Smooth deformations of piecewise expanding unimodal maps

In the space of C^k piecewise expanding unimodal maps, k>=1, we characterize the C^1 smooth families of maps where the topological dynamics does not change (the"smooth deformations") as the families tangent to a continuous distribution of codimension-one subspaces (the"horizontal"directions) in that space. Furthermore such codimension-one subspaces are defined as the kernels of an explicit class of linear functionals. As a consequence we show the existence of C^{k-1+Lip} deformations tangent to every given C^k horizontal direction, for k>=2.


Introduction
The topological class of a dynamical system f is the set of all maps topologically conjugate with f . A smooth deformation of a dynamical system f 0 is a smooth family of dynamical systems t → f t inside the topological class of f 0 . We also say that a smooth deformation f t is a family with "no bifurcations." Deciding whether or not there are bifurcations in a family is one of the primary problems concerning dynamical systems.
In the theory of complex dynamical systems, specially for rational functions, this type of study was very sucessful. One of the most powerful tools in complex dynamics are the quasiconformal methods: quasiconformal maps, quasiconformal vector fields, and holomorphic motions. For example, they allow us to easily find a holomorphic deformation between two holomorphic dynamical systems which are conjugate by a quasiconformal map, using the so-called Beltrami path. Beltrami paths are examples of holomorphic motions, whose importance in complex dynamics can not be overstated since the time they were introduced in the seminal work by Mañé, Sad and Sullivan [9]. Holomorphic motions are a key tool in the characterization of structurally stable rational maps and families of rational maps with no bifurcations [9] (see also [10]). The study of the regularity of hybrid classes of quadratic-like maps [7] and topological classes of analytic unimodal maps [2] also depends heavily on quasiconformal methods.
Unfortunately quasiconformal methods do not seem to be applicable for real onedimensional maps which do not have a holomorphic extension to the complex plane, like piecewise expanding C 2 unimodal maps (see Section 2 for formal definitions).
In our first main result, Theorem 1 (Section 4), we characterize all smooth families in the space of piecewise expanding unimodal maps which are smooth deformations: they are precisely the families tangent to a continuous distribution of codimension-one subspaces in that space. Following the notation in [7], these subspaces will be called "horizontal directions." See Section 3 for the definition of the linear functional J(f, ·) whose kernel defines the horizontal directions.
We observe that for families of smooth unimodal maps, the condition is the "nondegeneracy condition" for a family f t at a Collet-Eckmann parameter f t0 that appeared in a generalization of Jakobson's Theorem by Tsujii [11]. On the other hand, the condition J(f, v) = 0 for v to be horizontal is the same which is well-known for smooth unimodal maps satisfying certain summability condition (see e.g. [1], [2]). This condition first appeared in the context of piecewise expanding maps in [3]. One can wonder if there exist deformations of a given piecewise expanding unimodal map which are non trivial, i.e., so that the f t are not smoothly conjugate to f 0 . (Not only because this is an intrisically natural question, but also because it recently became clear that this is crucial to understand some dynamically defined quantities, see below. ) We answer this question in Theorem 2 (Section 5). In particular, for each "good" piecewise expanding unimodal map f 0 and each horizontal direction v, we construct a smooth deformation of f 0 tangent to v at f 0 , i.e., ∂ t f t | t=0 = v. In other words, Theorem 2 shows that the theory of smooth deformations is very rich, since there are plenty of deformations of a piecewise expanding unimodal map in "horizontal" directions.
In both theorems we heavily use "smooth motions," that is, we exploit the fact that the conjugacies h t depend smoothy on t. In Theorem 1, B ⇒ C (see [4]), we use them in the phase space, and in Theorem 2, in the parameter space.
Given a smooth family of dynamical systems, one can ask how dynamically defined quantities, such as the average of a given observable with respect to the SRB measures, the Lyapunov exponents, and the Hausdorff dimension of invariant sets, change along this family. Studying smoothness of these quantities can be a tricky issue. For example, SRB measures are often described as eigenvectors of Ruelle-Perron-Frobenius operators acting on infinite-dimensional spaces with a complicated structure.
