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MULTIFRACTAL ANALYSIS FOR MULTIMODAL MAPS

MIKE TODD

Abstract. Given a multimodal interval map f : I → I and a Holder potential

' : I → ℝ, we study the dimension spectrum for equilibrium states of '. The

main tool here is inducing schemes, used to overcome the presence of criticalpoints. The key issue is to show that enough points are ‘seen’ by a class of

inducing schemes. We also compute the Lyapunov spectrum. We obtain the

strongest results when f is a Collet-Eckmann map, but our analysis also holdsfor maps satisfying much weaker growth conditions along critical orbits.

1. Introduction

Given a metric space X and a probability measure � on X, the pointwise dimensionof � at x ∈ X is defined as

d�(x) := limr→0+

log�(Br(x))

log r

if the limit exists, where Br(x) is a ball of radius r around x. This tells us how con-centrated a measure is around a point x; the more concentrated, the lower the valueof d�(x). For an endomorphism f : X → X, we will study the pointwise dimensionof f -invariant measures �. In particular we will be interested in equilibrium states�' for ' : X → ℝ in a certain class of potentials (see below for definitions).

For any A ⊂ X, we let dimH(A) denote the Hausdorff dimension of A. We let

K'(�) :=

{x : lim

r→0+

log�'(Br(x))

log r= �

}, DS'(�) := dimH(K'(�)),

and

K′' :=

{x : lim

r→0+

log�'(Br(x))

log rdoes not exist

}.

Then we can make a multifractal decomposition:

X = K′' ∪ (∪�∈ℝK'(�)) .

The function DS' is known as the dimension spectrum of �'. The study of thisfunction fits into the more general theory of thermodynamic formalism which alsogives us information on the statistical properties of the system such as return timestatistics, large deviations and decay of correlations.

2000 Mathematics Subject Classification. 37E05, 37D25, 37D35, 37C45,Key words and phrases. Multifractal spectra, thermodynamic formalism, interval maps, non-

uniformly hyperbolicity, Lyapunov exponents, Hausdorff dimension.This work was supported by FCT grant SFRH/BPD/26521/2006 and also by FCT through

CMUP.

1

2 MIKE TODD

These ideas are generally well understood in the case of uniformly hyperbolic sys-tems, see [P]. The dimension spectrum can be described in terms of the pressurefunction, which we define below. A common way to prove this in uniformly hy-perbolic cases is to code the system using a finite Markov shift, and then exploitthe well developed theory of thermodynamic formalism and dimension spectra forMarkov shifts, see for example [PW2]. For non-uniformly hyperbolic dynamicalsystems this approach can be more complicated since we often need to code bycountable Markov shifts. As has been shown by Sarig [S1, S3], Iommi [I1, I2] andPesin and Zhang [PZ] among others, in going from finite to countable Markov shifts,more exotic behaviour, including ‘phase transitions’, appears.

The coding used in non-uniformly hyperbolic cases usually arises from an ‘inducingscheme’: that is, for some part of the phase space, iterates of the original mapare taken, and the resulting ‘induced map’ is considered. The induced maps areMarkov, and so the theory of countable Markov shifts as in [HMU, I1] can be used.In some cases the induced map can be a first return map to an interval, but this isnot always so.

There has been a lot of success with the inducing approach in the case of Manneville-Pomeau maps. These are interval maps which are expanding everywhere, exceptat a parabolic fixed point. The presence of the parabolic point leads to phasetransitions as mentioned above. Multifractal analysis, of the dimension spectrumand the Lyapunov spectrum (see below), of these examples has been carried outby Pollicott and Weiss [PoWe], Nakaishi [Na] and Gelfert and Rams [GR]. In thefirst two of these papers, inducing schemes were used (in the third one, the factthat the original system is Markov is used extensively). The inducing schemes usedare first return maps to a certain natural domain. The points of the original phasespace which the inducing schemes do not ‘see’ is negligible, consisting only of the(countable) set preimages of the parabolic point. We also mention a closely relatedtheory for certain Kleinian groups by Kessebohmer and Stratmann [KeS].

In the case of multimodal maps with critical points, if the critical orbits are densethen there is no way that useful inducing schemes can be first return maps tointervals. Moreover, the set of points which the inducing schemes do not ‘see’ can,in principle, be rather large. In these cases the thermodynamic formalism has alot of exotic behaviour: phase transitions brought about due to some polynomialgrowth condition were discussed by Bruin and Keller in [BK] and shown in moredetail by Bruin and Todd [BT4]. Multiple phase transitions, which are due torenormalisations rather than any growth behaviour, were proved by Dobbs [D2].

In this paper we develop a multifractal theory for maps with critical points by defin-ing inducing schemes which provide us with sufficient information on the dimensionspectrum. The main idea is that points with large enough pointwise Lyapunov ex-ponent must be ‘seen’ by certain inducing schemes constructed in [BT4]. Theseinducing schemes are produced via the Markov extension known as the Hofbauerextension, also known as the Hofbauer tower. This structure was developed by Hof-bauer and Keller, see for example [H1, H2, K2]. Their principle applications werefor interval maps. The theory for higher dimensional cases was further developed byBuzzi [Bu]. Once we have produced these inducing schemes, we can use the theoryof multifractal analysis developed by Iommi in [I1] for the countable Markov shiftcase. Note that points with zero pointwise Lyapunov exponent cannot be ‘seen’ by

MULTIFRACTAL ANALYSIS FOR MULTIMODAL MAPS 3

measures which are compatible to an inducing scheme, but in our case such setsturn out to be negligible.

There is a further property which useful inducing schemes must have: not onlymust they see sufficiently many points, but also they must be well understood fromthe perspective of the thermodynamic formalism. Specifically, given a potential ,we need its induced version on the inducing scheme to fit into the framework ofSarig [S2]. In [PSe, BT2, BT4] this was essentially translated into having ‘good tailbehaviour’ of the equilibrium states for the induced potentials.

Our main theorem states that, as in the expanding case, for a large class of multi-modal maps, the multifractal spectrum can be expressed in terms of the Legendretransform of the pressure function for important sets of parameters �. The Collet-Eckmann case is closest to the expanding case, and here we indeed get exactly thesame kind of graph for � 7→ DS'(�) as in the expanding case for the values of �we consider. In the non-Collet Eckmann case, we expect the graph of DS' to bequalitatively different from the expanding case, as shown for the related Lyapunovspectrum in [Na] and [GR]. We note that singular behaviour of the Lyapunovspectrum was also observed by Bohr and Rand [BoR] for the special case of thequadratic Chebyshev polynomial.

The results presented here can be seen as an extension of some of the ideas in [H3],in which the full analysis of the dimension spectrum was only done for uniformlyexpanding interval maps. See also [Y] for maps with weaker expansion properties.Moreover, Hofbauer, Raith and Steinberger [HRS] proved the equality of variousthermodynamic quantities for non-uniformly expanding interval maps, using ‘es-sential multifractal dimensions’. However, the full analysis in the non-uniformlyexpanding case, including the expression of the dimension spectrum in terms ofsome Legendre transform, was left open.

1.1. Key definitions and main results. Given a dynamical system f : X → X,we let

ℳ =ℳ(f) := {f -invariant probability measures on X}and

ℳerg =ℳerg(f) := {� ∈ℳ : � is ergodic}.For a potential ' : X → ℝ, the pressure is defined as

P (') := sup�∈ℳ

{ℎ� +

∫' d� : −

∫' d� <∞

}where ℎ� denotes the metric entropy with respect to �. Note that by the ergodicdecomposition, we can just take the above supremum over ℳerg. We let ℎtop(f)denote the topological entropy of f , which is equal to P (0), see [K4]. A measure� which ‘achieves the pressure’, i.e., ℎ� +

∫' d� = P ('), is called an equilibrium

state.

Let ℱ be the collection of C3 multimodal interval maps f : I → I where I = [0, 1],satisfying:

a) the critical set Crit = Crit(f) consists of finitely many critical point c with crit-ical order 1 < ℓc <∞, i.e., f(x) = f(c)+(g(x−c))ℓc for some diffeomorphismsg : ℝ→ ℝ with g(0) = 0 and x close to c;

4 MIKE TODD

b) f has no parabolic cycles;

c) f is topologically transitive on I;

d) fn(Crit) ∩ fm(Crit) = ∅ for m ∕= n.

Remark 1. Conditions c) and d) are for ease of exposition, but not crucial. Inparticular, Condition c) excludes that f is renormalisable. For multimodal mapssatisfying a) and b), the set Ω consists of finitely many components Ωk, on each ofwhich f is topologically transitive, see [MS, Section III.4]. In the case where thereis more than one transitive component in Ω, for example the renormalisable case,the analysis presented here can be applied to any one of the transitive componentsconsisting of intervals. We also note that in this case Ω contains a (hyperbolic)Cantor set outside components of Ω which consist of intervals. The work of Dobbs[D2] shows that renormalisable maps these hyperbolic Cantor sets can give riseto singular behaviour in the thermodynamic formalism (phase transitions in thepressure function t 7→ P (t')) not accounted for by the behaviour of critical pointsthemselves. For these components we could apply a version of the usual hyperbolictheory to study the dimension spectra.

We include condition b) in order to apply the distortion theorem, [SV, Theorem C].Alternatively, we could assume negative Schwarzian derivative, since this added tothe transitivity assumption implies that there are no parabolic points.

Condition d) rules out one critical point mapping onto another. Alternatively, itwould be possible to consider these critical points as a ‘block’, but to simplify theexposition, we will not do that here. Condition d) also rules out critical points beingpreperiodic.

We define the lower/upper pointwise Lyapunov exponent as

�f (x) := lim infn→∞

1

n

n−1∑j=0

log ∣Df(f j(x))∣, and �f (x) := lim supn→∞

1

n

n−1∑j=0

log ∣Df(f j(x))∣

respectively. If �f (x) = �f (x), then we write this as �f (x). For a measure � ∈ℳerg, we let

�f (�) :=

∫log ∣Df ∣ d�

denote the Lyapunov exponent of the measure. Since our definition of ℱ willexclude the presence of attracting cycles, [Pr] implies that �f (�) ⩾ 0 for all f ∈ ℱand � ∈ℳ.

For � ⩾ 0, we denote the ‘good Lyapunov exponent’ sets by

LG� := {x : �f (x) > �} and LG� := {x : �f (x) > �}.

We define

K'(�) := K'(�) ∩ LG0 and DS'(�) := dimH(K'(�)).

As well as assuming that our maps f are in ℱ , we will also sometimes imposecertain growth conditions on f :


∙ An exponential growth condition (Collet-Eckmann): there exist CCE, �CE > 0,

∣Dfn(f(c))∣ ⩾ CCEe�CEn for all c ∈ Crit and n ∈ ℕ. (1)

∙ A polynomial growth condition: There exist CP > 0 > 0 and �P > 2ℓmax(f)so that

∣Dfn(f(c))∣ ⩾ CPn�P for all c ∈ Crit and n ∈ ℕ. (2)

∙ A simple growth condition:

∣Dfn(f(c))∣ → ∞ for all c ∈ Crit. (3)

In all of these cases, [BRSS] implies that there is a unique absolutely continuousinvariant probability measure (acip). This measure has positive entropy by [MS,Exercise V.1.4] and [SV, Proposition 7].

We will consider potentials −t log ∣Df ∣ and also �-Holder potentials ' : I → ℝsatisfying

sup'− inf ' < ℎtop(f). (4)

Without loss of generality, we will also assume that P (') = 0. Note that our resultsdo not depend crucially on � ∈ (0, 1], so we will ignore the precise value of � fromhere on.

Remark 2. We would like to emphasise that (4) may not be easy to remove as anassumption on our class of Holder potentials if all the results we present here areto go through. For example, in the setting of Manneville-Pomeau maps, in [BT2,Section 6] it was shown that for any " > 0, there exists a Holder potential ' withsup'− inf ' = ℎtop(f) + " and for which the equilibrium state is a Dirac measureon the fixed point (which is not seen by any inducing scheme).

We briefly sketch some properties of these maps and potentials. For details, seePropositions 2 and 3. As we will see below, we are interested in potentials of theform −t log ∣Df ∣+ '. By [BT4] if f satisfies (1) then there exist t1 < 1 < t2 suchthat for each t ∈ (t1, t2) there is an equilibrium state �−t log ∣Df ∣ for −t log ∣Df ∣. Iff only satisfies (2) then we take t2 = 1. Combining [BT4] and [BT2], for Holderpotentials ' we have equilibrium states �−t log ∣Df ∣+ ' for −t log ∣Df ∣ + ' if t isclose to 1 and is close to 0. Also, by [BT2], if (3) holds and ' is a Holder potentialsatisfying (4), then there are equilibrium states �−t log ∣Df ∣+ ' for −t log ∣Df ∣+ 'if t is close to 0 and is close to 1. These equilibrium states are unique. Asexplained in the appendix, (3) is assumed in [BT2] in order to ensure that theinduced versions of ' are sufficiently regular, so if this regularity can be shownanother way, for example in the simple case that ' is a constant everywhere, thiscondition can be omitted.

