⁻ⁿ^k for some 0≤i≤nk the chain rule would imply

c⁻¹2^M^+N^k⁺ⁿ^k ≤ |DP_a^N^k+1⁻¹(a)| ≤c2^M⁺¹²^N^k⁺ⁿ^k⁻²ⁱ leading to the contradictionNk≤ −2i. Therefore,

lim inf

n→∞

1

nlnDP_aⁿ(a)≥lim inf

k→∞ min

0≤i≤n_k

1

klnc2^M⁺¹²^N^k⁻ⁿ^k ≥c(1−ρ)>0 as 2k≤Nk ≤n≤Nk+1<(M+ρ)k.

6. The Hausdorﬀ dimension

Consider an at most countable family of contractionsGk : Dk → F where eachDk is a closed subset of a compact invariant set

F =

Gk(Dk)

invariant with respect to allGk, where the closure is taken with respect to the Hausdorﬀ topology. Let

0< bk≤ |Gk(x)− Gk(y)|

|x−y| ≤ck <1 for all x = y ∈ Dk. Assume

ck ≤ 1 and let s+, s₋ ∈ R be such that c^s_k⁺= 1 =

b^s_k⁻.

Lemma 6.1. — AssumedimH(∂F)≤s+ andGi(Di)∩ Gj(Dj) =∅. Then s₋≤dim_H(F)≤s₊

Proof. — In the finite case, see [7]. For the upper bound of the infinite case, consider s > s+. By definition, for everyδ > 0 and η > 0 there is a δ-cover {Ui}ofF with

(diamUi)^s≤ Hδ^s+η. Then {Ui,k}i∈N:=

Gk(Ui∩F)

i∈N

o

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is a ckδ-cover of Gk(F) for each k ∈ N. As dimH(F_∞)≤ s+ < s, there is a δ-cover{Vj}of∂F such that{Ui,k, Vj} is aδ-cover ofF and

H^sδ ≤

i,k

(diamU_i,k)^s+

j

(diamV_j)^s≤c(H^sδ+η) +η for c =

kc^s_k < 1. As η > 0 and δ > 0 were chosen arbitrary we get dimH(F)≤sfor everys > s+, hence dimH(F)≤s+. The lower bound follows from the finite case, as F is also invariant for any finite subset of the infinite family of contractions. For details see [15].

Theorem 6.2. — For strongly regular parameters there is a constant c > 0 depending only on M such that

dimH(E) =log₂M M

1 +O(log₂log₂M

M )

.

Proof. — By Deﬁnition 2.4, dimH(E) = dimH(E). For the upper bound, Kk

and Γ are countable by Proposition 2.6 so Corollary 4.3 insures we can apply Lemma 6.1 toE. Write

2xk = Nk−nk−c fork≤q, 2xk = Nk−2nk−2ρk−c fork > q,

2xk,I = Nk+ord(I)−4ρk−c fork > qandI∈ Kk

and

H(s, x0, . . .) := 1−2⁻¹²^sM+1 ^q

k=0

2⁻^sx^k+

k>q

2⁻^sx^k+

k>q I∈Kk

2⁻^sx^k,I

.

Inserting the upper bounds of Proposition 5.1 yields 1−

G∈Γc^s_G ≥H(s, x₀, . . .).

As ∂H/∂s >0 and H(0, x0,· · ·)<0 valuess >0 with H(s, x0, . . .)≥0 give upper bounds ons+. The values ofxk are larger than 2xk ≥M−M+ 2k−c for k ≤ q, 2xk ≥ M + 2(1 − 2ρ)k − c for k > q as nk < ρk and 2xk,I ≥ M + 2(1−2ρ)k−c for k > q and I ∈ Kk as ord(I) ≥ 1. For strongly regular parameters Card(Kk)< nk < ρk fork > qby Proposition 2.6 and Proposition 4.2. Then, ass >0 and (1 +ρk)2⁻⁽¹⁻^2ρ)sk <2⁻^csk we have

H(s, x0, . . .) (2)

≥1−2⁽¹²^(c+M)⁻^M^)s+1

1−2⁻^(q+1)s

1−2⁻^s +2⁻¹²^(M^+cq)s 1−2⁻^cs

:=h(s, x0, . . .) Again∂h/∂s > 0 and the Implicit Function Theorem insures the existence of a C¹ solution s^∞ = s^∞(x0, x1, . . .) of h(s, x0, . . .) = 0 with s^∞ ≥ s+. F or our choice of constants ρand M we haveqM, and ass >0 there is some constant c >0

(3) c⁻¹s⁻¹≤c⁻¹(1−2⁻^cs)⁻¹≤c1−2⁻^cs(q+1)

1−2⁻^cs ≤c(1−2⁻^cs)⁻¹≤cs⁻¹.