In the case of piecewise expanding maps, Hölder continuity of SRB measures (for all exponents < 1) has been known for a long time [5]. However any hope of higher regularity for families transversal to the topological class was annihilated by the examples in [3] (see also [8]). In order to have a satisfactory theory about smooth variation of dynamically defined quantities, at least in the case of the SRB measure of piecewise expanding unimodal maps, it was recently discovered that we need to restrict ourselves to families tangent to topological classes [4]. Theorem 2 and its corollaries imply the result announced as Theorem 2.8 in [4]. This result was not used to obtain the other claims in [4], but it shows that there are plenty of families satisfying the restriction of tangency to the topological class needed there.

Preliminaries
Denote I = [−1, 1] and N = Z + . For k ≥ 0, we define the set B k (I) of piecewise C k functions to be the linear space of continuous functions f : I → R such that f is C k on the intervals [−1, 0] and [0, 1], with f (1) = f (−1). Then B k (I) is a Banach space for the norm For k ≥ 1, we define the set U k of piecewise expanding C k unimodal maps to be the set of maps f ∈ B k (I) such that The set U k is a convex subset of the affine subspace and U k ∩ {f ∈ B k (I) : f (0) < 1} is a convex and open set of the same affine subspace. We call elements of U 1 simply piecewise expanding unimodal maps. The point c = 0 is called the critical point of a piecewise expanding unimodal map. Set The itinerary of x ∈ I for a piecewise expanding unimodal map f is the sequence Let 1 ≤ j ≤ k, with k an integer, and j either an integer or j = k − 1 + Lip. A C j family f t of piecewise expanding C k unimodal maps is a C j map c) and i < j}, the set of critical relations of f t . Note that if the forward orbit of c (also called postcritical orbit) is infinite then R t is empty.
We say that a piecewise expanding unimodal map f is good if either c is not periodic or, writing p ≥ 2 for the prime period of c, if A piecewise expanding unimodal map f 0 is stably -expansive if every piecewise expanding unimodal map f close enough to f 0 (for | · | 1 ) is -expansive. We give the easy proof of the following useful result for completeness: 1. Let f be a piecewise expanding unimodal map. Then there exists > 0 so that f is -expansive. If we assume furthermore that f is good, then there exists > 0 so that f is stably -expansive.
Proof. Choose N 0 such that 1 2 λ N0−1 f > λ f , and such that Then for every interval Q ⊂ [− , ] we have If the turning point is not periodic it is easy to see that (3) remains true for any small enough perturbation of f . Consider an interval Q ⊂ I and suppose that |f i (Q)| < for every i ∈ N. Define n 0 = 0 and n 1 , n 2 , n 3 , . . . in the following way: If c ∈ Q s := f n0+n1+···+ns (Q) define n s+1 = 1. In this case so |Q s | ≥ λ s f |Q|, which implies that |Q| = 0, proving that f is -expansive. If c is periodic but (2) holds, the argument above can be easily modified to show stable -expansiveness.

The linear functional J(f, ·)
3.1. Definition and relation with the twisted cohomological equation. We shall associate a bounded linear functional J(f, ·) ∈ (L ∞ (I)) * to each piecewise expanding unimodal map f . This functional will play a main role in this work. Let v : I → R be a bounded function. If the critical point c is not periodic, we define The above expression is not well defined if the critical point c is periodic, since the derivative at the critical point does not exist. If c has prime period p we set Note that in both cases (non periodic and periodic critical points) we have It is easy to see that v → J(f, v) is not the zero functional on C(I), so for every k ∈ N, by the density of C k (I) in C(I), there exists v ∈ C k (I) with J(f, v) = 0. The meaning of the expression for J(f, v) can be clarified by the following comments. Let f t be a C 1 family of piecewise expanding C 1 unimodal maps such that So, if c has prime period k, then and if c is not periodic for f , then In other words, the derivatives in the phase and parameter spaces along the critical orbit are related by J(f, v).
We also mention that if , where s 1 < 0 is the jump s 1 = − lim x→1 ρ(x) at 1 of the invariant density ρ of f , and where J (f, X) was introduced in [3] and used in [4]. It was observed in [4] (see also Proposition 3.1 below) that elements v of the kernel of J (f, ·) satisfy Df i (f (c)) = 0 if c has prime period p. Such v ∈ Ker ((J (f, ·)) deserve to be called horizontal vector fields, by analogy with the theory for smooth unimodal maps ( [7], [2]) and in view of the results in [4] (in particular Corollary 2.6 and Remark 2.7 there). Our Theorems 1 and 2 also justify this terminology.