We define the auxiliary function

T'(q) := inf{t : P (−t log ∣Df ∣+ q') = 0}. (5)

If T'(q) is finite, we set

q := −T'(q) log ∣Df ∣+ q'.

If P (') = 0 then T'(1) = 0. As we will show in Lemma 2, the map q 7→ T'(q) isstrictly decreasing on [0, 1]. Moreover, by Ledrappier [L, Theorem 3], if there is an

6 MIKE TODD

acip then it is an equilibrium state for x 7→ − log ∣Df(x)∣ and so T'(0) = 1. It maybe the case that for some values of q, T'(q) = ∞. For example, let f ∈ ℱ be aunimodal map not satisfying (1). Then as in [NS], P (−t log ∣Df ∣) = 0 for all t ⩾ 1.If we set ' to be the constant potential, then P (') = 0 implies ' ≡ −ℎtop(f) sincethen P (') = P (0)− ℎtop(f) = 0. For such ' and for q < 0, then T'(q) =∞.

For ℎ a convex function, we say that (ℎ, g) form a Fenchel pair if

g(p) = supx{px− ℎ(x)}.

In this case, g is known as the Legendre-Fenchel transform of ℎ. If ℎ is convex andC1 then the function g is called the Legendre transform of ℎ and

g(�) = q�− ℎ(q) were q is such that � = −Dℎ(q).

If f ∈ ℱ satisfies (3) then [BRSS] guarantees the existence and uniqueness of anacip �− log ∣Df ∣ and we let

�ac :=−∫' d�− log ∣Df ∣

�f (�− log ∣Df ∣).

Theorem A. Suppose that f ∈ ℱ is a map satisfying (3) and ' : I → ℝ is aHolder potential satisfying (4), and with P (') = 0. If the equilibrium state �' isnot equal to the acip then there exist open sets U, V ⊂ ℝ so that T' is differentiable

on V and for � ∈ U , the dimension spectrum � 7→ DS'(�) is minus the Legendretransform of q 7→ T'(q). Moreover,

(a) U contains a neighbourhood of dimH(�'), and DS'(dimH(�')) = dimH(�');(b) if f satisfies (2), then U contains both a neighbourhood of dimH(�'), and

a one-sided neighbourhood of �ac, where DS'(�ac) = 1;(c) if f satisfies (1), then U contains both a neighbourhood of dimH(�') and

of �ac.

Furthermore, for all � ∈ U there is a unique equilibrium state � q for the potential

q so that � q (K�) = 1, where � = −DT'(q). This measure has full dimension on

K�, i.e., dimH(� q ) = dimH(K�).

Note that by Hofbauer and Raith [HR], dimH(�') =ℎ�'

�f (�') , and as shown by

Ledrappier [L, Theorem 3], dimH(�− log ∣Df ∣) =ℎ�− log ∣Df∣

�f (�− log ∣Df∣)= 1.

In Section 6 we consider the situation where ' is the constant potential, which werecall that since P (') = 0, must be of the form ' ≡ −ℎtop(f). In that setting, asnoted above T' is infinite for q < 0 when f is unimodal and does not satisfy (1).

Therefore, in that case we would expect DS' to behave differently to the expandingcase for � > �ac. This is why we only deal with a one-sided neighbourhood of �acin (b). See also Remark 8 for more information on this.

If, contrary to the assumptions of Theorem A, �' = �− log ∣Df ∣ then DS'(�) is zerofor every � ∈ ℝ, except at � = dimH(�'), where it takes the value 1. As in Re-mark 6 below, for a multimodal map f and a constant potential, this can only occur


when f has preperiodic critical points, for example when f is the quadratic Cheby-shev polynomial. In view of Livsic theory for non-uniformly hyperbolic dynamicalsystems, in particular the results in [BHN, Section 5], we expect �' ∕= �− log ∣Df ∣for multimodal maps with infinite critical orbit for more general Holder potentials'.

According to [BS], if (1) holds then there exists � > 0 so that the nonwanderingset Ω is contained in LG� ∪ (∪n⩾0f

−n(Crit)). Therefore we have the followingcorollary. Note that here the neighbourhood U is as in case (c) of Theorem A.

Corollary B. Suppose that f ∈ ℱ satisfies the Collet-Eckmann condition (1) and' : I → ℝ is a Holder potential satisfying (4) and with P (') = 0. If the equilibriumstate �' is not equal to the acip then there exist open sets U, V ⊂ ℝ so that T'is differentiable on V , U contains dimH(�') and 1, and so that for � ∈ U thedimension spectrum DS'(�) is minus the Legendre transform of T'.

In fact, to ensure that DS'(�) = DS'(�) it is enough to show that ‘enough pointsiterate into a finite set of levels of the Hofbauer extension infinitely often’. Asin [K2], one way of guaranteeing this is to show that a large proportion of thesets we are interested in ‘go to large scale’ infinitely often. Graczyk and Smirnov[GS] showed that for rational maps of the complex plane satisfying a summabilitycondition, this is true. Restricting their result to real polynomials, we have thefollowing Corollary, which we explain in more detail in Section 5.1.

Corollary C. Suppose that f ∈ ℱ extends to a polynomial on ℂ with no parabolicpoints, all critical points in I, and satisfying (2). Moreover, suppose that ' : I → ℝis a Holder potential satisfying (4) and P (') = 0. If the equilibrium state �' isnot equal to the acip then there exist sets U, V ⊂ ℝ such that U contains a one-sided neighbourhood of �ac, T' is differentiable on V , and for � ∈ U the dimensionspectrum DS'(�) is minus the Legendre transform of T'. Moreover, if dimH(�') >ℓmax(f)�P−1 then the same is true for any � in a neighbourhood of dimH(�').

Barreira and Schmeling [BaS] showed that in many situations the set K′' has fullHausdorff dimension. As the following proposition states, this is also the case inour setting. The proof follows almost immediately from [BaS], but we give somedetails in Section 5.

Proposition 1. Suppose that f ∈ ℱ satisfies (3) and ' : I → ℝ is a Holderpotential satisfying (4) and with P (') = 0. Then dimH(K′') = 1.

Theorem A also allows us to compute the Lyapunov spectrum. The results in thiscase are in Section 6.

For ease of exposition, in most of this paper the potential ' is assumed to be Holder.In this case existence of an equilibrium state �' was proved by Keller [K1]. However,as we show in the appendix, all the results here hold for a class of potentials (SV I)considered in [BT2]. Since we need information on the corresponding '-conformalmeasures for our potentials ', as an auxiliary result, we prove the existence ofconformal measures m' for potentials ' in the set SV I. Moreover, we show that

for the corresponding equilibrium states �', the densityd�'dm'

is uniformly bounded

away from 0 and∞. This is used here in order to compare d�Φ(x) and d�'(x), where

8 MIKE TODD

�Φ is the equilibrium state for an inducing scheme (X,F ), with induced potentialΦ : X → ℝ (see below for more details). The equality of d�Φ(x) and d�'(x) forx ∈ X is not immediate in either the case ' is Holder or the case ' satisfies SV I.This is in contrast to the situation where the inducing schemes are simply firstreturn maps, in which case �Φ is simply a rescaling of the original measure �' andhence d�Φ

(x) = d�'(x). However, we will prove that for the inducing schemes usedhere, this rescaling property is still true of the conformal measures m' and mΦ,which then allows us to compare d�Φ(x) and d�'(x). It is interesting to note thatthe proof of existence of a conformal measure also goes through for potentials ofthe form x 7→ −t log ∣Df(x)∣.

Note: After this work was completed, it was communicated to me that J. Rivera-Letelier and W. Shen have proved a result ([RS, Corollary 6.3]) which implies that

we can replace DS'(�) with DS'(�) throughout. For some details on this seeSection 5.1.

Acknowledgements: I would like to thank H. Bruin, N. Dobbs, G. Iommi, T. Jordan,W. Shen and an anonymous referee for useful comments on earlier versions of thispaper. I would also like to thank them and D. Rand for fruitful conversations.

2. The maps, the measures and the inducing schemes

Let (X, f) be a dynamical system and ' : X → [−∞,∞] be a potential. For uselater, we let

Sn'(x) := '(x) + ⋅ ⋅ ⋅+ ' ∘ fn−1(x).

We say that a measure m, is conformal for (X, f, ') if m(X) = 1, and for any Borelset A so that f : A→ f(A) is a bijection,

m(f(A)) =

∫A

e−' dm

(or equivalently, dm(f(x)) = e−'(x)dm(x)).

2.1. Hofbauer extensions. We next define the Hofbauer extension, sometimesalso known as a Hofbauer tower. The setup we present here can be applied togeneral dynamical systems, since it only uses the structure of dynamically definedcylinders. An alternative way of thinking of the Hofbauer extension specifically forthe case of multimodal interval maps, which explicitly makes use of the critical set,is presented in [BB].

We first consider the dynamically defined cylinders. We let P0 := I and Pn denotethe collection of maximal intervals Cn so that fn : Cn → fn(Cn) is a homeomor-phism. We let Cn[x] denote the member of Pn containing x. If x ∈ ∪n⩾0f

−n(Crit)there may be more than one such interval, but this ambiguity will not cause us anyproblems here.

The Hofbauer extension is defined as

I :=⊔k⩾0

⊔Ck∈Pk

fk(Ck)/ ∼


where fk(Ck) ∼ fk′(Ck′) as components of the disjoint union I if fk(Ck) =

fk′(Ck′) as subsets in I. Let D be the collection of domains of I and � : I → I

be the natural inclusion map. A point x ∈ I can be represented by (x,D) where

x ∈ D for D ∈ D and x = �(x). Given x ∈ I, we can denote the domain D ∈ D itbelongs to by Dx.

The map f : I → I is defined by

f(x) = f(x,D) = (f(x), D′)

if there are cylinder sets Ck ⊃ Ck+1 such that x ∈ fk(Ck+1) ⊂ fk(Ck) = D andD′ = fk+1(Ck+1). In this case, we write D → D′, giving (D,→) the structure of a

directed graph. Therefore, the map � acts as a semiconjugacy between f and f :

� ∘ f = f ∘ �.

We denote the ‘base’ of I, the copy of I in I by D0. For D ∈ D, we define lev(D)to be the length of the shortest path D0 → ⋅ ⋅ ⋅ → D starting at the base D0. Foreach R ∈ ℕ, let IR be the compact part of the Hofbauer extension defined by thedisjoint union

IR := ⊔{D ∈ D : lev(D) ⩽ R}.

For maps in ℱ , we can say more about the graph structure of (D,→) since Lemma1 of [BT4] implies that if f ∈ ℱ then there is a closed primitive subgraph DT of D.That is, for any D,D′ ∈ DT there is a path D → ⋅ ⋅ ⋅ → D′; and for any D ∈ DT ,if there is a path D → D′ then D′ ∈ DT too. We can denote the disjoint union of

these domains by IT . The same lemma says that if f ∈ ℱ then �(IT ) = Ω and f

is transitive on IT .

Given � ∈ ℳerg, we say that � lifts to I if there exists an ergodic f -invariant

probability measure � on I such that � ∘ �−1 = �. For f ∈ ℱ , if � ∈ ℳerg and

�(�) > 0 then � lifts to I, see [K2, BK].

For convenience later, we let � := �∣−1D0

. Note that there is a natural distancefunction dI within domains D (but not between them) induced from the Euclideanmetric on I.

2.2. Inducing schemes. We say that (X,F, �) is an inducing scheme for (I, f) if

∙ X is an interval containing a finite or countable collection of disjoint intervals Xi

such that F maps each Xi diffeomorphically onto X, with bounded distortionon all iterates (i.e. there exists K > 0 so that for if x, y are in the same domainof Fn then 1/K ⩽ DFn(x)/DFn(y) ⩽ K);

∙ � ∣Xi = �i for some �i ∈ ℕ and F ∣Xi = f�i . If x /∈ ∪iXi then �(x) =∞.

The function � : ∪iXi → ℕ is called the inducing time. It may happen that �(x) isthe first return time of x to X, but that is certainly not the general case. For easeof notation, we will often write (X,F ) = (X,F, �). In this paper we can alwaysassume that every inducing scheme is uniformly expanding.

10 MIKE TODD

Given an inducing scheme (X,F, �), we say that a measure �F on X is a lift of �on I if for all �-measurable subsets A ⊂ I,

�(A) =1∫

X� d�F

∑i

�i−1∑k=0

�F (Xi ∩ f−k(A)). (6)

Conversely, given a measure �F for (X,F ), we say that �F projects to � if (6)holds. We denote

(X,F )∞ :={x ∈ X : �(F k(x)) is defined for all k ⩾ 0

}.

