ANALYTICITY OF THE SRB MEASURE FOR HOLOMORPHIC FAMILIES OF QUADRATIC-LIKE COLLET-ECKMANN MAPS 8 0 VIVIANEBALADIANDDANIELSMANIA 0 2 Abstract. Weshowthatifft isaholomorphicfamilyofquadratic-likemaps n with all periodic orbits repelling so that for each real t the map ft is a real a Collet-Eckmann S-unimodal map then, writing µt for the unique absolutely J continuous invariantprobabilitymeasureofft,themap 2 2 t7→Z ψdµt isrealanalyticforanyrealanalyticfunction ψ. ] S D h. 1. Introduction and statement of the theorem t a If t f is a smooth one-parameter family of dynamics f so that f admits t t 0 m 7→ a unique SRB measure µ , it is natural to ask whether the map t µ , where t 0 t 7→ [ rangesoverasetΛofparameterssuchthatf has(atleast)oneSRBmeasureµ ,is t t differentiable at 0 (in the sense of Whitney if Λ does not contain a neighbourhood 1 v of 0, as suggested by Ruelle [13]). Katok, Knieper, Pollicott, and Weiss [6] gave a 6 positive answer to this question in the setting of C3 families of transitive Anosov 5 flows (here, Λ is a neighbourhood of 0), showing that t ψdµ is differentiable, t 3 for all smooth ψ. If f is a C3 mixing Axiom A attrac7→torRand the family t f 3 0 7→ t is C3, Ruelle [12] not only proved that t ψdµ is differentiable, but also gave 1. an explicit formula (the linear response7→forRmula)tfor the derivative. Ruelle [13] 0 suggestedthatthisformula,appropriatelyinterpreted,shouldholdinmuchgreater 8 generality. Indeed,Dolgopyat[5]obtainedthelinearresponseformulaforaclassof 0 : partiallyhyperbolicdiffeomorphisms. Inapreviouswork[3,4],wefoundthatinthe v (nonstructurallystable)settingofpiecewiseexpandingunimodalintervalmaps,the i X SRBmeasureisdifferentiableifandonlyifthepathf istangenttothetopological t r classoff0,thatis,ifandonlyif∂tft t=0 ishorizontal. Whendifferentiabilityholds, a | Ruelle’s candidate for the derivative, as interpretedin [2], gives the linear response formula. (We refer to [2, 3, 4], which also contain conjectures about smooth, not necessarily analytic, Collet–Eckmann maps, for more information and additional references.) Then, Ruelle [14] proved the linear response formula for a class of nonrecurrent 1 analytic unimodal interval maps f , assuming that all f stay in t t the topological class of f . In the present work, we consider holomorphic (that is, 0 Date:February2,2008. 1991 Mathematics Subject Classification. 37C4037C3037D2537E05. V.B. is partially supported by ANR-05-JCJC-0107-01. D.S. is partially supported by CNPq 470957/2006-9 and310964/2006-7, FAPESP2003/03107-9. D.S.thanks theDMAofE´coleNor- male Sup´erieure for hospitality during a visit where a crucial part of this work was done. V.B. wrote part of this paper while visiting the Universidad Cat´olica del Norte, Antofagasta, Chile, whosehospitalityisgratefullyacnowledged. WethankD.Sandsforveryhelpfulcomments. 1I.e.,infkd(ftk(c),c)>0,wherecdenotes thecriticalpoint. 