Normal forms for non-uniform contractions

Let $f$ be a measure-preserving transformation of a Lebesgue space $(X,\mu)$ and let $\f$ be its extension to a bundle $\E = X \times\Rm$ by smooth fiber maps $\f_x : \E_x \to \E_{fx}$ so that the derivative of $\f$ at the zero section has negative Lyapunov exponents. We construct a measurable system of smooth coordinate changes $\h_x$ on $\E_x$ for $\mu$-a.e. $x$ so that the maps $\p_x =\h_{fx} \circ \f_x \circ \h_x ^{-1}$ are sub-resonance polynomials in a finite dimensional Lie group. Our construction shows that such $\h_x$ and $\p_x$ are unique up to a sub-resonance polynomial. As a consequence, we obtain the centralizer theorem that the coordinate change $\h$ also conjugates any commuting extension to a polynomial extension of the same type. We apply our results to a measure-preserving diffeomorphism $f$ with a non-uniformly contracting invariant foliation $W$. We construct a measurable system of smooth coordinate changes $\h_x: W_x \to T_xW$ such that the maps $\h_{fx} \circ f \circ \h_x ^{-1}$ are polynomials of sub-resonance type. Moreover, we show that for almost every leaf the coordinate changes exist at each point on the leaf and give a coherent atlas with transition maps in a finite dimensional Lie group.