We call a measure � compatible to the inducing scheme (X,F, �) if

∙ �(X) > 0 and �(X ∖ (X,F )∞) = 0; and

∙ there exists a measure �F which projects to � by (6), and in particular∫X� d�F <

∞.

For a potential ' : I → ℝ, we define the induced potential Φ : X → ℝ for aninducing scheme (X,F, �) as

Φ(x) := S�(x)'(x) = '(x) + . . .+ ' ∘ f�(x)−1(x)

whenever �(x) < ∞. We denote Φi := supx∈Xi Φ(x). Note that sometimes wewill abuse notation and write (X,F,Φ) when we are particularly interested in theinduced potential for the inducing scheme. The following is known as Abramov’sformula, see for example [Z, PSe].

Lemma 1. Let �F be an ergodic invariant measure on (X,F, �) such that∫� d�F <

∞ and with projected measure �. Then ℎ�F (F ) =(∫� d�F

)ℎ�(f). Moreover,

if ' : I → ℝ is a potential, and Φ the corresponding induced potential, then∫Φ d�F =

(∫� d�F

) ∫' d�.

Fixing f , we let

ℳ+ := {� ∈ℳerg : �f (�) > 0}, and for " > 0, ℳ" := {� ∈ℳerg : ℎ� ⩾ "}.

For a proof of the following result, see [BT4, Theorem 3].

Theorem 1. If f ∈ ℱ and � ∈ ℳ+, then there is an inducing scheme (X,F, �)and a measure �F on X such that

∫X� d�F < ∞. Here �F is the lifted measure

of � (i.e., � and �F are related by (6)). Moreover, (X,F )∞ = X ∩ Ω.

Conversely, if (X,F, �) is an inducing scheme and �F an ergodic F -invariant mea-sure such that

∫X�d�F <∞, then �F projects to a measure � ∈ℳ+.

The proof of the above theorem uses the theory of [B, Section 3]. The main ideais that the Hofbauer extension can be used to produce inducing schemes. We pickX ⊂ IT and use a first return map to X to give the inducing scheme on X := �(X).We will always choose X to be a cylinder in Pn, for various values of n ∈ ℕ. Asin [BT4], sets X, and thus the inducing schemes they give rise to, will be of twotypes.

Type A: The set X is an interval in a single domain D ∈ DT . Then for x ∈ Xthere exists a unique x ∈ X so that �(x) = x. Then �(x) is defined as the first

return time of x to X. We choose X so that X ∈ Pn for some n, and X is compactly


contained in D. These properties mean that (X,F, �) is an inducing scheme whichis extendible. That is to say, letting X ′ = �(D), for any domain Xi of (X,F ) thereis an extension of f�i to X ′i ⊃ Xi so that f�i : X ′i → X ′ is a homeomorphism. Bythe distortion [SV, Theorem C(2)], this means that (X,F ) has uniformly bounded

distortion, with distortion constant depending on � := dI(X, ∂D).

Type B: We fix � > 0 and some interval X ∈ Pn for some n. We say thatthe interval X ′ is a �-scaled neighbourhood of X if, denoting the left and rightcomponents of X ′ ∖ X by L and R respectively, we have ∣L∣, ∣R∣ = �∣X∣. We fix

such an X ′ and let X = ⊔{D ∩ �−1(X) : D ∈ DT , �(D) ⊃ X ′}. Let rX denote

the first return time to X. Given x ∈ X, for any x ∈ X with �(x) = x, we set�(x) = rX(x). In [B] it is shown that by the setup, this time is independent of the

choice of x in �∣−1

X(x). Also for each Xi there exists X ′i ⊃ Xi so that f�i : X ′i → X ′

is a homeomorphism, and so, again by the Koebe Lemma, F has uniformly boundeddistortion, with distortion constant depending on �.

We will need to deal with both kinds of inducing scheme since we want informationon the tail behaviour, i.e., the measure of {� ⩾ n} for different measures. As inPropositions 2 and 3 below, for measures close to �' we have good tail behaviourfor schemes of type A; and for measures close to the acip �− log ∣Df ∣ we have goodtail behaviour for schemes of type B. We would like to point out that any typeA inducing time �1 can be expressed as a power of a type B inducing time �2,i.e., �1 = �p2 where p : X → ℕ. Moreover,

∫p d�1 <∞ for the induced measure �1

for the type A inducing scheme. This type of relation is considered by Zweimuller[Z].

2.3. Method of proof. The main difficulty in the proof of Theorem A is to get anupper bound on the dimension spectrum in terms of T'. To do this, we show thatthere are inducing schemes which have sufficient multifractal information to give

an upper bound on DS'. Then we can use Iommi’s main theorem in [I1], whichgives upper bounds in terms of the T for the inducing scheme. It is the use of theseinducing schemes which is the key to this paper.

We first show in Section 3 that for a given range of � there are inducing schemeswhich are compatible to any measure � which has ℎ� +

∫ q d� sufficiently large,

where q depends on �. In doing this we will give most of the theory of thermo-dynamic formalism needed in this paper. For example, we show the existence ofequilibrium states on K� which will turn out to have full dimension (these also givethe lower bound for DS').

In Section 4, we prove that given � > 0, there is a finite set of inducing schemes that‘sees’ all points x ∈ I with �f (x) ⩾ �, up to set of small Hausdorff dimension. Thismeans that we can fix inducing schemes which contain all the relevant measures,as above, and also contain the multifractal data. Then in Section 5 we proveTheorem A and Proposition 1. In Section 6 we show how our results immediatelygive us information on the Lyapunov spectrum. In the appendix we show thatpointwise dimensions for induced measures and the original ones are the same, alsoextending our results to potentials in the class SV I.

12 MIKE TODD

3. The range of parameters

In this section we determine what U is in Theorem A. In order to do so, wemust introduce most of the theory of the thermodynamical properties for inducingschemes required in this paper. The first step is to show that if �(q) ∈ U , then theequilibrium states for q are forced to have positive entropy. By Theorem 1, thisensures that the equilibrium states must be compatible to some inducing scheme,and thus we will be able to use Iommi’s theory. In order to do this we need to showthat T'(q) is finite for q ⩾ 0.

Lemma 2. Let f ∈ ℱ and ' : I → ℝ be a potential satisfying (4) and withP (') = 0. If q ⩾ 0 then the function T'(q) is finite. If (1) holds then T'(q) isalso finite for all q in a neighbourhood of 0. In any case, T' is strictly decreasingon (0, 1).

Proof. We begin without needing to assume (1). We first show that our assumptionsimply that ' < 0. By (4) and P (') = 0, we have

0 = P (') ⩾ ℎtop(f) +

∫' d�−ℎtop(f) ⩾ ℎtop(f) + inf ' > sup'

where �−ℎtop(f) denotes the measure of maximal entropy (for more details of thismeasure, see Section 6). Hence ' < 0 as required.

If q ⩾ 0 then q' ⩽ 0. Therefore,

P (−t log ∣Df ∣+ q') ⩽ P (−t log ∣Df ∣).

Since P (− log ∣Df ∣) ⩽ 0 and t 7→ P (−t log ∣Df ∣) is decreasing, this implies thatT'(q) ⩽ 1. It remains to check T'(q) ∕= −∞.

We have

P (−t log ∣Df ∣+ q') ⩾ P (−t log ∣Df ∣+ q inf ') = P (−t log ∣Df ∣) + q inf '.

It is easy to show that

limt→−∞

P (−t log ∣Df ∣) =∞.

Hence there exists t0 < 0 such that

P (−t0 log ∣Df ∣)− q inf ' > 0.

That is

P (−t0 log ∣Df ∣+ q') > 0,

so T'(q) ∈ [t0, 1]. Since the function t → P (−t log ∣Df ∣ + q') is continuous, theIntermediate Value Theorem implies that there exists T'(q) ∈ [t0, 1] such that

T'(q) = inf{t ∈ ℝ : P (−t log ∣Df ∣+ q') = 0},

as required.

If (1) holds then there exists t1 > 1 such that P (−t1 log ∣Df ∣) = −" < 0 and so forq ∈ ("/ inf ', 0)

P (−t log ∣Df ∣+ q') ⩽ P (−t1 log ∣Df ∣) + q inf ' < 0,

so T'(q) ⩽ t1. The lower bound on T'(q) follows as above.


To show that T' is decreasing, note that since ' < 0, if q, q + " ∈ (0, 1),

P (−T'(q) log ∣Df ∣+ (q + ")') ⩽ P (−T'(q) log ∣Df ∣+ q') + " sup' < 0.

Since by [Pr], for any measure � ∈ ℳ, �(�) ⩾ 0 this implies that T'(q + ") <T'(q). □

We let

G"(') :=

{q : ∃� < 0 such that ℎ� +

∫ q d� > � ⇒ ℎ� > "

}.

The next lemma shows that most of the relevant parameters q which we are inter-ested in must lie in G"(').

Lemma 3. Let f ∈ ℱ and ' : I → ℝ be a potential satisfying (4) and withP (') = 0. Suppose that (3) holds for f . There exist " > 0, q1 < 1 < q2 sothat (q1, q2) ⊂ G"('). If we take " > 0 arbitrarily close to 0 then we can take q1

arbitrarily close to 0. If (1) holds then [0, 1] ⊂ (q1, q2).

Proof. We first prove the existence of q1 < 1 such that (q1, 1] ⊂ G"('). As in theproof of Lemma 2, we have ' < 0. Let q1 be any value in (0, 1]. Then suppose thatfor some � < 0, a measure � ∈ℳerg has

ℎ� +

∫−T'(q1) log ∣Df ∣+ q1' d� > �

for T' as in (5). Recall that by [Pr], �(�) ⩾ 0 since we excluded the possibility ofattracting cycles for maps f ∈ ℱ . Then

ℎ� > � +

∫T'(q1) log ∣Df ∣ − q1' d� ⩾ � + q1∣ sup'∣ > 0.

If q1 was chosen very close to 0 then " > 0 must be chosen small too.

We can similarly show the existence of q2 > 1 such that [1, q2) ⊂ G"('), the onlydifference in this case being that q > 1 implies that T'(q) < 0. So we can take �as above and obtain

ℎ� > � +

∫T'(q) log ∣Df ∣ − q' d� ⩾ � +

∫T'(q) log ∣Df ∣ d�+ q∣ sup'∣.

Since T'(q) is close to 0 for q close to 1 and since∫

log ∣Df ∣ d� < log supx∈I ∣Df(x)∣,for q2 > 1 close to 1, the above must be strictly positive.

Suppose now that (1) holds. Then by [BS], there exists � > 0 so that any invariantmeasure � ∈ℳerg must have �f (�) > �. So if ℎ�+

∫−T'(q) log ∣Df ∣+ q' d� > �,

then

ℎ� > � +

∫T'(q) log ∣Df ∣ − q' d� ⩾ � + T'(q)� + q∣ sup'∣.

For q close to 0, T'(q) must be close to 1, so we can choose � and q1 < 0 so thatthe lemma holds. □

The sets Cover(") and SCover("): Let " > 0. By [BT4, Remark 6] there exists

� > 0 and a compact set E ⊂ IT so that � ∈ℳ" implies that �(E) > �. Moreover

E can be taken inside IR ∖ B�(∂I) for some R ∈ ℕ and � > 0. (Here B�(∂I) is

a �-neighbourhood of ∂I with respect to the distance function dI). As in [BT4,

14 MIKE TODD

Section 4.2], E can be covered with sets X1, . . . , Xn so that each Xk acts as the set

which gives the inducing schemes (Xk, Fk) (where Xk = �(Xk)) as in Theorem 1.We will suppose that these sets are either all of type A, or all of type B. Thismeans that any � ∈ℳ" must be compatible to at least one of (Xk, Fk). We denote

CoverA(") = {X1, . . . , Xn} and the corresponding set of schemes by SCoverA(")if we are dealing with type A inducing schemes. Similarly we use CoverB(") andSCoverB(") for type B inducing schemes. If a result applies to schemes of bothtypes then we omit the superscript.

We let {Xk,i}i denote the domains of the inducing scheme (Xk, Fk) and we denotethe value of �k on Xk,i by �k,i. Given (Xk, Fk, �k), we let Ψq,k denote the inducedpotential for q.

From this setup, given q ∈ G"(') there must exist a sequence of measures {�n}n ⊂ℳ" and a scheme (Xk, Fk) so that ℎ�n +

∫ q d�n → P ( q) = 0 and �n are all

compatible to (Xk, Fk). Later this fact will allow us to use [BT4, Proposition 1] tostudy equilibrium states for q.

If � : I → ℝ is some potential and (X,F ) is an inducing scheme with inducedpotential Υ : X → ℝ, we let Υi := supx∈Xi Υ(x). We let PFk be the set of k-cylinders generated by (X,F ). We define the kth variation as

Vk(Υ) := supCk∈PFk

{∣Υ(x)−Υ(y)∣ : x, y ∈ Ck}.