We next recall the relation between J(f, v) and the twisted cohomological equation (7) from [4, Lemma 2.2].
Proposition 3.1. For every piecewise expanding unimodal map f and v ∈ L ∞ (I) the following holds: Let D be the set of x ∈ I with a forward orbit that does not contain c. There exists a unique bounded function α : D → R such that for every x ∈ D. There exists a unique bounded function α : I → R such that α(c) = 0 and (7) holds for every x = c. Furthermore Proof. We refer to [4, Lemma 2.2]. We just recall that for each x ∈ D is continuous at piecewise expanding unimodal maps with non periodic critical point: 2. Let f 0 ∈ U 1 be a piecewise expanding unimodal map. If the critical point of f 0 is not periodic then Reducing W , if necessary, we can assume that f i (f (c)) = c for every f ∈ W and i ≤ N , and that Then we get the claim from (8) and our choice of N .
Proof of Claim B. We can assume that |v 0 | ≤ 1. Fix η, and let W and N be like in the proof of Claim A. Reducing W if necessary, we have The calculations in the proof of Claim A imply that Continuity of f → J(f, v) fails at maps f 0 with periodic critical points. However, to prove Theorem 2, the next result (which, losely speaking, implies that when 3. Let f 0 be a good piecewise expanding C 1 unimodal map with periodic critical point of prime period p 0 . There exist C + , C − > 0, such that: A. For every η > 0, there exists a neighborhood W of f 0 in U 1 such that, setting the set W \ M has two connected components, W + and W − , so that, for (9)) has two connected components, A consequence of Propositions 3.2 and 3.3 (B.) is that Ker (J(f, ·)) is a continuous distribution of codimension-one subspaces for any good f : ). Let f be a good piecewise expanding C 1 unimodal map. Suppose that f n is a sequence of piecewise expanding C 1 unimodal maps with |f n − f | 1 → 0, and that v n ∈ B 0 (I) and v ∈ B 0 (I) are such that where p 0 denotes the prime period of c for f 0 .
Note that F is a C 1 Fréchet differentiable function and (recall (4)) So by the Implicit Function Theorem for C 1 Fréchet differentiable functions on Banach spaces (see [6, page 17 (9), is a C 1 Banach submanifold and so that W \ M has two connected components.
Let W + be the connected component containing the maps f satisfying and let W − be the other component. We claim that our assumption that f 0 is stably -expansive implies that for every n there exist neighborhoods W n ⊂ W n−1 of f 0 with the following properties: the sets W n ∩ W + and W n ∩ W + are connected and the critical point c of every map f ∈ W n \ M is either non periodic or periodic with prime period ≥ n. Indeed, consider open intervals I 0 , I 1 , . . . , I p0−1 , with pairwise disjoint closures, such that In particular, if f has a critical point with prime period p < n then (11) holds for every i. We claim that f p0 (c) = c. It is enough to show that for every i. Since f is -expansive, this implies that y = f (c), so that = 1, as desired.
Consequently, the itinerary of the critical point up to the n-th iteration is the same for all maps in W n ∩ W + . The same statement holds for W n ∩ W − . In particular there exist sequences L}, such that the itinerary of the critical point of a map f ∈ W + converges to σ + (in the product topology of {C, R, L} N ) when the map converges to f 0 , and an analogous statement holds for and Fix δ > 0 and let m ≥ 1 be such 2mλ −p0m < δ. If we assume, as in Claim A, This proves Claim A for W + .
To show Claim B for W + , consider, without loss of generality, v ∈ B 0 (I) with |v| 0 ≤ 1. Then we can find W δ,p0m such that holds for every f ∈ W δ,p0m ∩ W + ∩ W p0m . Then We can apply a similar argument to f ∈ W p0m ∩ W − and The proof of the claims for f ∈ M is easier.

Bifurcations in families of expanding unimodal maps
We are going to see in this section that if a C 1 family f t of good piecewise expanding C 1 unimodal maps is tangent to the distribution of codimension-one subspaces f → Ker (J(f, ·)) then there are no bifurcations in this family, that is, there are homeomorphisms h t such that h t • f 0 = f t • h t for every t. In other words, the family is a smooth deformation of f 0 . The reverse statement also holds: If f t is a family such that J(f 0 , ∂ t f t | t=0 ) = 0 and if f 0 is good, then there are bifurcations in this family.