1 2 VIVIANEBALADIANDDANIELSMANIA complex analytic)families f ofquadratic-likeholomorphic Collet–Eckmannmaps. t Our assumptions imply (using classical holomorphic motions) that all f lie in the t sameconjugacyclass. Generalisingoneoftheargumentsin[6],weareabletoshow that t ψdµ is real analytic for any real analytic function ψ. t 7→ Let usRnow state our resultmoreprecisely. Let I =[ 1,1]. AC3 mapf :I I − → isanS-unimodalmapifithasc=0asuniquecriticalpoint,andf hasnonpositive Schwarzianderivative,thatis f′′′ 3 f′′ 2 0exceptatc. AnS-unimodalmapis f′ −2 f′ ≤ called Collet-Eckmann if there exist C(cid:0) >(cid:1)0 and λ >1 so that (fn)′(f(c)) Cλn c | |≥ c foralln 1. Inthispaper,weshallonlyconsiderS-unimodalmapswithf′′(c)=0. ≥ 6 In Section 2 we shall define precisely the notion of a holomorphic (complex analytic)family of quadratic-like maps in a neighbourhood of I and provethe main result of this work: Theorem 1.1. Let t f be a holomorphic family of quadratic-like maps in a t 7→ neighbourhood of I, with all periodic orbits repelling. Assume in addition that for each small real t the map f restricted to I is a (real) Collet-Eckmann S-unimodal t map. Then there exists ǫ>0 so that for each real analytic ψ :I C, the map → t ψρ dx, 7→Z t where ρ is the invariant density of f , is real analytic on ( ǫ,ǫ). t t − The quadratic-like assumption implies that f′′(c) < 0. The fact that periodic t orbits are repelling implies that f is topologically conjugated with f (see our use t 0 ofMan˜´e-Sad-Sullivan[8]inthebeginning oftheproofofthe theoreminSection2). BesidesMan˜´e-Sad-Sullivan[8]theothermainingredientofourproofaretheresults andconstructionsofKellerandNowicki[7] whichallowustoexploitdynamicalzeta functions, following the argument in the work of Katok–Knieper–Pollicott–Weiss [6, First proof of Theorem 1]. The extension from quadratic-like to polynomial-like is straightforward,and we stick to the nondegenerate case f′′(c) = 0 for the sake of simplicity of exposition. 6 As the proof uses only real-analyticity of the holomorphic motions t h , it is t 7→ conceivable that the conclusion of the theorem holds if f is a real analytic family t of quadratic-like maps, using ideas of [1], but this generalisation appears to be nontrivial. 2. Proof of the Theorem Beforewe provethe theorem,letus define preciselythe objectswe arestudying: Definition. We say that f is a holomorphic family of quadratic-like maps in a t neighbourhood of I ifthereexistsacomplexneigbourhoodU ofI sothatt f isa t 7→ holomorphicmapfromacomplexneighbourhoodofzerototheBanachspaceB(U) of holomorphic functions on U extending continously to U (with the supremum norm), such that: Forrealt,themapf isrealon U,withf (I) I andf ( 1)=f (1)= 1. t t t t • ℜ ⊂ − − There existsimply connectedcomplex domainsW andV, whoseboundaries • areanalyticJordancurves,withI W,I V,V U,V W,andsothat ⊂ ⊂ ⊂ ⊂ f : V W is a double-branched ramified covering, with c = 0 as a unique 0 7→ critical point. (That is, f :V W is a quadratic-like restriction of f .) 0 0 7→ ANALYTICITY OF THE SRB MEASURE FOR FAMILIES OF COLLET-ECKMANN MAPS 3 Iff isaholomorphicfamilyofquadratic-likemapsinaneighbourhoodofI then t itiseasytosee2thatforsmallcomplext,denotingbyV theconnectedcomponent t of f−1(W) containing 0, then f :V W is a quadratic-like restriction of f . We t t t 7→ t may then give another definition: Definition. We say that f is a holomorphic family of quadratic-like maps in a t neighbourhood of I with all periodic orbits repelling, if f is a holomorphic family t of quadratic-like maps in a neighbourhood of I so that, for each small complex t, the map f only has repelling periodic orbits in V . t t Proof. Sinceweassumedthatallperiodicpointsoff