We say that Υ is locally Holder continuous if there exists � > 0 so that Vk(Υ) =O(e−�k). We let

Z0(Υ) :=∑i

eΥi , and Z∗0 (Υ) :=∑i

�ieΥi . (7)

As in [S2], if Υ is locally Holder continuous, then Z0(Υ) <∞ implies P (Υ) <∞.

We say that a measure � satisfies the Gibbs property with constant P ∈ ℝ for(X,F,Υ) if there exists KΦ, P ∈ ℝ so that

1

KΦ⩽

�(Cn)

eSnΥ(x)−nP ⩽ KΦ

for every n-cylinder Cn ∈ PFn and all x ∈ Cn.

The following is the main result of [BT2] (in fact it is proved for a larger class ofpotentials there).

Proposition 2. Given f ∈ ℱ satisfying (3) and ' : I → ℝ a Holder potentialsatisfying (4) and with P (') = 0, then for any " > 0 and any (X,F ) ∈ SCoverA(")with induced potential Φ:

(a) There exists �Φ > 0 such that∑�i=n

eΦi = O(e−n�Φ);

(b) Φ is locally Holder continuous and P (Φ) = 0;(c) There exists a unique Φ-conformal measure mΦ, and a unique equilibrium state

�Φ for (X,F,Φ).

(d) There exists CΦ so that 1CΦ⩽ d�Φ

dmΦ⩽ CΦ;

(e) There exists a unique equilibrium state �' for (I, f, ');

(f) The map t 7→ P (t') is analytic for t ∈(−ℎtop(f)

sup'−inf ' ,ℎtop(f)

sup'−inf '

).


The existence of the equilibrium state �' under even weaker conditions than thesewas proved by Keller [K1]. However, we need all of the properties above to completeour analysis of the dimension spectrum of �'.

The following is proved in [BT4]. For the same result for unimodal maps satisfying(1) see [BK], which used tools from [KN].

Proposition 3. Suppose that f ∈ ℱ satisfies (2) and let

(x) = t(x) := −t log ∣Df(x)∣ − P (−t log ∣Df(x)∣).

Then there exists t0 < 1 such that for any t ∈ (t0, 1) there is " = "(t) > 0 so thatfor any (X,F ) ∈ SCoverB(") with induced potential Ψ:

(a) There exists �DF > 0 such that∑�i=n

eΨi = O(e−n�DF );

(b) Ψ is locally Holder continuous and P (Ψ) = 0;(c) There exists a unique Ψ-conformal measure mΨ, and a unique equilibrium state

�Ψ for (X,F,Ψ);

(d) There exists CΨ so that 1CΨ⩽ d�Ψ

dmΨ⩽ CΨ;

(e) There exists a unique equilibrium state � for (I, f, ) and thus for (I, f,−t log ∣Df ∣);(f) The map t 7→ P (−t log ∣Df ∣) is analytic in (t0, 1).

If f ∈ ℱ satisfies (1), then this proposition can be extended so that t can be takenin a two-sided neighbourhood of 1.

In Proposition 2 both mΦ and �Φ satisfy the Gibbs property, and in Proposition 3both mΨ and �Ψ satisfy the Gibbs property: in all these cases, the Gibbs constant Pis 0. By the Gibbs property, part (a) of Proposition 2 and 3 imply that �Φ({� = n})and �Ψ({� = n}) respectively decay exponentially. These systems are referred toas having exponential tails.

One consequence of the first item in both of these propositions, as noted in [BT2,Theorem 10] and [BT4, Theorem 5], is that we can consider combinations of thepotentials above: x 7→ −t log ∣Df(x)∣+ s'(x)−P (−t log ∣Df ∣+ s'). We can derivethe same results for this potential for t close to 1 and s sufficiently close to 0, oralternatively for s close to 1 and t sufficiently close to 0. Note that by [KN, BK] thiscan also be shown in the setting of unimodal maps satisfying (1) with potentials 'of bounded variation.

If (X,F ) is an inducing scheme with induced potential Φ : X → ℝ, we define

PB"(Φ) := {q ∈ G"(') : ∃� > 0 s.t. Z∗0 (Ψq + ��) <∞} .

Lemma 4. For (Xk, Fk) ∈ SCover("), if q ∈ PB"(Φk) then P (Ψq,k) = 0. More-over, there is an equilibrium state �Ψq,k for (Xk, Fk,Ψq,k) and the correspondingprojected equilibrium state � q is compatible to any (Xj , Fj) ∈ SCover(").

In this lemma, SCover(") can be SCoverA(") or SCoverB("). Note that by [BT4,Proposition 1], if for any (X,F ) ∈ SCover(") and q ∈ PB"(Φ), then there existsan equilibrium state �Ψq for (X,F,Ψq), as well as a unique equilibrium state � qfor (I, f, q).

16 MIKE TODD

Proof. Firstly we have P (Ψq,k) = 0 for the inducing scheme (Xk, Fk) by Case 3 of[BT4, Proposition 1]. Secondly we can replace (Xk, Fk) with any inducing scheme(Xj , Fj) ∈ SCover(") by [BT4, Lemma 9]. □

This lemma means that if q ∈ PB"(Φk) for (Xk, Fk) ∈ SCoverA("), then q ∈PB"(Φj) for any (Xj , Fj) ∈ SCoverA("). Therefore, we can denote this set of q byPBA" ('). Since the same argument holds for inducing schemes of type B, we cananalogously define the set PBB" ('). Note that "′ < " implies PB"′(') ⊃ PB"(').We define

PB(') := ∪">0PB"(').

Remark 3. The structure of inducing schemes here means that we could just fix asingle inducing scheme which has all the required thermodynamic properties in thissection. However, in Section 4 we need to consider all the inducing schemes herein order to investigate the dimension spectrum.

In [I1], the following conditions are given:

q∗ := inf{q : there exists t ∈ ℝ such that P (−t log ∣DF ∣+ qΦ) ⩽ 0},(note that for the inducing schemes it is possible to find measures with arbitrarilylarge Lyapunov exponent so pressure can be infinite), and

TΦ(q) :=

{inf{t ∈ ℝ : P (−t log ∣DF ∣+ qΦ) ⩽ 0} if q ⩾ q∗,

∞ if q < q∗.

The following is the main result of [I1, Theorem 4.1]. We can apply it to ourschemes (X,F ) since they can be seen as the full shift on countably many symbols(Σ, �). In applying this theorem, we choose the metric dΣ on Σ to be compatiblewith the Euclidean metric on X.

Theorem 2. Suppose that (Σ, �) is the full shift on countably many symbols andΦ : Σ → ℝ is locally Holder continuous. The dimension spectrum � 7→ DSΦ(�) isminus the Legendre transform of q 7→ TΦ(q).

If we know that an inducing scheme has sufficiently high, but not infinite, pressurefor the potential Ψq then, as we will show, the measures we are interested in are allcompatible to this inducing scheme. This leads to TΦ defined above being equal toT' as defined in (5), as in the following proposition.

Proposition 4. Suppose that f ∈ ℱ is a map satisfying (3) and ' : I → ℝ isa Holder potential satisfying (4). Let " > 0. For all q ∈ PBA" ('), if (X,F ) ∈SCoverA(") with induced potential Φ, then TΦ(q) = T'(q). Similarly for type Binducing schemes.

Moreover,

(a) there exists " > 0 and q0 < 1 < q1 so that (q0, q1) ⊂ PBA" (');(b) if f satisfies (2), then for all " > 0 there exist 0 < q2 < q3 so that (q2, q3) ⊂

PBB" (') (taking " small, q2 can be taken arbitrarily close to 0);(c) if f satisfies (1), for all small " > 0 there exist q2 < 0 < q3 so that

(q2, q3) ⊂ PBB" (').


In this proof, and later in the paper, given a set A and a function g : A→ ℝ we let

∣g∣∞ = supx∈A∣g(x)∣.

Proof. By Lemma 4, for q ∈ PB"('), and any (X,F ) ∈ SCover("), P (Ψq) = 0.The Abramov formula in Lemma 1 implies that

0 = ℎ� q (f) +

∫−T'(q) log ∣Df ∣+ q' d� q

=

(1∫

� d�Ψq

)(ℎ�Ψq

(F ) +

∫−T'(q) log ∣DF ∣+ qΦ d�Ψq

)and hence TΦ(q) ⩽ T'(q) on PB"('). Since log ∣DF ∣ is uniformly positive, we alsoknow that t 7→ P (−t log ∣DF ∣ + qΦ) is strictly decreasing in t and hence TΦ(q) =T'(q) on PB"(').

By Lemma 4, for " > 0, in order to check if q ∈ PB"(') and thus prove (a), (b) and(c), we only need to check if q ∈ PB"(Φ) for one scheme (X,F ) ∈ SCover("). Wewill show that the estimate for Z∗0 (Ψq) is a sum of exponentially decaying terms,which is enough to show that there exists � > 0 so that Z∗0 (Ψq + ��) <∞.

As in the proof of Lemma 3, (4) and P (') = 0 imply that ' < 0. Recall that bydefinition, P (−T'(q) log ∣Df ∣ + q') = 0. Given (X,F ) ∈ SCover("), by the localHolder continuity of every Ψq, there exists C > 0 such that for Z∗0 as in (7),

Z∗0 (Ψq) :=∑i

�ie−T'(q) log ∣DFi∣+qΦi ⩽ C

∑n

n∑�i=n

∣Xi∣T'(q)eqΦi .

We will first assume only that f satisfies (3) and that q is close to 1. In this casewe work with inducing schemes of type A. By Proposition 2(a), there exists �Φ > 0so that

∑�i=n

eΦi = O(e−n�Φ).

Case 1: q near 1 and q > 1. In this case T'(q) < 0. Since ∣Xi∣ ⩾ (∣Df ∣∞)−�i ,

Z∗0 (Ψq) ⩽ C∑n

n(∣Df ∣∞)n∣T'(q)∣∑�i=n

eqΦi ⩽ C ′∑n

n(∣Df ∣∞)n∣T'(q)∣e−nq�Φ .

Since for q near to 1, T'(q) is close to 0, the terms on the right decay exponentially,proving the existence of q1 > 1 in part (a).

Case 2: q near 1 and q < 1. In this case T'(q) > 0. By the Holder inequality thereexists C ′ > 0 such that

Z∗0 (Ψq) ⩽ C∑n

n

(∑�i=n

eΦi

)q (∑�i=n

∣Xi∣T'(q)

1−q

)1−q

⩽ C ′∑n

ne−qn�Φ

(∑�i=n

∣Xi∣T'(q)

1−q

)1−q

⩽ C ′∑n

ne−qn�Φ#{�i = n}1−q.

As explained in [BT4], for any � > 0 there exists C� > 0 such that #{�i = n} ⩽C�e

n(ℎtop(f)+�). For q close to 1, 1 − q is close to 0 so the terms e−nq�Φ dominatethe estimate for Z∗0 (Ψq), which completes the proof of part (a) of the proposition.

18 MIKE TODD

Next we assume that f satisfies (2) and q > 0 is close to 0. In this case we workwith inducing schemes of type B.

Case 3: q near 0 and q > 0. In this case T'(q) < 1 and T'(q) is close to 1. By[BT4, Proposition 3], if t is close to 1 then

∑�i=n

∣Xi∣t is uniformly bounded. Thus,as in Case 2,

Z∗0 (Ψq) ⩽ C∑n

n

(∑�i=n

eΦi

)q (∑�i=n

∣Xi∣T'(q)

1−q

)1−q

= O

(n∑�i=n

eΦi

)q.

As in Case 2, there exists �Φ > 0 so that∑�i=n

eΦi = O(e−n�Φ), which implies

Z∗0 (Ψq) can be estimated by exponentially decaying terms, proving (b).

Case 4: q near 0 and q < 0. This can only be considered when f satisfies (1). Inthis case T'(q) > 1. Note that by Proposition 3(a), there exists �DF > 0 so that∑�i=n

∣Xi∣ = O(e−n�DF ). Thus,

Z∗0 (Ψq) ⩽ C∑n

neqn inf '

(∑�i=n

∣Xi∣

)T'(q)

= O

(∑n

nen[q inf '−T'(q)�DF ]

).

For q close to 0 we have q inf ' − T'(q)�DF < 0 and so Z∗0 (Ψq) can be estimatedby exponentially decaying terms, proving (c). □

Corollary 1. The map T' is convex analytic on PBA" (') ∪ PBB" (').

Proof. As shown in [I1, Proposition 4.3], for Φ the induced potential for ' withrespect to an inducing scheme (X,F ), TΦ, when it is finite, is analytic and convex.Since T'(q) = TΦ(q) for PBA" (') ∪ PBB" ('), these properties pass to T'. □

4. Inducing schemes see most points with positive Lyapunov Exponent

The purpose of this section is to show that if we are only interested in those setsfor which the Lyapunov exponent is bounded away from 0, then there are inducingschemes which contain all the multifractal data for these sets. This is the contentof the following proposition.