Theorem 1 (Characterization of smooth deformation). Let f t , t ∈ (−δ, δ), be a C k family of piecewise expanding C k unimodal maps, with k ≥ 1. Then the following properties are equivalent: A. For small t, the set of critical relations R t is constant. B. For small t, there exists a family h t : I → I of homeomorphisms so that h t is a conjugacy between f 0 and f t , C. For small t, there are conjugacies h t , as in B, and we have that is continuous and for each Furthermore if we restrict t to a compact interval Q ⊂ (− , ) we have that this family is a bounded subset in C k−1+Lip (Q). (In fact, there is a universal constant C so that the diameter of this subset is ≤ C sup t∈Q Furthermore A, B and C imply D. For small t we have that J(f t , ∂ s f s | s=t ) = 0. Note that for these implications we do not assume that f 0 is good. But if we assume in addition that f 0 is stably -expansive, then D is equivalent to A, B, and C.
Proof. Note that C trivially implies B and A.
Proof of A implies B. This implication is a consequence of Milnor-Thurston theory of kneading invariants, but we will give a self-contained argument.
Let δ > 0 be so that R t is constant for t ∈ (−δ, δ). Note that the itinerary σ t of the critical point of f t is constant for t ∈ (−δ, δ). Indeed, if f i t (c) = c for some i and some t, then by definition (0, i) ∈ R t , and assumption A implies f i s (c) = c for every small s. By the continuity of the family f t , this implies that the itinerary of c is constant (also if R t = ∅).
Let P t be the set of points which are either periodic or eventually periodic points of f t , and whose forward orbit does not contain the critical point. It is easy to see that P t is dense in I. We claim that up to taking a smaller δ > 0, each point p ∈ P 0 has an analytic continuation h t (p), defined for every |t| < δ. Moreover h t : P 0 → P t is a bijection. In fact, since the forward orbit of p does not contain the critical point, we can find a maximal open interval Q, where the analytic continuation h t (p) is (uniquely) defined. If there exists t ∞ ∈ ∂Q ∩ (−δ, δ), choose t n ∈ Q, with lim n→∞ t n = t ∞ .
Since Q is maximal, every accumulation point q of the sequence h tn (p) has a priori the itinerary of p, replacing at least one of its symbols by C. But note that since the f t are piecewise expanding, and since we proved that the itinerary of the critical point under f t is constant, every itinerary obtained by replacing C by either R or L symbols in the itinerary of the critical point is forbidden for p. So ∂Q∩(−δ, δ) = ∅ and h t (p) is defined for every t. Of course h t (p) ∈ P t . Furthermore h t is injective, since h t (p) has the same itinerary as p and distinct points in P 0 have distinct itineraries.
It remains to prove that h t0 (P 0 ) = P t0 , for every t 0 . This can be achieved by considering a smooth re-parametrization g u of the family f t such that g 0 = f t0 and and applying the argument above to construct h −1 t0 . Due the uniqueness of the analytic continuation (14) h t • f = f t • h t on P 0 . Moreover p < q implies h t (p) < h t (q) for every t. By the density of P t , for every t, we can extend h t to a homeomorphism h t : I → I. The continuity of h t and Eq. (14) imply that f 0 is conjugate to f t by h t .

Proof of B implies C. See [4, Proposition 2.4] (the proof there works for k ≥ 1).
Proof of A, B, C implies D. It is enough to show that A. implies D. First, suppose that R t = ∅. Then f t has a periodic critical point with prime period p, for all small t, that is, f p−1 t (f t (c)) = c for small t. Differentiating with respect to t, we obtain Now assume that R t = ∅ for small t and suppose for a contradiction that J(f t0 , ∂ t f t | t=t0 ) = 0 for some small t 0 . By Proposition 3.2, Claim B., either J(f t , ∂ t f t ) ≥ ξ > 0 for every t close to t 0 , or J(f t , ∂ t f t ) ≤ ξ < 0 for every t close to t 0 . Without loss of generality, assume the first case. Using (6) for f t and ∂ t f t , and the fact that θ = inf t,x |Df t (x)| > 1, we find δ > 0 and k 0 ≥ 1 so that for every k ≥ k 0 , which is absurd since θ > 1.
