Analyticity of the SRB measure for holomorphic families of quadratic-like Collet-Eckmann maps

We show that if f_t is a holomorphic family of quadratic-like maps with all periodic orbits repelling so that for each real t the map f_t is a real Collet-Eckmann S-unimodal map then, writing m_t for the unique absolutely continuous invariant probability measure of f_t, the map t ->\int g dm_t is real analytic for any real analytic function g.


Introduction and statement of the theorem
If t → f t is a smooth one-parameter family of dynamics f t so that f 0 admits a unique SRB measure µ 0 , it is natural to ask whether the map t → µ t , where t ranges over a set Λ of parameters such that f t has (at least) one SRB measure µ t , is differentiable at 0 (in the sense of Whitney if Λ does not contain a neighbourhood of 0, as suggested by Ruelle [13]). Katok, Knieper, Pollicott, and Weiss [6] gave a positive answer to this question in the setting of C 3 families of transitive Anosov flows (here, Λ is a neighbourhood of 0), showing that t → ψ dµ t is differentiable, for all smooth ψ. If f 0 is a C 3 mixing Axiom A attractor and the family t → f t is C 3 , Ruelle [12] not only proved that t → ψ dµ t is differentiable, but also gave an explicit formula (the linear response formula) for the derivative. Ruelle [13] suggested that this formula, appropriately interpreted, should hold in much greater generality. Indeed, Dolgopyat [5] obtained the linear response formula for a class of partially hyperbolic diffeomorphisms. In a previous work [3,4], we found that in the (non structurally stable) setting of piecewise expanding unimodal interval maps, the SRB measure is differentiable if and only if the path f t is tangent to the topological class of f 0 , that is, if and only if ∂ t f t | t=0 is horizontal. When differentiability holds, Ruelle's candidate for the derivative, as interpreted in [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 t , assuming that all f t stay in the topological class of f 0 . In the present work, we consider holomorphic (that is, complex analytic) families f t of quadratic-like holomorphic Collet-Eckmann maps. Our assumptions imply (using classical holomorphic motions) that all f t lie in the same conjugacy class. Generalising one of the arguments in [6], we are able to show that t → ψ dµ t is real analytic for any real analytic function ψ.
Let us now state our result more precisely. Let I = [−1, 1]. A C 3 map f : I → I is an S-unimodal map if it has c = 0 as unique critical point, and f has nonpositive for all n ≥ 1. In this paper, we shall only consider S-unimodal maps with f ′′ (c) = 0. 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 prove the main result of this work: Let t → f t be a holomorphic family of quadratic-like maps in a neighbourhood of I, with all periodic orbits repelling. Assume in addition that for each small real t the map f t restricted to I is a (real) Collet-Eckmann S-unimodal map. Then there exists ǫ > 0 so that for each real analytic ψ : The quadratic-like assumption implies that f ′′ t (c) < 0. The fact that periodic orbits are repelling implies that f t is topologically conjugated with f 0 (see our use of Mañé-Sad-Sullivan [8] in the beginning of the proof of the theorem in Section 2). Besides Mañé-Sad-Sullivan [8] the other main ingredient of our proof are the results and constructions of Keller and Nowicki [7] which allow us to exploit dynamical zeta 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. As the proof uses only real-analyticity of the holomorphic motions t → h t , it is conceivable that the conclusion of the theorem holds if f t is a real analytic family of quadratic-like maps, using ideas of [1], but this generalisation appears to be nontrivial.