Proposition 5. For all �, s > 0 there exist " = "(�, s) > 0, a set LG′� ⊂ LG�, and

an inducing scheme (X,F ) ∈ SCoverA(") so that dimH(LG� ∖ LG′�) ⩽ s and for

all x ∈ LG′� there exists k ⩾ 0 so that fk(x) ∈ (X,F )∞. There is also an inducingscheme in SCoverB(") with the same property.

By the structure of the inducing schemes outlined above, we can replace " withany "′ ∈ (0, "). This means that if there is a set A ⊂ I and � > 0 so thatdimH(A ∩ LG�) > 0 then there is an inducing scheme (X,F ) so that dimH(A ∩LG�∩(X,F )∞) = dimH(A∩LG�). Hence the multifractal information for A∩LG�can be found using (X,F ). We remark that by Lemma 4, for � > 0 and q ∈ PB('),if dimH(K� ∩ LG�) > 0 then we can fix an inducing scheme (X,F ) such that

dimH(K� ∩ LG� ∩ (X,F )∞) = dimH(K� ∩ LG�).

For the proof of Proposition 5 we will need two lemmas.


Partly for completeness and partly in order to fix notation, we recall the definitionof Hausdorff measure and dimension. For E ⊂ I and s, � > 0, we let

Hs� (E) := inf

{∑i

diam(Ai)s

}where the infimum is taken over collections {Ai}i which cover E and with diam(Ai) <�. Then the s-Hausdorff measure of E is defined as Hs(E) := lim sup�→0H

s� (E).

The Hausdorff dimension is then dimH(E) := sup{s : Hs(E) =∞}.

Lemma 5. For all �, s > 0 there exists � > 0, R ∈ ℕ and LG′� ⊂ LG� so that

dimH(LG� ∖ LG′�) ⩽ s, and x ∈ LG′� implies

lim supk

1

k#{

1 ⩽ k ⩽ n : fk(�(x)) ∈ IR}> �.

Note on the proof: It is important here that we can prove this lemma for LG�rather than LG�. Otherwise Proposition 5 and, for example, our main corollarieswould not hold. We would like to briefly discuss why we can prove this result forLG� rather than LG�. The argument we use in the proof is similar to argumentswhich show that under some condition on pointwise Lyapunov exponents for m-almost every point, there is an invariant measure absolutely continuous with respectto m. Here m is usually a conformal measure. For example in [BT1, Theorem 4] weshowed that if m(LG�) > 0 for a conformal measure m then ‘most points’ spend apositive frequency of their orbit in a compact part of the Hofbauer extension, andhence there is an absolutely continuous invariant measure � ≪ m. In that caseit was convenient to use LG� rather than LG�. In [K3], and in a similar proofin [MS, Theorem V.3.2], m is Lebesgue measure and the ergodicity of m is usedto allow them to weaken assumptions and to consider LG� instead. In our casehere, we cannot use a property like ergodicity, but on the other hand we do notneed points to enter a compact part of the extension with positive frequency (whichis essentially what is required in all the above cases), but simply infinitely often.Hence we can use LG� instead.

For the proof of the lemma we will need the following result from [BRSS, Theorem4]. Here m denotes Lebesgue measure, and as above ℓmax(f) is the maximal criticalorder of all critical points of f .

Proposition 6. If f ∈ ℱ satisfies (3) then there exists K0 > 0 so that for anyBorel set A,

m(f−n(A)) ⩽ K0m(A)1

2ℓmax(f) .

Remark 4. For f ∈ ℱ , such a theorem holds whenever there is an acip � withdensity � = d�

dm ∈ L1+� for � > 0. Standard arguments show that transitivity implies

that there exists " > 0 such that � ⩾ ". Then

m(f−n(A)) ⩽1

"�(f−n(A)) =

1

"�(A) =

1

"

∫1A� dm

⩽1

"

(∫1A dm

) �1+�(∫

�1+� dm

) 11+�

⩽ Cm(A)�

1+� ,

for some C > 0.

20 MIKE TODD

Proof of Lemma 5. For this proof we use ideas of [K2], see also [BT1]. We also usethe notation ∣ ⋅ ∣ to denote the length of a connected interval. We suppose thatdimH(LG�) > 0, otherwise there is nothing to prove. We fix s ∈ (0,dimH(LG�)).Throughout this proof, we write ℓmax = ℓmax(f).

For ⩾ 0 and n ∈ ℕ, let LGn := {x : ∣Dfn(x)∣ ⩾ e n}.

For x ∈ I, we define

freq(R, �, n) :=

{x :

1

n#{

0 ⩽ k < n : fk(�(x)) ∈ IR}⩽ �

}and

freq(R, �) :=

{y : lim sup

k

1

k#{

1 ⩽ k ⩽ n : fk(�(y)) ∈ IR}< �

}.

For �0 ∈ (0, �), R,n ⩾ 1 and � > 0 we consider the set

E�0,R,n(�) := LGn�0∩ freq(R, �, n).

If x ∈ LG�∩ freq(R, �) then there exists arbitrarily large n ∈ ℕ so that ∣Dfn(x)∣ ⩾e�0n, and x ∈ freq(R, �, n). Hence

freq(R, �) ∩ LG� ⊂∩k

∪n⩾k

E�0,R,n(�).

This means we can estimate the Hausdorff dimension of freq(R, �) ∩ LG� throughestimates on dimH(E�0,R,n(�)).

We let PE,n denote the collection of cylinder sets of Pn which intersect E�0,R,n(�).We will compute Hs

� (E�0,R,n(�)) using the natural structure of the dynamical cylin-ders Pn. First note that by [H2, Corollary 1] (see also, for example, the proof of[BT1, Theorem 4]), for all > 0 there exist R ⩾ 1 and � > 0 so that #PE,n ⩽ e nfor all large n. In [BT1] this type of estimate was sufficient to show that conformalmeasure ‘lifted’ to the Hofbauer extension. The Hausdorff measure is more diffi-cult to handle, since distortion causes more problems. Here we use an argument of[BT3] to deal with the distortion. We will make some conditions on , dependingon s and � below.

Let n(�) ∈ ℕ be so that n ⩾ n(�) implies ∣Cn∣ < � for all Cn ∈ Pn.

We choose any ∈ (0, �s/16ℓ2max) and � := 4 ℓ2max/s. For x ∈ I, let

Vn[x] :={y ∈ Cn[x] : d(fn(y), ∂fn(Cn[x])) < e−�n∣fn(Cn[x])∣

}.

For a point x ∈ E�0,R,n, we say that x is in Case 1 if x ∈ Vn[x], and in Case 2otherwise. We consider the measure of points in these different sets separately.

Case 1: For x ∈ I, we denote the part of fn(Cn[x]) which lies within e−�n∣fn(Cn[x])∣of the boundary of fn(Cn[x]) by Bdn[x]. We will estimate the Lebesgue measureof the pullback f−n(Bdn[x]). Note that this set consists of more than just the pairof connected components Cn[x] ∩ Vn[x].


Clearly, m(Bdn[x]) ⩽ 2e−�nm(fn(Cn[x])). Hence from Proposition 6, we have the(rather rough) estimate

m(Vn[x]) ⩽ m(f−n(Bdn[x])) ⩽ K0

[2e−�nm(fn(Cn[x]))

] 12ℓ2max

⩽ 2K0e− �n

2ℓ2max = 2K0e− 2 n

s .

Case 2: Let Cn[x] := Cn[x] ∖ Vn[x]. As in the proof of [BT3, Lemma 15], theintermediate value theorem and the Koebe lemma allow us to estimate

∣Cn[x]∣∣fn(Cn[x])∣

⩽

(1 + e−n�

e−n�

)21

∣Dfn(x)∣.

Hence for all large n,∣Cn[x]∣ ⩽ 2e2�ne−�n.

By our choice of ,

∣Cn[x]∣ ⩽ 2e−n�2 .

If we assume that n ⩾ n(�), the sets Vn[x] ⊂ Cn[x] ∈ PE,n in Case 1 and Cn[x] ⊂Cn[x] ∈ PE,n in Case 2 form a �-cover of E�0,R,n(�). This implies that for n large,

Hs� (E�0,R,n(�)) ⩽ 4e n

(e−n

�s2 +K0e

−2 n).

By our choice of , this is uniformly bounded in n. Since we can make the aboveestimate for all small �, we get that

dimH

(LG� ∩ freq(R, �)

)⩽ s.

So the set LG′� := LG� ∖ freq(R, �) has the required property. □

Let {"n}n be a positive sequence decreasing to 0 and let Bn := B"n(∂I), where weuse the distance function dI as described in Section 2.1.

Lemma 6. For any R ∈ ℕ and � > 0, there exists N(R, �) ∈ ℕ so that for x ∈ I,if

lim supk

1

k#{

1 ⩽ j ⩽ k : f j(�(x)) ∈ IR}> �,

then f j(�(x)) ∈ IR ∖BN infinitely often.

Proof. In a Hofbauer extension, if a point x ∈ I is very close to ∂I then its f -orbit shadows a point in ∂I for a very long time, and so it must spend a long timehigh up in the Hofbauer extension. Therefore we can choose p,N ∈ ℕ so thatx ∈ BN (∂I) ∩ IR implies that

fp(x) ∈ I ∖ IR and1

p#{

1 ⩽ j ⩽ p : f j(x) ∈ IR}< �. (8)

Suppose, for a contradiction, that k is the last time that, for x ∈ I, fk(�(x)) ∈IR ∖ BN . Then if f j(�(x)) ∈ IR for j > k then f j(�(x)) must be contained in BN .Hence by (8), we have

lim supk

1

k#{

1 ⩽ j ⩽ k : f j(�(x)) ∈ IR}< �,

a contradiction. □

22 MIKE TODD

Proof of Proposition 5. We choose R,N ∈ ℕ, LG′� as in Lemmas 5 and 6 so that

for any x ∈ LG′�, �(x) enters IR ∖BN infinitely often.

In the following we can deal with either inducing schemes of type A or type B. Wecan choose " > 0 so small that IR ∖ BN ⊂ ∪X∈Cover(")X. We denote the set of

points x ∈ I so that the orbit of x enters X ⊂ I infinitely often by X∞. Therefore,

for x ∈ LG′�, there exists Xk ∈ Cover(") so that �(x) ∈ X∞k . Thus

LG′� =

n∪k=1

{x ∈ LG′� : �(x) ∈ X∞k

}.

Therefore, we can choose a particular Xk so that

dimH(LG′�) = dimH

{x ∈ LG′� : �(x) ∈ X∞k

},

as required. □

5. Proof of main results

For a potential ' : I → ℝ, if the Birkhoff average limn→∞Sn'(x)n exists, then we

denote this limit by S∞'(x). If Φ is some induced potential, we let S∞Φ(x) be theequivalent average for the inducing scheme.

Remark 5. Let f ∈ ℱ satisfy (3) and ' be a Holder potential satisfying (4) andP (') = 0. Proposition 2 implies that there exists an equilibrium state �', but alsofor an inducing scheme (X,F ), it must have P (Φ) = 0 for the induced potential Φ.In fact this is only stated for type A inducing schemes in Proposition 2, but will weprove this for type B schemes as well in Lemma 11.

For x ∈ X, we define

d�Φ(x) := lim

n→∞

log�Φ(CFn [x])

− log ∣DFn(x)∣

if the limit exists. Here CFn [x] is the n-cylinder at x with respect to the inducing

scheme (X,F ). Since P (Φ) = 0, the Gibbs property of �Φ implies

d�Φ(x) = limn→∞

SnΦ(x)

− log ∣DFn(x)∣whenever one of the limits on the right exists. Also note that if both S∞Φ(x) and�F (x) exist then d�Φ

(x) also exists. Suppose that S∞Φ(x) exists. It was shown byPollicott and Weiss [PoWe, Proposition 3] that if we also know

∙ d�Φ(x) exists, then d�Φ

(x) and �F (x) exist and d�Φ(x) = d�Φ

(x) = S∞Φ(x)−�F (x) ;

∙ d�Φ(x) exists, then d�Φ

(x) and �F (x) exist and d�Φ(x) = d�Φ

(x) = S∞Φ(x)−�F (x) .

Note that for x ∈ (X,F )∞ we can write

SnΦ(x)

− log ∣DFn(x)∣=

('nk (x)

nk

)(− log ∣Dfnk (x)∣

nk

)


where nk = �k(x). Hence we can replace any assumption on the existence of S∞Φ(x)and �F (x) above by the existence of S∞'(x) and �f (x).

Let

�(q) := −∫' d� q∫

log ∣Df ∣ d� q= −

∫Φ d�Ψq∫

log ∣DF ∣ d�Ψq

.