If g(t 0 ) = c then by (17) we have dg dt (t 0 ) = 0. Therefore, using that the f t are piecewise uniformly Lipschitz and the family is C 1 , we get Consequently g n (t) = f n t (g(t))) is a solution of (17), for every n. Of course the constant function c 0 (t) = c is a solution of (17). Since the functions α t can be uniformly bounded by a constant which is independent of t, we conclude by (17) that the set of functions c n (·) is equicontinuous.
Suppose now that there is t 0 so that c is a periodic point of f t0 . If the prime period of c is p, choose open intervals I 0 , I 1 , . . . I p−1 , with pairwise disjoint closures, |I i | < , and such that f i t0 (c) ∈ I i mod p ∀i.
Since {c n (·)} is an equicontinuous set of functions, there exists δ 0 > 0 such that (18) f i t (c) ∈ I i mod p for every i and every t such that |t − t 0 | < δ 0 . We claim that if |t − t 0 | < δ 0 then the map f t has a periodic critical point with the same itinerary as that of c for f t0 . By (18), it is enough to show that By the stable -expansivity of f 0 we must have y = f t (c) which implies N = 1.
So we conclude that for every itinerary σ of length p, the set of parameters O such that f t has a p-periodic critical point with itinerary σ is an open set. Of course for all parameters in the closure of O, f t has a p-periodic critical point, but, a priori, not with prime period p. But if we apply the same argument to this boundary parameter, we conclude that its critical point has the same itinerary as points in O. This implies that either O = ∅ or O = {t : |t| < δ}. If O = ∅ for each finite orbit, then each f t has an infinite postcritical orbit and an empty R t . So the set of critical relations R t does not depend on t.
We mention an easy consequence of Theorem 1 which will be useful in the proof of Theorem 2: Corollary 4.1 (Unstable families). Let f t be a C 1 family of piecewise expanding C 1 unimodal maps such that f 0 is good and A. If f 0 has a periodic critical point then there exists a sequence of parameters t n → 0 such that the critical point of f tn is not periodic. B. If f 0 has a non periodic critical point then there exists a sequence of parameters t n → 0 such that the critical point of f tn is periodic.
Proof of Claim A. By Corollary 3.1 and the continuity of t → ∂ t f t in the B 0 (I) norm, there existsδ > 0 such that Suppose by contradiction that for all parameters |t| ≤ δ 1 < δ 0 the critical point of f t is periodic. Define P n := {t : f n t (c) = c and |t| ≤ δ 1 }. Of course P n is closed. By the Baire Theorem there exists n 0 ≥ 1 so that P n0 contains a nonempty connected open set Q ⊂ P n0 . For each 1 ≤ i ≤ n 0 , let P i ⊂ P n0 be set of parameters for which f t has a critical point whose prime period is equal or larger than i. Of course each P i is an open subset of P n0 . Let p = max{i : P i = ∅}.
Then there exists an open set U such that the critical point of f t has prime period p if t ∈ U . In particular the set of critical relations R t is constant on U . By the implication A ⇒ D in Theorem 1, J(f t , ∂ t f t ) = 0 for every t ∈ U , which contradicts (19).
Proof of Claim B. The proof in this case is even easier. By Proposition 3.2 B., we have (19) for some δ 0 > 0. If there are non periodic critical points for f t for all small enough t, then the set of critical relations R t is empty for those t. By A ⇒ D in Theorem 1, J(f t , ∂ t f t ) = 0 for all small enough t, which contradicts (19).

Finding or approximating families tangent to a given horizontal direction
We can now state and prove our second main result: is topologically conjugate with f for all |t| < δ. Furthermore this unique function b is in fact C k−1+Lip and satisfies b (0) = 0 (in particular ∂ tft | t=0 = v), and the familyf t is a C k−1+Lip -family of piecewise expanding C k unimodal maps.
In addition, there exists a sequence of C k families of piecewise expanding C k unimodal maps t → g t,n (t ∈ (−δ, δ)) such that -the map g t,n is topologically conjugate with g 0,n , for each t and n, -the critical point of g 0,n is periodic for each n, -For each t the map g t,n converges to the mapf t in the B k−1 (I) topology.