arerepelling,[8,TheoremB] t (theresultthereisquotedforpolynomialmaps,buttheproofimmediately extends to polynomial-like) implies that there exists a holomorphic motion of the Julia set K(f ) of f , that is, a map h : D K(f ) C where D = z C z < ǫ for 0 0 0 0 × → { ∈ | | | } some ǫ >0, such that for each x K(f ) the map t h (x) is holomorphic, and 0 0 t ∈ 7→ for every t D the function x h (x) is continuous and injective on K(f ), with t 0 ∈ 7→ h f =f h . t 0 t t ◦ ◦ (In particular, h is a homeomorphism from K(f ) to K(f ).) Our assumptions t 0 t implythat[f2(0),f (0)]=K(f ) Randh (K(f ) R)=K(f ) R=[f2(0),f (0)]. 0 0 0 ∩ t 0 ∩ t ∩ t t From now on, we only use real analyticity of t f (x) and t h (x) for x t t 7→ 7→ ∈ [f2(0),f(0)]. Wenextclaimthatourassumptionsguaranteethateachf satisfiesthetechnical t requirement needed by Keller and Nowicki [7, (1.2)]. Denoting by var φ the total J variation of a function φ on an interval J, and writing f = f , we need to check t that there is that a constant M >0 such that: a. M−1 <sup |x−c| +var |x−c| <M, I |f′(x)| I|f′(x)| b. var |f(x)−f(u)| <M where J =[ 1,u] if u<c and =[u,1] if u>c. Ju|x−u||f′(x)| u − Let δ > 0 be so that f′′(y) > f′′(c)/2 if y c < δ . It suffices to prove (a.) 1 1 | | | | | − | and (b.) for x c < δ and u c < δ , and we restrict to such points. Noting 1 1 | − | | − | that for every such x=c there exist y , z , and z˜ , between x and c, so that x x x 6 x c x c 1 | − | = − = , f′(x) −f′(x) f′(c) −f′′(y ) x | | − and(use f′′(x)=f′′(c)+f(3)(z )(x c)andf′(x)=f′′(c)(x c)+f(3)(z˜ )(x−c)2) x − − x 2 x c f′(x)+(x c)f′′(x) (x c)2 f(3)(z˜ ) ∂ | − | = − − = − f(3)(z ) x , x f′(x) (f′(x))2 (f′(x))2 x − 2 | | (cid:0) (cid:1) the first two conditions hold because f is C3. For the third condition, consider x u>c (the other case is symmetric). Since ≥ f(x) f(u) x uf′′(z ) x u f′′(z ) x x − =1+ − =1+ − , (x u)f′(x) f′(x) 2 f′(x) 2f′′(y ) x − 2Indeed, ∂W is an analytic Jordan curve, and f0 has no critical point on ∂V. If ft ∈ B(U) is close to f0, there is a simplyconnected domain Vt close to V such that ft(Vt)=W, and the boundary of ∂Vt is a Jordan curve, by the implicit function theorem. Then ft : Vt → W is a quadratic-likeextension. 4 VIVIANEBALADIANDDANIELSMANIA and 0 < x−u < x−c , we get that f(x)−f(u) is bounded on [u,1], uniformly −f′(x) −f′(x) (x−u)f′(x) in u. Finally, since (cid:12) (cid:12) (cid:12) (cid:12) x u f′(x) (x u)f′′(x) ∂ − = − − , xf′(x) (f′(x))2 analyticity of f implies that ∂ x−u changes signs finitely many times, uniformly xf′(x) in u, proving (b.). Also, the results of Nowicki–Sands [11] and Nowicki–Przytycki [10] ensure (see Appendix A)thatthere existλ >1,λ >1,λ >1,andǫ >0sothat,foreach c per η 1 t <ǫ , there is C >0 with 1 t | | (1) (fn)′(f (0) C λn, n 1, | t t |≥ t c ∀ ≥ and so that for each x I so that fp(x)=x for some p 1, we have ∈ t ≥ (2) (fp)′(x) C λp , | t |≥ t per and, finally, setting λ (t):=liminf η −1/n η I is the biggest monotonicity interval of fn , η n→∞ {| | | ⊂ t } (3) inf λ (t)>λ . η η |t|<ǫ1 In other words, the hyperbolicity constants are uniform in t, guaranteeing unifor- mity when applying the results of Keller and Nowicki [7]. (We choose ǫ <ǫ .) 