For the proof Theorem A we will need two propositions relating the pointwisedimension for the induced measure and the original measure. The reason we needto do this here is that the induced measure �Φ is not, as it would be if the inducingscheme were a first return map, simply a rescaling of �'.

Proposition 7. Given f ∈ ℱ and a Holder potential ' : I → ℝ satisfying (4) andP (') = 0, then there exists an equilibrium state �' and a '-conformal measure m'

and C' > 0 so that1

C'⩽

d�'dm'

⩽ C'.

Notice that this implies that dm' = d�' and, by the conformality of m', d�'(x) =d�'(fn(x)) for all n ∈ ℕ.

This proposition follows from [K1]. However, as we mentioned in the introduc-tion, we can also prove the existence of conformal measures under slightly dif-ferent hypotheses on the map and the potential. The class of potentials we candeal with include discontinuous potentials satisfying (4), as well as potentials x 7→−t log ∣Df(x)∣ for t close to 1. Since this is of independent interest, we will providea proof of this in the appendix. A generalised version of the following result is alsoproved in the appendix.

Proposition 8. Suppose that f ∈ ℱ satisfies (3) and ' : I → ℝ is a Holderpotential satisfying (4) and P (') = 0. For any inducing scheme (X,F ) either oftype A or type B, with induced potential Φ : X → ℝ, for the equilibrium states �'for (I, f, ') and �Φ for (X,F,Φ), there exists C ′Φ > 0 so that

1

C ′Φ⩽d�Φ

d�'⩽ C ′Φ.

Our last step before proving Theorem A is to show that the function T' as in (5)is strictly convex, which will mean that DS' is strictly convex also, and the sets Uwill contain non-trivial intervals.

Lemma 7. Suppose that f ∈ ℱ satisfies (3) and ' is a Holder potential satisfying(4). Then either there exists � > 0 such that T' is strictly convex in

PB(') ∩ ((−�, �) ∪ (1− �, 1 + �)) ,

or �' = �− log ∣Df ∣.

Remark 6. For the particular case when f ∈ ℱ and ' is a constant potential, inwhich case P (') = 0 implies ' ≡ −ℎtop(f), Lemma 7 says that T' is not convex ifand only if �− log ∣Df ∣ = �−ℎtop(f). By [D1, Proposition 3.1], this can only happenif f has finite postcritical set. We have excluded such maps from ℱ .

24 MIKE TODD

Proof of Lemma 7. Suppose that T' is not strictly convex on some interval U in-tersecting a neighbourhood of PB(') ∩ [0, 1]. Since T' is necessarily convex, in Uit must be affine. We will observe that for all q ∈ U , the equilibrium state for qis the same. We will then show that [0, 1] ⊂ U . Since (3) holds, and hence there isan acip �− log ∣Df ∣, this means that �' ≡ �− log ∣Df ∣.

Our assumptions on U imply that there exists q0 ∈ U so that for a relevant inducingscheme (X,F ), there exists � > 0 so that �Ψq0

{� ⩾ n} = O(e−�n). Moreover,

DT'(q) is some constant ∈ ℝ for all q ∈ U . As in for example [PW1, Section

II] or [P, Chapter 7 p.211] the differentiability of T' implies that∫' d� q�(� q ) = −

for all q ∈ U . Since by definition P ( q) = 0, for q0 ∈ U , any measure � with∫' d��(�) =

T'(q0)q0

must be an equilibrium state for q0 . Since there is a unique

measure for q0 we must have =T'(q0)q0

and � q = � q0 for all q ∈ U .

By Proposition 4 there exists � > 0 such that (1 − �, 1 + �) ⊂ PB(') and (0, �) ⊂PB('). If, moreover, PB(') contains a neighbourhood of 0 then we can adjust� > 0 so that (−�, �) ⊂ PB(').

Case 1: Suppose that U ∩PB(')∩ (1− �, 1 + �) ∕= ∅. Since by Proposition 4, T' isanalytic in this interval, T' must be affine in the whole of (1− �, 1 + �). Therefore1 ∈ U . We will prove that 0 ∈ U . By Proposition 4 we can choose a type A inducingscheme (X,F ) so that � q is compatible with (X,F ) for all q ∈ (1−�, 1+�). Recall

from Proposition 2 that there exists �Φ > 0 so that �Ψ1{� ⩾ n} = O(e−�Φn).

We suppose that 0 ⩽ q < 1, and hence T'(q) ⩾ 0. We choose q0 > 1− � very closeto 1− �. Then by convexity T'(q) ⩾ T'(q0) + (q − q0). Hence, for Z∗0 as in (7),

Z∗0 (Ψq) =∑n

n∑�i=n

∣Xi∣T'(q)eqΦi ⩽∑n

n∑�i=n

∣Xi∣T'(q0)+ (q−q0)eqΦi

⩽∑n

n sup�i=n

(∣Xi∣ (q−q0)e(q−q0)Φi

) ∑�i=n

∣Xi∣T'(q0)eq0Φi

⩽∑n

nen(q−q0) inf '∑�i=n

∣Xi∣T'(q0)eq0Φi .

By the Gibbs property of �Ψq0, we can estimate

∑�i=n

∣Xi∣T'(q0)eq0Φi by �Ψq0{� =

n} = �Ψ1{� = n} ⩽ e−�Φn. So if (q − q0) inf ' < �Φ then similarly to the proof of

Proposition 4, q ∈ PB('). Since T' is analytic in PB('), this means that T' isstill affine at q and therefore that U was not the largest domain of affinity ‘to theleft’. We can continue doing this until we hit the left-hand boundary of PB('). Inparticular, this means that 0 ∈ U .

Case 2: Suppose that PB(')∩(−�, �)∩U ∕= ∅. As in Case 1, this implies [0, �] ⊂ U .We will prove that 1 ∈ U .

By Proposition 4 we can choose a type B inducing scheme (X,F ) so that � q iscompatible with (X,F ) for all q ∈ (�′, �) where �′ := �/2. Recall from Proposition 2that there exists �DF > 0 so that �Ψ�′{� ⩾ n} = O(e−n�DF ).


We let � < q ⩽ 1 and q0 < � be very close to �. Again by convexity T'(q) ⩾T'(q0) + (q − q0). Similarly to Case 1,

Z∗0 (Ψq) =∑n

n∑�i=n

∣Xi∣T'(q)eqΦi ⩽∑n

n∑�i=n

∣Xi∣T'(q0)+ (q−q0)eqΦi

⩽∑n

n sup�i=n


) ∑�i=n

∣Xi∣T'(q0)eq0Φi .

Since ∣Xi∣ ⩾ e−�i∣Df ∣∞ ,

sup�i=n


)⩽ en(q−q0)(− ∣Df ∣sup+sup').

So if (q − q0)(− ∣Df ∣∞ + sup') < �DF then similarly to Case 1 we can concludethat all points in PB(') to the right of q0 are in U . In particular 1 ∈ U .

In both cases 1 and 2, we concluded that [0, 1] ⊂ U . Therefore �' ≡ �− log ∣Df ∣, asrequired. □

Proof of Theorem A. Let L' be the minus Legendre transform of T' as in (5) wher-ever these functions are well defined.

The upper bound: 퓓퓢' ⩽ L'. To get this bound, we first pick a suitable

inducing scheme. Given q ∈ PB('), since K'(�(q)) = ∪n⩾1LG 1n∩K'(�(q)), for all

� > 0 there exists � > 0 so that dimH(LG′�∩K'(�(q))) ⩾ dimH(K'(�(q)))−�. For

some s < dimH(K'(�(q))), we take an inducing scheme (X,F ) as in Proposition 5(this can be for schemes of type A or B, whichever we need).

We next show that DS' ⩽ DSΦ and then use Theorem 2 and Proposition 4 to

conclude the proof of the bound. Let x ∈ K'(�)∩LG′�. By transitivity there exists

j so that x ∈ f j(X). Let y ∈ X be such that f j(y) = x. Since x ∈ LG′�, we

must also have y ∈ (X,F )∞ by Proposition 5. By Propositions 7 and 8, d�'(x) =d�'(y) = d�Φ

(y), so y ∈ KΦ(�). Therefore,

K'(�) ∩ LG′� ⊂ ∪∞k=0fk(KΦ(�)).

Hence

DS' − � ⩽ dimH(K'(�) ∩ LG′�) ⩽ dimH

(∪∞k=0f

k(KΦ(�))).

Since f is clearly Lipschitz, dimH

(∪∞k=0f

k(KΦ(�)))

= dimH(KΦ(�)), so DS'(�)−� ⩽ DSΦ(�). Theorem 2 says that DSΦ(�(q)) is LΦ(�), minus the Legendre

transform of TΦ. Therefore, DS'(�)−� ⩽ LΦ(�) = L'(�), where the final equality

follows from Proposition 4. Since � > 0 was arbitrary, we have DS'(�) ⩽ L'(�).

The lower bound: 퓓퓢' ⩾ L'. We will use the Hausdorff dimension of theequilibrium states for q to give us the required upper bound here. For � ∈ ℳ+,by Theorem 1 there exists an inducing scheme (X,F ) which � is compatible to.This can chosen to be of type A or type B. By Proposition 8, d�'(x) = d�Φ

(x) forany x ∈ (X,F )∞, where Φ is the induced potential for (X,F ). Now suppose that∫' d��f (�) = −�. Then for �-a.e. x, S∞'(x) and �(x) exist, and by the above and

26 MIKE TODD

Remark 5, since we may choose X so that for x ∈ (X,F )∞, we have

d�'(x) = d�Φ(x) =

S∞'(x)

−�f (x)= �.

Hence �-a.e. x is in K'(�). Therefore,

DS'(�) ⩾ sup

{ℎ�

�f (�): � ∈ℳ+ and

∫' d�

�f (�)= −�

}.

By Lemma 4, we know that there is an equilibrium state � q for q. Then by

definition, ℎ� q +∫−T (q) log ∣Df ∣+ q' d� q = 0. Therefore, for � = �(q),

ℎ� q�f (� q )

= T (q) + q� = L'(�).

And hence DS'(�) ⩾ L'(�). Putting our two bounds together, we conclude that

DS'(�) = L'(�).

We next show (a), (b) and (c). First note that since we have assumed that �' ∕=�− log ∣Df ∣, Lemma 7 means that T' is strictly convex in PB('). This implies thatU will contain non-trivial intervals. For example, if (3) holds then P (') = 0 and[HR] imply that

�(1) = −∫' d�'�f (�')

=ℎ�'

�f (�')= dimH(�').

By Proposition 4 and Lemma 7, for any � close to dimH(�') there exists q such

that DT'(q) = �. Hence by the above, DS'(�) = L'(�).

Similarly, let us assume that (2) holds. We have

�(0) = −∫' d�− log ∣Df ∣

�f (�− log ∣Df ∣)= �ac.

So the arguments above, Proposition 4 and Lemma 7 imply that for any � < �acthere exists q such that DT'(q) = �, and also DS'(�) = L'(�). The same holdsfor all � in a neighbourhood of �ac when (1) holds. □

Proof of Proposition 1. It was pointed out in [I1, Remark 4.9] that by [BaS], for aninducing scheme (X,F ) with potential Φ : X → ℝ, the Hausdorff dimension of theset of points with d�Φ

(x) not defined has the same dimension as the set of pointsfor which the inducing scheme is defined for all time. So we can choose (X,F ) tobe any inducing scheme which is compatible to the acip to show that the Hausdorffdimension of this set of points is 1. In fact any type A or type B inducing schemeis compatible to the acip. By Proposition 8, if d�Φ

(x) not defined then neither isd�'(x), so the proposition is proved. □

5.1. Going to large scale: the proof of Corollary C. Suppose that f ∈ ℱextends to a polynomial on ℂ with no parabolic points and all critical points inI. In the context of rational maps, Graczyk and Smirnov [GS] prove numerousresults for such maps satisfying (2). For � > 0, we say that x goes to �-large scaleat time n if there exists a neighbourhood W of x such that f : W → B�(f

n(x))is a diffeomorphism. [GS, Theorem 3] says that there exists � > 0 such that theset of points which do not go to �-large scale for an infinite sequence of times has


Hausdorff dimension less than ℓmax(f)�P−1 < 1 where �P is defined in (2). Here we will

sketch how this implies Corollary C.

By [K2], if f ∈ ℱ and x ∈ I goes to �-large scale with frequency , then there

exists N = N(�) so that iterates of �(x) by f enter IN with frequency at least .In [K2, BT1], this idea was used to prove that for � ∈ ℳerg, if �-a.e. x goes to�-large scale with some frequency greater than > 0, then there exists � an ergodic

f -invariant probability measure on I, with �(IN ) > (so also �-a.e. x enters INwith positive frequency), and � = �∘�−1. By the arguments above this means that

we can build an inducing (X,F ) scheme from a set X ∈ IN which is compatible to�.