Proof of the existence off t . Note that if f 0 (c) = +1, then either f t (c) = +1 for all small enough t (in which case we may take b(t) ≡ 0, so that existence off t is proved) or f t (c) < 1 for all nonzero small enough t. Denote v t = ∂ s f s | s=t . For small η > 0 set M η = {|t| < η, |θ| < η} and consider (In fact, if f 0 (c) = +1 but f t (c) < 1 for all small nonzero t we must take M η = {|t| < η, |θ| < Θ(t)} with Θ a C 1 function so that Θ(0) = 0, Θ(t) > 0 for t = 0.) Since J(f 0 , v 0 ) = 0 but J(f 0 , w) = 0, Propositions 3.3 and 3.2 and Corollary 3.1 imply that if η is small enough then for all (t, θ) ∈ M η the linear space is a one-dimensional subspace of R 2 , which depends continuously on (t, θ) and never coincides with the vertical line {0} × R. In other words: There exists a uniquely defined function d : M η → R so that ,θ) , ·)).
By D ⇒ B in Theorem 1,f t is topologically conjugate with f 0 for small t.
Proof of the uniqueness off t . Suppose that b,b are two continuous functions with b(0) =b(0) = 0 and such that both maps are topologically conjugate to f for each small |t| < δ. Using the map d : M η → R from (21) in the proof of the existence off t , choose 0 <η <η < η such that if (t 0 , θ 0 ) ∈ Mη then the ordinary differential equation with initial condition b(t 0 ) = θ 0 , has a C 1 -solution defined for every |t| <η, and, moreover, we have |b(t)| <η for |t| <η. Suchη,η exist, since d(0, 0) = 0. Suppose there is |t 0 | <η such that b(t 0 ) =b(t 0 ). Sinceb and b are continuous, up to taking a smaller t 0 we may assume that max(|b(t 0 )|, |b(t 0 )|) <η. To fix ideas, assume 0 ≤ b(t 0 ) <b(t 0 ) (the other cases are similar). Then for every θ 0 ∈ (b(t 0 ),b(t 0 )) we can find a solution b : (−η,η) → R for (23) such that b(t 0 ) = θ 0 . By the Intermediate Value Theorem, there exists Note that by (23) and the definition of d(·, ·) Thus, by D ⇒ B in Theorem 1 for b, and by assumption for b,b, all maps are in the topological class of f . As a consequence, for every θ ∈ (b(t 0 ),b(t 0 )), the map f t0 + θw is topologically conjugate with f . Consequently there is no change of combinatorics in the C ∞ family of piecewise expanding C k unimodal maps and B ⇒ D in Theorem 1 gives J(f t0 + θ 0 w, w) = 0. Taking a sequence t n → 0 such that b(t n ) =b(t n ), the argument above gives a sequence θ n → 0 so that J(f tn +θ n w, w) = 0, and Corollary 3.1 implies J(f 0 , w) = 0, contradicting the assumption on w.
Proof of the C k−1+Lip regularity, construction of g t,n . Recall (20) and the characterization (21) of d(·, ·). By Corollary 4.1.B, there exists a sequence θ n → 0 such that f (0,θn) has a periodic critical point (if f 0 has a periodic critical point, define θ n = 0, for every n). Consider the C 1 -integral curves b n of the ordinary differential equation Note that since d(0, 0) = 0, if η is small, then the solution b n is defined for |t| < η, provided n is large enough. As a consequence, for all large enough n, Let p n be the prime period of the turning point of g n,0 = f (0,θn) . By (27) and D ⇒ B in Theorem 1 we have that g t,n is topologically conjugate with g n,0 , so g t,n has a critical point with the same prime period p n .
We shall first prove that each g t,n ∈ U k (I), by showing that each function b n is C k . Indeed consider the non-linear functional Then F n is C k on M η , and if f (t,θ) has a periodic point with prime period p n our assumption on w gives (recalling (4), as for (10)) Since F n (t, b n (t)) = 0, the Implicit Function Theorem implies that t → b n (t) is C k . For further use, note also that ∂ t F n (t, b n (t)) + ∂ θ F (t, b n (t))b n (t) = 0, and since ∂ t F n (t, θ) = Df pn−1 (t,θ) (f (t,θ) (c))J(g t,n , v t ), we obtain (28) b n (t) = − J(g t,n , v t ) J(g t,n , w) .
We shall next show that the families {(t, x) → g t,n (x), n ∈ N} form a bounded subset of C 2 (in the sense of families (1)) as the first step in the inductive proof that this set is bounded for C k . Let λ = inf t,n λ gt,n . We have λ > 1.