1 0 We now adapt the strategy used in the first proof of [6, Theorem 1]. Fix ψ and, for x I so that fp(x)=x for p 1, and for small real s and t, consider ∈ 0 ≥ esψ(ht(x)) (4) g (x)= . s,t f′(h (x)) | t t | Since ψ is real analytic, the analyticity of t h and of t f together with (2) t t 7→ 7→ imply that there is ǫ > 0 so that, for every periodic point x I of period p 1 2 ∈ ≥ for f, the function (t,s) g(p)(x):= esPpk−=01ψ(ht(fk(x)) 7→ s,t (fp)′(h (x)) | t t | is real analytic in s <ǫ and t <ǫ , uniformly in x. We take ǫ <ǫ . 2 2 2 1 | | | | Therefore, the dynamical zeta function defined by ∞ zp (5) ζ(s,t,z):=exp g(p)(x) p s,t Xp=1 x∈I:Xf0p(x)=x hasthefollowingproperty: Thereexistsδ >0sothatforeach z <δ thefunction 2 2 | | ζ(s,t,z)is realanalyticin t <ǫ , s <ǫ , andsothatforeach(s,t)with t <ǫ , 2 2 2 | | | | | | s <ǫ the map ζ(s,t,z) is holomorphic and nonvanishing in z <δ . 2 2 | | | | Now, h f =f h immediately implies t 0 t t ◦ ◦ ∞ zp esPpk−=01ψ(ftk(y)) (6) ζ(s,t,z)=exp . p (fp)′(y) Xp=1 y∈I:Xfp(y)=y | t | t Recall (1, 2, 3) and take Θ (0,1) with ∈ Θ−1 <min λ , min(λ ,λ ) . η c per { q } ANALYTICITY OF THE SRB MEASURE FOR FAMILIES OF COLLET-ECKMANN MAPS 5 KellerandNowicki[7,Theorem2.1]provethat,ifǫ (0,ǫ )issmallenough,then 3 2 ∈ for s <ǫ and t <ǫ the transfer operator 3 3 3 | | | | ω (y)exp(sψ(y)) t ϕ(x)= ϕ(y), Ls,t fˆtX(y)=xωt(x) |fˆt′(y)| acting on functions of bounded variation on a suitable Hofbauer tower extension fˆ : Iˆ Iˆ of f [7, Section 3], endowed with an appropriate [7, 6.2] cocycle ω t t t → § (which embodies the singularities along the postcritical orbit of f ), is a bounded t operator. If s = 0 then the spectral radius λ of is equal to 1, it is a simple 0,t s,t L eigenvalue (whose eigenvector gives the invariant density ρ of f ), and the rest t t of the spectrum is contained in a disc of strictly smaller radius. In addition, the essential spectral radius θ of satisfies sup θ < Θ, and for each s,t Ls,t |t|<ǫ3,|s|<ǫ3 s,t t <ǫ the spectral radius 4 λ > Θ of is an analytic function [7, Prop. 4.2] 3 s,t s,t | | L of s. Also, perturbation theory gives (see [7, (5.2)]) (7) ∂ logλ = ψρ dx. s s,t|s=0 Z t Keller and Nowicki also show [7, Theorem 2.2] that for t <ǫ and s <ǫ the 3 3 | | | | power series ζ(s,t,z) defined by (6) extends meromorphically to the disc of radius Θ−1 (where it does not vanish, by [7, Prop. 4.3 and Lemma 4.5]), and its poles z in this disc are in bijection with the eigenvalues λ of , via λ =z−1. (The k k Ls,t k k order of the zero coincides with the algebraic multiplicity of the eigenvalue.) It follows that z ζ(s,t,z)−1 is holomorphic in the disc of radius Θ−1. This disc 7→ contains λ−1, which is a simple zero. s,t Toendtheproof,recalling(7),itsufficestoseethat(s,t) λ isrealanalytic, s,t 7→ but this easily follows from Shiffman’s [15] real analytic Hartogs’ theorem (see Appendix B or [6, Thm p. 589]) applied to d(s,t,z) = ζ(s,t,z)−1, which implies that for each (s,t) ( ǫ ,ǫ ) ( ǫ ,ǫ ) the map z d(s,t,z) is holomorphic in 3 3 3 3 ∈ − × − 7→ z < Θ−1. Indeed, by the implicit function theorem, the simple zeroes of d(s,t, ) | | · depend real analytically on s and t. (We used the same ǫ discs for the s and t i variable, but a more careful analysis shows that ǫ in the statement of the theorem may be selected independently of ψ.) (cid:3) Appendix A. Uniformity of the hyperbolicity constants We start with a preliminary observation5: Let