However, to prove Corollary C, we only need that sufficiently many points x havek ⩾ 0 such that fk(x) ∈ (X,F )∞, which does not necessarily mean that thesepoints must go to large scale with positive frequency. (Note that we already know

that all the measures � we are interested in can be lifted to I.) We only need to usethe fact, as above, that if A is the set of points which go to �-large scale infinitelyoften, then there exists R ∈ ℕ so that for all x ∈ A, �(x) enters IR infinitely often.Hence the machinery developed above ‘sees’ all of A, up to a set of Hausdorff

dimension < ℓmax(f)�P−1 . Since this value is < 1, for our class of rational maps, we

have DS'(�) = DS'(�) for � close to �ac. Similarly, if ℓmax(f)�P−1 < dimH(�') then

the same applies for � close to dimH(�').

Note that for rational maps as above, but satisfying (1), the same argument givesanother proof of Corollary B.

It seems likely that the analyticity condition can be weakened to include all mapsin ℱ satisfying (2).

Note: Corollary 6.3 of [RS], which was written after this work was completed,states that for all f ∈ ℱ satisfying (3), the complement of the set of points letgoing to large scale infinitely often has Hausdorff dimension 0. Therefore we can

replace DS'(�) with DS'(�) throughout.

5.2. Points with zero Lyapunov exponent can be seen. In this section wediscuss further which points can and cannot be seen by the inducing schemes weuse here.

Suppose that (X,F, �) is an inducing scheme of type A. Then there is a correspond-

ing set X ⊂ I such that �(y) is rX(y) where y ∈ X is such that �(y) = y and rX is

the first return time to X. Then there exist points x ∈ X so that �(fk(x)) ∈ Crit

and f j(x) /∈ X for all 1 ⩽ j < k. This implies that from iterate k onwards, this

orbit is always in the boundary of its domain D ∈ D. Since X is always chosen tobe compactly contained inside its domain DX ∈ D, this means that x never returns

to X. Hence for x = �(x), �(x) = ∞. On the other hand, there are precritical

points x with x = �∣−1

X(x) which returns to X before it hits a ‘critical line’ �−1(c)

for c ∈ Crit. For such a point, �(x) < ∞, but for all large iterates k, we musthave �(fk(x)) =∞. Hence precritical points in X cannot have finite inducing time

28 MIKE TODD

for all iterates. This can be shown similarly for type B inducing schemes. We canextend this to show that no precritical point is counted in our proof of Theorem A.

Moreover, in this paper we are able to find DS'(�) through measures on K�. In factwe can only properly deal with measures which are compatible to some inducingscheme. As in Theorem 1, the only measures we can consider are in ℳ+. Thismeans that the set of points x with �(x) = 0 is not seen by these measures. Aspointed out above Corollary B, [BS] shows that in the Collet-Eckmann case, theset of points with �(x) = 0 is countable and thus has zero Hausdorff dimension.(Note that even in this well-behaved case it is not yet clear that the set of pointswith �(x) = 0 has zero Hausdorff dimension.) The general question of what isthe Hausdorff dimension of I ∖ LG0 for topologically transitive maps is, to ourknowledge, open.

On the other hand, it is not always the case that given an inducing scheme (X,F, �),all points x ∈ X for which �(F k(x)) < ∞ for all k ⩾ 0 have positive Lyapunovexponent. For example, we say that f has uniform hyperbolic structure if inf{�f (p) :p is periodic} > 0. Nowicki and Sands [NS] showed that for unimodal maps in ℱthis condition is equivalent to (1). If we take f ∈ ℱ without uniform hyperbolicstructure, then it can be shown that for any inducing scheme (X,F, �) as above,there is a sequence {nk}k such that

sup{log ∣DF (x)∣ : x ∈ Xnk}�nk

→ 0.

There exists x ∈ X so that F k(x) ∈ Xnk for all k. Thus �(x) ⩽ 0, but �(F k(x)) <∞ for all k ⩾ 0. In the light of the proof of Corollary C, we note that x goes to∣X∣-large scale infinitely often, but with zero frequency.

6. Lyapunov spectrum

For � ⩾ 0 we let

L� = L�(f) := {x : �f (x) = �} and L′ = L′(f) := {x : �f (x) does not exist} .

The function � 7→ dimH(L�) is called the Lyapunov spectrum. Notice that by [BS],if f ∈ ℱ satisfies (3) then if the Lyapunov exponent at a given point exists thenit must be greater than or equal to 0. In this section we explain how the resultsabove for pointwise dimension are naturally related to the Lyapunov spectrum. Aswe show below, the equilibrium states �−t log ∣Df ∣ found in [PSe, BT4] for certainvalues of t, depending on the properties of f , are the measures of maximal dimensionsitting on the sets L� for some � = �(t).

Recall that �− log ∣Df ∣ is the acip for f . We denote the measure of maximal entropyby �−ℎtop(f) since it is the equilibrium state for a constant potential 'a(x) = a forall x ∈ I; and in order to ensure P ('a) = 0, we can set a = −ℎtop(f). We letDS−ℎtop(f)(�) = dimH(K−ℎtop(f)(�)) where K−ℎtop(f) is defined for the measure�−ℎtop(f) as above.

Proposition 9. If f ∈ ℱ then there exists an open set U ⊂ ℝ containingℎtop(f)

�f (�−ℎtop(f))

so that for each � ∈ U the values of dimH

(Lℎtop(f)

�

)= DS−ℎtop(f)(�) are given as


minus the Legendre transform of T−ℎtop(f) at �. If f satisfies (2), thenℎtop(f)

�f (�− log ∣Df∣)

is in the closure of U , and if f satisfies (1) thenℎtop(f)

�f (�− log ∣Df∣)is contained in U .

As observed by Bohr and Rand, this proposition would have to be adapted slightlywhen we are dealing with quadratic Chebyshev polynomial (which is not in our classℱ). In this case, �−ℎtop(f) = �− log ∣Df ∣, so the Lyapunov spectrum can not analyticin a neighbourhood of 1. Note that this agrees with Lemma 7 and Remark 6.

Note that the first part of the proposition makes no assumption on the growth of∣Dfn(f(c))∣ for c ∈ Crit. We can rephrase the statement of this proposition as: forthe range of Lyapunov exponents � close to that of �(�−ℎtop) and �(�− log ∣Df ∣),

1

�inft∈ℝ

(P (−t log ∣Df ∣) + t�) .

The proof of this proposition follows almost exactly as in the proof of Proposition 4,so we only give a sketch.

Proof. Given an inducing scheme (X,F ), by Remark 5, for each x ∈ (X,F )∞ if�f (x) exists then

�f (x) =ℎtop(f)

d�−�ℎtop(f)(x)

.

Here the potential is ' ≡ −ℎtop(f), and the induced potential is −�ℎtop(f). Thismeans that we can get the Lyapunov spectrum directly from d�−�ℎtop(f)

. As in

Proposition 8, d�−�ℎtop(f)(x) = d�−ℎtop(f)

(x) for all x ∈ X.

Therefore it only remains to discuss the interval U , i.e. the equivalent of Propo-sition 4. First we note that Lemma 7 holds in this case without any assumptionon the proof of ∣Dfn(f(c))∣ for c ∈ Crit. We fix an inducing scheme (X,F ). ThatZ∗0 (Ψq + �q�) < ∞ for some small �q > 0, for q in some open interval U can beproved exactly in the same way as in the proof of Proposition 4. □

Note that similarly to Proposition 1, the set of points for which the Lyapunovexponent is not defined has Hausdorff dimension 1.

Remark 7. For t ∈ ℝ, let Pt := P (−t log ∣Df ∣). It follows that PT−ℎtop(f)(q) =

qℎtop(f). Since � q is an equilibrium state for −T−ℎtop(f)(q) log ∣Df ∣ − qℎtop(f),then it is also an equilibrium state for −T−ℎtop(f)(q) log ∣Df ∣. Therefore, the mea-sures for q are precisely those found for the potential −t log ∣Df ∣ in Proposition 3and in [BT2, Theorem 6].

Remark 8. If (1) does not hold, then Proposition 9 does not deal with L� for � <�(�− log ∣Df ∣). This is because, at least in the unimodal case, we have no equilibriumstate with positive Lyapunov exponent for the potential x 7→ −t log ∣Df(x)∣ for t > 1(i.e., there is a phase transition at 1).

Nakaishi [Na] and Gelfert and Rams [GR] consider the Lyapunov spectrum forManneville-Pomeau maps with an absolutely continuous invariant measure, whichhas polynomial decay of correlations. Despite there being a phase transition fort 7→ Pt at t = 1, they are still able to compute the Lyapunov spectrum in the regime� ∈ [0, �(�− log ∣Df ∣)). Indeed they show that dimH(L�) = 1 for all these values of

30 MIKE TODD

�. In forthcoming work we will show that we have the same phenomenon in oursetting when (2), but not (1), holds.

Remark 9. If (1) holds then it can be computed that in the above proof, Z∗0 (Ψq +��) < ∞ whenever (1 − T−ℎtop(f)(q) − q)ℎtop(f) − �T−ℎtop(f)(q), where � is therate of decay of �− log ∣DF ∣{� > n} and � is some constant > 0. If f is a Collet-Eckmann map very close to the Chebyshev polynomial, then t 7→ P (−t log ∣Df ∣)is close to an affine map, and thus T−ℎtop(f) is also close to an affine map, thenZ∗0 (Ψq + �q�) <∞ for all q in a neighbourhood of [0, 1] and for some �q > 0.

The unimodal maps considered by Pesin and Senti [PSe] have the above propertyand so there exists " > 0 so that [0, 1] ⊂ PB"(−ℎtop(f)). However, this may not bethe whole spectrum.

In [PSe], they ask if it is possible to find a unimodal map f : I → I so that there isa equilibrium state for the potential x 7→ −t log ∣Df ∣ for all t ∈ (−∞,∞), and thatthe pressure function t 7→ P (−t log ∣Df ∣) is analytic in this interval. This wouldbe in order to implement a complete study of the thermodynamic formalism. AsDobbs points out in [D2], in order to show this, even in the ‘most hyperbolic’ cases,one must restrict attention to measures on a subset of the phase space: otherwisewe would at least expect a phase transition in the negative spectrum.

Appendix

In this appendix we introduce a class of potentials for which the results in the restof the paper hold. We will also prove slightly generalised versions of Propositions 7and 8.

Given a potential ', and an inducing scheme (X,F ) of type A or B, as usual welet Φ be the induced potential. If∑

n

Vn(Φ) <∞, (9)

then we say that ' satisfies the summable variations for induced potential condition,with respect to this inducing scheme. If ' satisfies this condition for every type Aor B inducing scheme (X,F ) with ∣X∣ sufficiently small, we write ' ∈ SV I. Notethat in [BT2, Lemma 3] it is proved that if ' is Holder and f ∈ ℱ satisfies (4) then' ∈ SV I. Also in [BT2] it was proved that Proposition 2 holds for all potentials inSV I satisfying (4), with no assumptions on the growth along the critical orbits.

Proposition 7 is already known in the case that ' is Holder. For interest, we willchange the class of potentials in that proposition to those in SV I satisfying (4), aswell as to potentials of the form x 7→ −t log ∣Df(x)∣. We also widen the class ofpotentials considered in Proposition 8. We will refer to Propositions 7 and 8, butwith only the assumptions that f ∈ ℱ and ' ∈ SV I, as Propositions 7’ and 8’.Note that Proposition 8’ plus [BT2, Lemma 3] implies Proposition 8. The proof ofthese propositions requires three steps:

∙ Proving the existence of a conformal measure m' for a potential ' ∈ SV Isatisfying (4) and P (') = 0. Since we do this using the measure mΦ fromProposition 2, we only really need to prove this for inducing schemes of type


A. However, it is of independent interest that this step can also be done forthe potential x 7→ −t log ∣Df(x)∣ − P (−t log ∣Df ∣), so we allow type B inducingschemes also.

∙ Proving that a rescaling of the measure m' is also conformal for our inducingschemes. This will be used directly in the proof of Proposition 7’, so must holdfor both type A and type B inducing schemes. Note that this step works for allof the types of potential mentioned above.

∙ Proving that the densityd�'dm'

is bounded. We will use type A inducing schemes

to prove this. In this step, we must assume that ' is in SV I, satisfies (4) andP (') = 0.

The necessary parts of the first and third of these steps are the content of Propo-sition 7’. As above, for the proof of this proposition, we only need to use typeA inducing schemes. But we will give the proof of the existence of the conformalmeasure for both types of schemes for interest. Our inducing scheme (X,F, �) is

derived from a first return map to a set X ⊂ I. Recall that if we have a typeA scheme, then X is an interval in a single domain X ⊂ D ∈ D in the Hofbauerextension. In the type B case, X may consist of infinitely many such intervals. We

let rX be the first return time to X and RX = frX . We let Xi denote the firstreturn domains of RX .