INTRODUCTION
The theory of normal forms for smooth maps originated in the works of Poincaré and Sternberg [26], and normal forms at fixed points and invariant manifolds have been extensively studied [4]. More recently, non-stationary normal form theory was developed in the context of a diffeomorphism f contracting a foliation W . The goal is to obtain a family of diffeomorphisms H x : W x → T x W such that the maps are as simple as possible, for example, linear maps or polynomial maps in a finite dimensional Lie group. Such a mapf x is called a normal form of f on W x .
The non-stationary normal form theory started with the linearization along one-dimensional foliations obtained by Katok and Lewis [15]. In a more general setting of contractions with narrow band spectrum, it was developed by Guysinsky and Katok [11,10], and a differential geometric point of view was presented by Feres [7]. For the linearization, further results were obtained by the second author in [25], and it was shown in [19] that the coordinates H x give a consistent affine atlas on each leaf of W . In [20] we extended these results to the general narrow band case. More precisely, we gave a construction of H x that depend smoothly on x along the leaves and proved that they define an atlas with transition maps in a finite dimensional Lie group. Non-stationary normal forms were used extensively in the study of rigidity of uniformly hyperbolic dynamical systems and group actions, see, for example, [17,18,19,5,6,9,8].
To obtain applications for non-uniformly hyperbolic systems and actions, one needs a similar theory of non-stationary normal forms for non-uniform contractions. The existence and centralizer theorems were stated without proof in [12] along with a program of potential applications. The theory, however, was not developed for quite a while. The linearization of a C 1+α diffeomorphism along a one-dimensional non-uniformly contracting foliation was constructed in [13] and used in the study of measure rigidity in [13,14]. Similar results for higher dimensional foliations with pinched exponents were obtained by Katok and Rodriguez Hertz in [16]. The existence of H x for a general contracting C ∞ extension was proved by Li and Lu [22] in the setting of random dynamical systems. Some results, such as existence of Taylor polynomial or formal series for H x , can be obtained for extensions more general than contractions, see [2,1,22].
In this paper we develop the theory of non-stationary polynomial normal forms for smooth extensions of measure preserving transformations by nonuniform contractions, described in the beginning of Section 2. This is a convenient general setting for the construction. The foliation setting reduces to it by locally identifying the leaf W x with its tangent space E x = T x W and viewing F x = f | W x : E x → E f x as an extension of the base system f : M → M by smooth maps on the bundle E = T W . The base system can then be viewed as just a measure preserving one. In the extension setting, the map H x is a coordinate change on E x , and we denote In Theorem 2.3 we construct coordinate changes H x for µ almost every x so that P x is a sub-resonance polynomial. For any regularity of F above the critical level, we obtain H in the same regularity class.
Our construction allows us to describe the exact extent of non-uniqueness in H x and P x . Essentially, they are defined up to a sub-resonance polynomial. As a consequence of this, we obtain the centralizer theorem that the coordinate change H also conjugates any commuting extension to a normal form of the same type. We just learned of similar results in differential geometric formulations by Melnick [23]. The approach in [23] is different from ours and it relies on ergodic theorems for higher jets of F x . Our results assume only temperedness of the higher derivatives of F x rather than certain integrability required in [23]. This allows us to obtain applications to the foliation setting without any assumptions on transverse regularity of the foliation. In particular, we consider a diffeomorphism f which preserves an ergodic measure with some negative Lyapunov exponents and take W to be any strong part of the stable foliation. In this setting Theorem 2.5 gives sub-resonance normal forms for f along the leaves of W . Moreover, we show that for almost every leaf the normal form coordinates H x exist at each point on the leaf and give a coherent atlas with transition maps in a finite dimensional Lie group G determined by subresonance polynomials. This yields an invariant structure of a G homogeneous space on almost every leaf.
We expect these results to be useful in the study of non-uniformly hyperbolic systems and group actions.
Sub-resonance polynomials. Let χ 1 < · · · < χ < 0 be the distinct Lyapunov exponents of F and let E x = E 1 x ⊕ · · · ⊕ E x be the splitting of E x for x ∈ Λ into the Lyapunov subspaces given by the Multiplicative Ergodic Theorem 3.1.
We say that a map between vector spaces is polynomial if each component is given by a polynomial in some, and hence every, bases. We consider a polynomial map P : E x → E y with P (0 x ) = 0 y and split it into components (P 1 (t ), . . . , P (t )), where P i : E x → E i y . Each P i can be written uniquely as a linear combination of polynomials of specific homogeneous types: we say that Q : E x → E i y has homogeneous type s = (s 1 , . . . , s ) if for any real numbers a 1 , . . . , a and vectors We denote by S x,y the space of all sub-resonance polynomial maps from E x to E y .
Clearly, for any sub-resonance relation we have that s j = 0 for j < i and that s j ≤ χ 1 /χ . It follows that sub-resonance polynomial maps have degree at most Sub-resonance polynomial maps P : E x → E x with P (0) = 0 with invertible derivative at the origin form a group with respect to composition [11]. We will denote this finite-dimensional Lie group by G χ x . All groups G χ x are isomorphic, moreover, any map P ∈ S x,y with P (0 x ) = 0 y and invertible derivative at 0 x induces an isomorphism between G χ x and G χ y by conjugation. We denote by B x,σ(x) the closed ball of radius σ(x) centered at 0 ∈ E x . For N ≥ 1 and 0 < α ≤ 1 we denote by σ(x) and with N th derivative satisfying α-Hölder condition at 0: (2.6) and that K is ε-tempered on a set if it is ε-tempered at each of its points.
We consider an extension F satisfying Assumptions 2.1 and denote by Λ the set of regular points and by χ 1 < · · · < χ < 0 the Lyapunov exponents of F given by the Multiplicative Ergodic Theorem 3.1. For N and α as above we define If N ≥ 2 we allow α = 0, in which case we understand C N,α as C N . THEOREM 2.3 (Normal forms for non-uniformly contracting extensions). Let F be an extension of f satisfying Assumptions 2.1. Suppose that Then there exist positive constants L = L(N , α) and ε * = ε * (N , α, χ 1 , . . . , χ ) so that for any 0 < ε ≤ ε * the following holds. If there exists a positive measurable function σ : Moreover, H x C N,α (B x,ρ(x)) is Lε-tempered on Λ and D (n) 0 H x is n 2 εtempered on Λ for n = 1, . . . , N , with respect to the ε-Lyapunov metric (3.2).
(2) SupposeH = {H x } x∈Λ is another measurable family of diffeomorphisms as in (1) conjugating F to a sub-resonance polynomial extensionP . Then for all x ∈ Λ there exists G x ∈ G χ x which is measurable and tempered in x (3) Let g : X → X be an invertible map commuting with f and let Λ be a subset of Λ which is both f and g invariant.
be an extension of g to E which preserves the zero section and commutes Normal forms on stable manifolds. Let M be a compact smooth manifold and let f be a diffeomorphism of M preserving an ergodic Borel probability measure µ. We assume that f is C N,α , that is C N with N th derivative α-Hölder on M. We denote by Λ the full measure set of Lyapunov regular points for (D f , µ). Let χ 1 < · · · < χ be the Lyapunov exponents of (D f , µ) and suppose is such that χ < 0. Then for each x ∈ Λ there exists the (strong) stable manifold W x tangent to  x : E x → E y are affine for all x ∈ X and y ∈ W x , and H y depends C N -smoothly on y along the stable manifolds.