Since f t and w are in B 2 (I), and b n , b n are uniformly bounded in n (use (25)), there exist by the definition (26) uniform upper bounds for the derivatives ∂ t g t,n , ∂ x g t,n , ∂ 2 x g t,n , ∂ 2 xt g t,n , ∂ 2 tx g t,n . So we must only estimate ∂ 2 tt g t,n . By (26), it is enough to show that {b n } n is a bounded subset in C 2 . Since we already know that each b n is C 2 , it is enough to get an n-uniform bound on the Lipschitz constant of b n . In view of (28), we first show that for every vector field u ∈ B 1 (I), the map t → J(g t,n , u) is Lipschitz, and its Lipschitz norm does not depend on n. By B ⇒ C in Theorem 1, the conjugacies h t,n •g 0,n = g t,n •h t,n are such that (the C k−1+Lip map) t → h t,n (x) is K-Lipschitz, with This implies in particular that t → u(g i t,n (c)) is Lipschitz uniformly in t, i and n.
So (we omit n in g t,n and h t,n to avoid a cumbersome notation) .
Note that 1 In the last step we used that ∂ 2 tx g t,n is bounded uniformly in n. So (29) gives |J(g t+δ,n , u) − J(g t,n , u)| ≤ which proves that t → J(g t,n , u) is C|u| 1 -Lipschitz, uniformly in n. Next, we obtain from (28) that We used that J(g, ·) is linear and that J(g t,n , w) is bounded away from zero and infinity, uniformly in n and |t| < η (by Propositions 3.2 and 3.3 since g t,n and f 0 are B 2 (I)-close, using that sup n,t |b n (t)| < ∞). This proves our claim that b n is C 1+Lip , and thus C 2 , uniformly in n, and thus the uniform C 2 claim on g t,n . Note that by B ⇒ C in Theorem 1 the maps t → h t,n (x) are C 1+Lip uniformly in n and x. If k = 2, we are done. If k ≥ 3, we have concluded the first step in the induction.
Before we perform the inductive step, we introduce some notation and terminology. Let X be a set and let f λ , λ ∈ Λ, be an indexed family F of functions on X. For each formal monomial λ 1 λ 2 . . . λ n , we can associate the function f λ1 f λ2 . . . f λn . This function is called a F-monomial combination of degree n. A F-polynomial combination of degree n is a finite sum of F-monomials whose maximal degree is n. Note that if Λ 1 is a finite subset of Λ and P n ∈ N n [Λ 1 ] is a polynomial with non negative integer coefficients, we can associate to it a F-polynomial combination of degree n. We will call this combination the P -combination of the family F.
If F 1 and F 2 are families indexed by Λ 1 and Λ 2 , we will denote by F 1 ∪ F 2 the disjoint union of these families, indexed by the disjoint union Λ 1 and Λ 2 .
If X is an open interval, all functions in F are differentiable and F is the indexed family of derivatives of functions in F, then the derivative of a sum of m S-monomials of degree n is a sum of m · n F ∪ F -monomials of degree n.
Suppose by induction that {b n } n is a bounded family in C q , with 2 ≤ q < k. Then g t,n is a C q family. By B ⇒ C in Theorem 1, the conjugacies h t,n • g 0,n = g t,n • h t,n are such that t → h t,n (x) ∈ C q−1+Lip , uniformly in n and x, and, setting is a bounded family in C Lip . Note also that for every for every a ≤ k and b ≤ q. In particular for each i < p n , if we define the indexed families of functions is also a bounded subset in C Lip . It is easy to show by induction on q that there exist C q and d q such that for each 1 ≤ < p n , there exists a P q, -combination ψ n, ,q of the family C q−1 n, ∪ G q n, such that ∂ q−1 t 1 Dg t,n (h t (g 0,n (c))) = ψ n, ,q (Dg t,n (h t (g 0,n (c))) 2 q−1 , and P q, is a sum of at most C q monomials of maximal degree d q . In particular if we define the indexed families F q−1 n,i := {∂ r t 1 Dg t,n (h t (g 0,n (c))) : 0 ≤ r ≤ q − 1, 1 ≤ < p n } then (33) ∪ n F q−1 n,pn−1 is a bounded set in C Lip . Let u ∈ B k (I). We claim that {J(g t,n , u)} n is a bounded subset of functions in C q−1+Lip . Indeed, for each n and i < p n , define the indexed families of functions O q−1 n,i = {D j x u(h t,n (g 0,n (c))) : 0 ≤ j ≤ q − 1, 1 ≤ ≤ i}, Of course (34) ∪ n O q−1 n,pn−1 is a bounded subset of functions in C Lip . Note that for each i < p n u 0,i,n := u(g i t,n (c)) Dg i t,n (g t,n (c)) = u(h t,n (g i 0,n (c))) i =1 1 Dg t,n (h t,n (g 0,n (c))) , in particular there exists a monomial P 0,i of degree i + 1 such that u 0,i,n is a P 0,icombination of the family C 0 n,i ∪ F 0 n,i ∪ O 0 n,i . It can be easily proven by induction on q that for every 1 ≤ i < p n there exists a polynomial P q,i such that ∂ q−1 t u(g i t,n (c)) Dg i t,n (g t,n (c)) is a P q,i -combination of the family C q−1 n,i ∪F q−1 n,i ∪O q−1 n,i . Furthermore P q,i is the sum of at most (i + q − 1)!/i! monomials with maximal degree i + q, and each monomial contains at least max{0, i − q + 1} indexes in F 0 n,i . So if n C q−1 n,pn−1 ∪ F q−1 n,pn−1 ∪ O q−1 n,pn−1 .