We let ' := '∘�, and �',X :=�'∣X�(X)

be the conditional measure on X. As explained

in [BT4], the measure �Φ is the same as �',X ∘ �−1. Proposition 2 implies that

for type A inducing schemes (X,F ), the induced potential Φ has P (Φ) = 0, andthere a conformal measure and equilibrium state mΦ and �Φ and CΦ > 0 so that

1CΦ⩽ d�Φ

dmΦ⩽ CΦ. We show in Lemma 11 that this is also true for type B inducing

schemes.

We define m'∣X := mΦ ∘ �∣X . We can propagate this measure throughout I asfollows.

For x ∈ X with rX(x) <∞, for 0 ⩽ k ⩽ rX(x)− 1, we define

dm'(fk(x)) = e−'k(x)dm'∣X(x).

Let (X, f) be a dynamical system and ' : X → ℝ be a potential. We say that ameasure m, is '-sigma-conformal for (X, f) if for any Borel set A so that f : A→f(A) is a bijection,

m(f(A)) =

∫A

e−' dm.

Or equivalently dm(f(x)) = e−'(x)dm(x). So the usual conformal measures arealso sigma-conformal, but this definition allows us to deal with infinite measures.The next two lemmas apply to potentials ' ∈ SV I satisfying (4) and P (') = 0, orof the form x 7→ −t log ∣Df(x)∣ − P (−t log ∣Df ∣) as in Proposition 3.

Lemma 8. Suppose that (X,F ) is a type A or type B system and P (Φ) = 0.

(a) m' as defined above is a '-sigma-conformal measure.

(b) Given a '-sigma-conformal measure m′' for (I , f), then up to a rescaling,m′' = m'.

32 MIKE TODD

Proof. We first prove (a). The Φ-conformality of mΦ implies that m'∣X is Φ-

conformal for the system (X, RX , Φ) for Φ(x) := Φ(�(x)).

Given x ∈ X, if 0 ⩽ j < rX(x)− 1, then the relation

dm' ∘ f(f j(x)) = e−'(x)dm'(f j(x))

is immediate from the definition. For j = rX(x) − 1, then f(f j(x)) = RX(x) and

we obtain, for x ∈ X,

dm' ∘ f(f j(x)) = e−'j(x)dm'(x) = dm'(R(x)) = e−Φ(x)dm'(x)

= e−'(frX

(x)−1(x))e

−'rX

(x)−2(x)dm'(x)

= e−'(frX

(x)−1(x))dm'(f rX(x)−1(x)) = e−'(fj(x))dm'(f j(x)),

as required.

For the proof of (b), for x ∈ X, by definition dm′'(RX(x)) = e−Φ(x)dm′'(x). Let

X ′ be some domain in X contained in some single domain D ∈ D (this is not anecessary step if the inducing scheme is of type A). This implies thatm′' := m′'∘�−1

X′

is Φ-conformal after rescaling. As in Proposition 2, there is only one Φ-conformalmeasure for (X,F ), which implies that m′' = m' up to a rescaling. □

Given X ⊂ I, we consider the system (X, RX) where RX is the first return map to

X. The measure �' is an invariant measure for (X, RX), see [K4]. Adding Kac’s

Lemma to (6), for any A ⊂ I we have

�'(A) :=∑i

∑0⩽k⩽rX ∣Xi−1

�'(f−k(A) ∩ Xi). (10)

This means we can compare m' and �' on domains f j(Xi), for 0 ⩽ k ⩽ rX ∣Xi − 1,in a relatively simple way.

We will project the measure m' to I. Although it is possible to show that for many

potentials we consider, m'(I) < ∞, we allow the possibility that our conformalmeasures are infinite. This leaves the possibility to extend this theory to a widerclass of measures open. So in the following lemma, we use another way to projectm'.

Lemma 9. Suppose that Y ⊂ IT is so that Y = ⊔nYn for Yn an interval containedin a single domain DYn ∈ DT and � : Y → I is a bijection. Then for �' :=

m' ∘ �∣−1

Y, we have �'(I) <∞. Moreover, m' :=

�'�(I) is a conformal measure for

(I, f, '), and m' is independent of Y .

Proof. We first prove that �' is independent of Y , up to rescaling. In doing so, the'-sigma-conformal property of �' become clear. The we show that �'(I) <∞.

Let us pick some Y , and let �' be as in the statement of the lemma. Let x /∈∪n∈ℕfn(Crit). Suppose that x1, x2 have �(x1) = �(x2) = x. By our condition

on x, we have xi /∈ ∂I for i = 1, 2. We denote D1, D2 ∈ D to be the domainscontaining x1, x2 respectively. The independence of the measure from Y follows if


we can show for any neighbourhood U of x such that for Ui := �−1(U) ∩Di such

that Ui ⋐ Di for i = 1, 2, we have m'(U1) = m'(U2).

As in [K2] there exists n ⩾ 0 so that fn(x1) = fn(x2). Since we are only interestedin the infinitesimal properties of our measures, we may assume that the same is

true of U1 and U2, i.e., fn(U1) = fn(U2). Therefore m'(fn(U1)) =∫U1e−'n dm'.

Since m'(fn(U1)) = m'(fn(U2)) and ' = ' ∘ �, we have m'(U1) = m'(U2), asrequired. So it only remains to show �'(I) <∞.

By the above, the '-sigma-conformality of m' passes to '-sigma-conformality of

�'. We can pick U ⊂ I such that U = �(U) for some U ⊂ D ∈ DT . Recall thatm' was obtained from a conformal measure mΦ for some inducing scheme (X,F ).

We may assume that U is such that U ⊂ fk(Xi) ∩D for some 0 ⩽ k ⩽ rX ∣Xi − 1

and some D ∈ D. This implies that m'(U) <∞, and so �'(U) <∞. Since f is inℱ , it is locally eventually onto, i.e., for any small open interval W ⊂ I there existsn ∈ ℕ so that fn(W ) ⊃ Ω. Therefore there exists n so that fn(U) ⊃ I. Then bythe '-sigma-conformality of �', we have

�'(I) = �'(fn(U)) =

∫U

e−'n d�' ⩽ �'(U)e− inf 'n <∞.

Hence m' is conformal. □

Note that combining Lemmas 8 and 9, we deduce that m' is independent of theinducing scheme that produced it. We next consider the density.

Lemma 10. For ' ∈ SV I satisfying (4) and P (') = 0,d�'dm'

is uniformly bounded

above.

Proof. Suppose thatd�'dm'

(x) > 0. We let �−1(x) = {x1, x2, . . .}, where the ordering

is by the level, i.e., lev(xj+1) ⩾ lev(xj) for all j ∈ ℕ. Then since �' = �' ∘ �−1,

d�'dm'

(x) =

∞∑j=1

d�'dm' ∘ �

(xj).

We will use this fact allied to equation (10) for return maps on the Hofbauerextension, and the bounded distortion of the measures for these first return mapsto get the bound on the density. We note that since for any R ∈ ℕ, there are atmost 2#Crit domains of D of level R (see for example [BB, Chapter 9]), there canbe at most 2#Crit elements xj of the same level.

We let (X,F ) be a type A inducing scheme with induced potential Φ : X → ℝ.

Let X be the interval in I for which the first return map RX defines the inducing

scheme (X,F ). Recall that �Φ can be represented as�'∘�∣−1

X

�'(X)and by Lemma 9, we

can express mΦ asm'

m'(X) . Moreover as in Proposition 2 there exists CΦ > 0 so

that d�Φ

dmΦ⩽ CΦ.

Since RX is a first return map, for each i there exists at most one point xj,i in

Xi so that fk(xj,i) = xj for 0 ⩽ k < rX ∣Xi . We denote this value k by rj,i. Let

kj := inf{rj,i : i ∈ ℕ}.

34 MIKE TODD

By (10), d�'(xj) =∑i d�'(xj,i). By conformality, for each i,

dm'(xj) = e−'rj,i (xj,i) dm'(xj,i) ⩾ e− sup'rj,i dm'(xj,i).

Therefore, letting xj,i = �(xj,i),

d�'dm'

(xj) ⩽∑i

d�'dm'

(xj,i)esup'rj,i ⩽

(m'(X)

�'(X)

)∑i

d�Φ

dmΦ(xj,i)e

sup'rj,i

⩽ CΦ

(m'(X)

�'(X)

)∑i

esup'rj,i ⩽ CΦ

(m'(X)

�'(X)

)∑n

#{i : rj,i = n}en sup'.

By [H1], if lev(xj) = R then there exist C > 0 and (R) > 0 so that (R) → 0

as R → ∞ and the number of n-paths terminating at Dxj ∈ D at most Cen (R).

Then #{i : rj,i = n} ⩽ Cen (lev(xj)). Also kj ⩾ lev(xj)− lev(X). Therefore,

d�'dm'

(xj) ⩽ CCΦ

(m'(X)

�'(X)

) ∑n⩾kj

en( (lev(xj))+sup')

⩽ CCΦ

(m'(X)

�'(X)

)e(lev(xj)−lev(X))( (lev(xj))+sup')

∑n⩾0

en( (lev(xj))+sup').

Since, as in Lemma 11, our conditions on ' ensure that sup' < 0, there exists� > 0, and j0 ∈ ℕ so that (lev(xj)) + sup' < −� for all j ⩾ j0. Since there areat most 2#Crit points xj of any given level R, there are only finitely many j with

lev(xj)− lev(X) ⩽ 0. Moreover, there exists C ′ > 0 so that

d�'dm'

(x) ⩽j0−1∑j=1

d�'dm' ∘ �

(xj) +

∞∑j=j0

d�'dm' ∘ �

(xj) ⩽ C′ + C ′

∞∑j=j0

e−j�

which is uniformly bounded. □

Proof of Proposition 7’. The existence of the conformal measurem' is proved in the

above lemmas. Lemma 10 implies that the densityd�'dm'

is uniformly bounded above.

The lower bound follows by a standard argument, which we give for completeness.Proposition 2 implies that we can take a type A inducing scheme (X,F,Φ) so thatd�Φ

dmΦis uniformly bounded below by some C−1

Φ ∈ (0,∞). Also, Lemma 8 implies

thatm'

m'(X) = mΦ. Since, as in the proof of Lemma 9, (I, f) is locally eventually

onto, there exists n ∈ ℕ so that fn(X) ⊂ Ω. So for a small interval A ⊂ Ω, thereexists some Ai ⊂ Xi so that fk(Ai) = A for some 0 ⩽ k ⩽ n. Then (6) implies that

�'(A)

m'(A)⩾

�'(Ai)

m'(Ai)einf 'n ⩾

(m'(X)∫� d�Φ

)(�Φ(Ai)

mΦ(Ai)

)einf 'n ⩾

(m'(X)∫� d�Φ

)(einf 'n

CΦ

).

Henced�'dm'

is uniformly bounded below. □

Lemma 11. Suppose that f ∈ ℱ satisfies (3) and ' ∈ SV I. Then there exists" > 0 so that for any inducing scheme (X,F ) ∈ SCoverB("), the induced potentialΦ has P (Φ) = 0.


Proof. We will apply Case 3 of [BT4, Proposition 1]. Firstly we need to show thatZ0(Φ) <∞. By Proposition 7’ there exists a conformal measure m', coming froman inducing scheme of type A in Proposition 2’. By the '-conformality of m' andthe local Holder continuity of Φ, as in Proposition 2(b), there exists C > 0 so thatZ∗0 (Φ) ⩽ C

∑i �im'(Xi). Then by Proposition 7’ and the facts that (X,F ) was

generated by a first return map to some X and �' = �' ∘ �−1,

Z∗0 (Φ) ⩽ CC ′'∑i

�i�'(Xi) = CC ′'∑i

rX ∣Xi �'(Xi).

By Kac’s Lemma this is bounded.

Now the fact that �' is compatible to (X,F ) follows simply, see for example Claim1 in the proof of [BT4, Proposition 2]. Then Case 3 of [BT4, Proposition 1] impliesP (Φ) = 0. □

Proof of Proposition 8’. Suppose that (X,F ) is an inducing scheme as in the state-ment, with induced potential Φ. If (X,F ) is of type A then by Lemma 8, themeasure m' works as a conformal measure for (X,F,Φ), up to renormalisation. ByProposition 2(c), m' is in fact equal to mΦ up to renormalisation. By Lemma 11,

this is also true for type B inducing schemes. Since by Proposition 7’,d�'dm'

is

bounded above and below, and as in Proposition 2, we have 1CΦ⩽ d�Φ

dmΦ⩽ CΦ, this

implies that d�Φ

d�'is also uniformly bounded above and below. □

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Centro de Matematica da Universidade do Porto, Rua do Campo Alegre 687, 4169-007

Porto, Portugal 1,

E-mail address: [email protected]

URL: http://math.bu.edu/people/mtodd/

1 Current address:

Department of Mathematics and StatisticsBoston University

111 Cummington Street

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