g be an S-unimodal Collet– Eckman map (with g′′(0) < 0, say). Denote by λ (g), λ (g), and λ (g) the c per η constants defined by (1, 2, 3) (replacing f by g). Nowicki and Sands [11] proved t that if g is an S-unimodal map and λ (g)>1 then λ (g)>1. A careful study of per c theirproofshowsthatλ (g)>λ (g)α,wheretheexponentα>0onlydependson c per the maximum length N(g) of “almost-parabolicfunnels” of g (see [11, Lemma 6.6] for a definition of N(g), which can be bounded by a function of 1/log(λ (g)) per and sup g′ ). Since N(g) is in fact invariant under topological conjugacy and f is t | | 3Our parameter s is called t in [7], the parameter β in [7] is β = 1, and our parameter t correspondstochangingthedynamics. 4Note that λs,t isthe exponential of the topological pressure of sψ−log|ft′| for ft, and that ρtdxistheequilibriumstateforft and−log|ft′|. 5WethankDuncanSands forhisexplanations. 6 VIVIANEBALADIANDDANIELSMANIA topologically conjugated to f , we conclude that λ (f ) > λ (f )α, with α > 0 0 c t per t uniform in small t. Next, recall that Nowicki and Przytycki [10] proved that if g and g˜ are S- unimodal maps (with g′′(c) = 0 and g˜′′(c) = 0, say) conjugated by a homeomor- 6 6 phism of the interval and g is Collet–Eckmann, then g˜ is Collet–Eckmann. Take g =f and g˜=f (in particular, f is C2 close to f and t h is smooth). Then 0 t t 0 t 7→ it is not very difficult to see that the constants M = M(f )> 0, P = P (f ) > 0, t 4 4 t and δ = δ (f ) > 0 from the topological characterisation (“finite criticality”) of 4 4 t Collet–Eckmannin [10, (4) p. 35]) are uniform in small t. Recallthatourassumptionsimply f′′(c)=0forallsmallt,sothatthe constant t 6 denoted l in [10] is l = 2. Section 2 of [10], and in particular the use of the c c Koebe principle there, implies that there exists a (universal) function q : R+ ∗ × (0,1) (0,1) with q(M,1/4)<1/2 for any M (see [10, Lemma 2.2]), and so that → −1 λ (f ) > 1 2q(M(f ),1/4) . Therefore, λ (f ) > 1 is uniformly bounded per t t per t − away from (cid:0)1 for small t. The p(cid:1)reliminary observation then implies that λ (f ) is c t also uniformly bounded in t. By [9, Proposition 3.2] (see also [10, p. 35]), this implies a uniform lower bound for λ (f ). (Indeed, in the notations of [9, 3], we η t § have λ =λ =λ λ =λ √λ .) η 5 4 3 1 c ≥ ≥ Appendix B. Shiffman’s real analytic Hartogs’ extension theorem Theorem B.1. [15] Let δ >0 and 0<r <R. Assume that d:( δ,δ)2 z C z <R C − ×{ ∈ || | }→ satisfies the following conditions: For each (s,t) ( δ,δ)2 the map z d(s,t,z) is holomorphic in z <R. • ∈ − 7→ | | For each z <r the map (s,t) d(s,t,z) is real analytic in ( δ,δ)2. • | | 7→ − Then d(s,t,z) is real analytic on ( δ,δ)2 z <R . − ×{| | } NotethattheabovetheoremfailsifrealanalyticityisreplacedbyCk fork . ≤∞ Thetheoremholdsbecause z <r isnotpluripolarin z <R. Shiffman’sresult | | | | is based on deep work of Siciak [16] References 1. A.Avila,M.Lyubich,andW.deMelo,Regularorstochasticdynamicsinrealanalyticfamilies of unimodal maps,Invent. 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D.M.A., UMR8553,E´coleNormaleSup´erieure,75005Paris, France E-mail address: [email protected] Departamento de Matema´tica, ICMC-USP, Caixa Postal 668, Sa˜o Carlos-SP, CEP 13560-970Sa˜oCarlos-SP, Brazil E-mail address: [email protected]