Proof of the Theorem
Before we prove the theorem, let us define precisely the objects we are studying: We say that f t is a holomorphic family of quadratic-like maps in a neighbourhood of I if there exists a complex neigbourhood U of I so that t → f t is a holomorphic map from a complex neighbourhood of zero to the Banach space B(U ) of holomorphic functions on U extending continously to U (with the supremum norm), such that: • For real t, the map f t is real on ℜU , with f t (I) ⊂ I and f t (−1) = f t (1) = −1.
• There exist simply connected complex domains W and V , whose boundaries are analytic Jordan curves, with I ⊂ W , I ⊂ V , V ⊂ U , V ⊂ W , and so that f 0 : V → W is a double-branched ramified covering, with c = 0 as a unique critical point. (That is, If f t is a holomorphic family of quadratic-like maps in a neighbourhood of I then it is easy to see 2 that for small complex t, denoting by V t the connected component We may then give another definition: Definition. We say that f t is a holomorphic family of quadratic-like maps in a neighbourhood of I with all periodic orbits repelling, if f t is a holomorphic family of quadratic-like maps in a neighbourhood of I so that, for each small complex t, the map f t only has repelling periodic orbits in V t . Proof. Since we assumed that all periodic points of f t are repelling, [8,Theorem B] (the result there is quoted for polynomial maps, but the proof immediately extends to polynomial-like) implies that there exists a holomorphic motion of the Julia set We next claim that our assumptions guarantee that each f t satisfies the technical requirement needed by Keller and Nowicki [7, (1.2)]. Denoting by var J φ the total variation of a function φ on an interval J, and writing f = f t , we need to check that there is that a constant M > 0 such that: Let δ 1 > 0 be so that |f ′′ (y)| > |f ′′ (c)|/2 if |y − c| < δ 1 . It suffices to prove (a.) and (b.) for |x − c| < δ 1 and |u − c| < δ 1 , and we restrict to such points. Noting that for every such x = c there exist y x , z x , andz x , between x and c, so that , the first two conditions hold because f is C 3 . For the third condition, consider x ≥ u > c (the other case is symmetric). Since is close to f 0 , there is a simply connected 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-like extension.
analyticity of f implies that ∂ x x−u f ′ (x) changes signs finitely many times, uniformly in u, proving (b.).
Also, the results of Nowicki-Sands [11] and Nowicki-Przytycki [10] ensure (see Appendix A) that there exist λ c > 1, λ per > 1, λ η > 1, and ǫ 1 > 0 so that, for each In other words, the hyperbolicity constants are uniform in t, guaranteeing uniformity when applying the results of Keller and Nowicki [7]. (We choose ǫ 1 < ǫ 0 .) We now adapt the strategy used in the first proof of [6, Theorem 1]. Fix ψ and, for x ∈ I so that f p 0 (x) = x for p ≥ 1, and for small real s and t, consider Since ψ is real analytic, the analyticity of t → h t and of t → f t together with (2) imply that there is ǫ 2 > 0 so that, for every periodic point x ∈ I of period p ≥ 1 for f , the function )| is real analytic in |s| < ǫ 2 and |t| < ǫ 2 , uniformly in x. We take ǫ 2 < ǫ 1 .
Keller and Nowicki [7, Theorem 2.1] prove that, if ǫ 3 ∈ (0, ǫ 2 ) is small enough, then for |s| < ǫ 3 and |t| < ǫ 3 the transfer operator 3 acting on functions of bounded variation on a suitable Hofbauer tower extension f t :Î →Î of f t [7, Section 3], endowed with an appropriate [7, §6.2] cocycle ω t (which embodies the singularities along the postcritical orbit of f t ), is a bounded operator. If s = 0 then the spectral radius λ 0,t of L s,t is equal to 1, it is a simple eigenvalue (whose eigenvector gives the invariant density ρ t of f t ), and the rest of the spectrum is contained in a disc of strictly smaller radius. In addition, the essential spectral radius θ s,t of L s,t satisfies sup |t|<ǫ3,|s|<ǫ3 θ s,t < Θ, and for each |t| < ǫ 3 the spectral radius 4 λ s,t > Θ of L s,t is an analytic function [7,Prop. 4.2] of s. Also, perturbation theory gives (see [7, (5.2)]) Keller and Nowicki also show [7, Theorem 2.2] that for |t| < ǫ 3 and |s| < ǫ 3 the 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 k in this disc are in bijection with the eigenvalues λ k of L s,t , via λ k = z −1 k . (The 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 contains λ −1 s,t , which is a simple zero. To end the proof, recalling (7), it suffices to see that (s, t) → λ s,t is real analytic, 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 3 , ǫ 3 ) × (−ǫ 3 , ǫ 3 ) the map z → d(s, t, z) is holomorphic in |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 i discs for the s and t variable, but a more careful analysis shows that ǫ in the statement of the theorem may be selected independently of ψ.) topologically conjugated to f 0 , we conclude that λ c (f t ) > λ per (f t ) α , with α > 0 uniform in small t.
Next, recall that Nowicki and Przytycki [10] proved that if g andg are Sunimodal maps (with g ′′ (c) = 0 andg ′′ (c) = 0, say) conjugated by a homeomorphism of the interval and g is Collet-Eckmann, theng is Collet-Eckmann. Take g = f 0 andg = f t (in particular, f t is C 2 close to f 0 and t → h t is smooth). Then it is not very difficult to see that the constants M = M (f t ) > 0, P 4 = P 4 (f t ) > 0, and δ 4 = δ 4 (f t ) > 0 from the topological characterisation ("finite criticality") of Collet-Eckmann in [10, (4) p. 35]) are uniform in small t.
Note that the above theorem fails if real analyticity is replaced by C k for k ≤ ∞. The theorem holds because |z| < r is not pluripolar in |z| < R. Shiffman's result is based on deep work of Siciak [16]