belongs to the ball of radius R in C Lip , it is easy to see that Thus, by (28), b n belongs to a bounded subset in C q+Lip , and thus in C q+1 . That concludes the induction step. In particular {g t,n (x), n} belongs to a bounded set of C k families (in the sense of families (1)). We can thus choose a convergent subsequence lim j→∞ g nj ,t = g ∞,t in the B k−1 (I)topology, where the family g ∞,t is a C k−1+Lip family of piecewise expanding C k unimodal maps. Note that g ∞,0 = f . Since J(g t,n , ∂ t g t,n ) = 0 and lim j→∞ ∂ t g nj ,t | t=t0 = ∂ t g ∞,t | t=t0 , in the B 0 (I) topology, we conclude, by Corollary 3.1, that J(g ∞,t , ∂ s g ∞,s | s=t ) = 0. By D ⇒ B in Theorem 1, the map g ∞,t is topologically conjugate with f . By uniqueness, g ∞,t =f t and lim n→∞ b n = b.
For k ≥ 2, let f ∈ U k (I) be a good map, and let w ∈ B k (I) be such that w(−1) = w(1) = 0 and J(f, w) = 0. Using a technique similar to the proof of Theorem 2, one can prove that there exist a neighborhood V 1 of f and a neighborhood V 2 of 0 in Ker J(f, ·) (we consider J(f, ·) : {v ∈ B k , v(−1) = v(1) = 0} → R) such that each topological class in V 1 is of the form {f + v + ψ(v)w : v ∈ V 2 }, where ψ is a C k−1+Lip real-valued function defined in V 2 . Moreover if f n ∈ V 1 for all integers n ≥ 0, with lim n→∞ f n = f ∞ ∈ V 1 , and if ψ n : V 2 → R defines the topological class of f n in V 1 , then lim n→∞ ψ n = ψ ∞ in C k−1 defines the topological class of f ∞ .
We end by mentioning two immediate corollaries of Theorem 2.
Corollary 5.1. Let f be a piecewise expanding C k unimodal map and v ∈ B k (I), with k ≥ 2, such that v(−1) = v(1) = 0 and J(f, v) = 0. Then there exists a C k−1+Lip family of piecewise expanding C k unimodal mapsf t such thatf 0 = f , ∂ tft | t=0 = v andf t is topologically conjugate with f , for every t.
Corollary 5.2. Let f t be a C k family of piecewise expanding C k unimodal maps, k ≥ 2, such that f t is topologically conjugate with f 0 , for every t. Then there exists a sequence of C k -families t ∈ (−δ, δ) → g t,n such that -The map g t,n is topologically conjugate with g 0,n , for every t.
-The critical point of g 0,n is periodic.
-The families g t,n converge to the family f t in the B k−1 topology.
Proof. Apply Theorem 2 to the family f t , noting that Theorem 1 implies that J(f 0 , ∂ t f t | t=0 ) = 0. So there exists a unique familyf t such thatf t is topologically conjugate with f 0 , for every small t. We conclude that f t =f t . To finish, apply the last claim in Theorem 2.