LYAPUNOV EXPONENTS AND LYAPUNOV NORM
In this section we review some basic definitions and facts of the Oseledets theory of linear extensions. We use [3] as a general reference. For a linear extension F of a map f we will use the notation

THEOREM 3.1 (Oseledets Multiplicative Ergodic Theorem, see [3] Theorem 3.4.3).
Let f be an invertible ergodic measure-preserving transformation of a Lebesgue probability space (X , µ), and let F be a measurable linear extension satisfying log F x , log F −1 x ∈ L 1 (X , µ). Then there exist numbers χ 1 < · · · < χ , an finvariant set Λ with µ(Λ) = 1, and an F -invariant Lyapunov decomposition The numbers χ 1 , . . . , χ are called the Lyapunov exponents of F and the points of the set Λ are called regular. We denote the standard scalar product in R m by 〈·, ·〉. For a fixed ε > 0 and a regular point x, the ε-Lyapunov scalar product (or metric) 〈·, ·〉 x,ε in E x = R m is defined as follows. For u ∈ E i x and v ∈ E j x with i = j , 〈u, v〉 x,ε := 0, and for i = 1, . . . , and u, v Note that the series converges exponentially for any regular x. The constant m in front of the conventional formula is introduced for more convenient comparison with the standard scalar product. Usually, ε will be fixed and we will denote 〈·, ·〉 x,ε simply by 〈·, ·〉 x and call it the Lyapunov scalar product. The norm generated by this scalar product is called the Lyapunov norm and is denoted by · x,ε or · x .
Below we summarize the basic properties of the Lyapunov scalar product and norm, for more details see [3,. A direct calculation shows [3, Theorem 3.5.5] that for any regular x and any u ∈ E i and exp(nχ − εn) ≤ F n x f n x←x ≤ exp(nχ + εn) for all n ∈ N, (3.4) where · f n x←x is the operator norm with respect to the Lyapunov norms. It is defined for any points x, y ∈ Λ and any linear map F : E x → E y as follows: We emphasize that Lyapunov scalar product and norm are defined only for regular points and depend measurably on the point. Thus, a comparison with the standard norm is important. The uniform lower bound follows easily from the definition: u x,ε ≥ u . The upper bound is not uniform, but it changes slowly along the regular orbits [3, Proposition 3.5.8]: there exists a measurable function Using (3.5) we obtain that for any point x, y ∈ Λ and any linear map F : When ε is fixed we will usually omit it and write K (x) = K ε (x) and u x = u x,ε .
Similarly, we will consider the Lyapunov norm of a homogeneous polynomial map R : E x → E y of degree n defined as For a homogeneous polynomial map R : E x → E y of degree n we have This formula allows us to switch between the standard and Lyapunov norms in spaces of polynomials and smooth functions.

PROOF OF THEOREM 2.3
We note that (2.8) implies N ≥ d . We give the proof for the case α > 0. The proof for α = 0, and hence N ≥ 2, is similar but avoids difficulties of estimating the Hölder constant at 0. We will use the notation F n Now we define constants that will be used throughout the proof. We set We define λ < 0 as the largest value of −χ i + j =1 s j χ j over all i ∈ {1, . . . , } and non-negative integers s 1 , . . . , s such that sub-resonance condition (2.2) is not satisfied: The maximum exists since there are only finitely many values of −χ i + s j χ j greater than any given number. Next we recall that N + α > χ 1 /χ and set The proof of part (1) of Theorem 2.3 works for any L ≥ L(N , α) and any ε < ε 0 , where For parts (2) and (3) of Theorem 2.3 we will use smaller bounds on ε, namely, ε 1 = ε 0 /(N + 1) and ε * = ε 0 /3(N + 1), respectively.

Construction of P and of the Taylor polynomial for H .
For each x ∈ Λ and map F x : E x → E f x we consider the Taylor polynomial at t = 0: As a function of t , F (n) x (t ) : E x → E f x is a homogeneous polynomial map of degree n. First we construct the Taylor polynomials at t = 0 for the desired coordinate change H x (t ) and the polynomial extension P x (t ). We use similar notations for these Taylor polynomials: For the first derivative we choose We will inductively construct the terms H (n) x and P (n) x for all in x ∈ Λ so that P (n) x is of sub-resonance type and they are measurable in x and n 2 ε-tempered, i.e., The base of the induction is the linear terms chosen above. Now we assume that the terms of order less than n are constructed. Using these notations in the conjugacy equation Considering the terms of degree N ≥ n ≥ 2, we obtain where the summations are over all i and j such that i j = n and 1 < i , j < n. We rewrite the equation as We note that Q x is composed only of terms H (i ) and P (i ) with 1 < i < n, which are already constructed, and terms F (i ) with 1 < i ≤ n, which are given. Thus by the inductive assumption Q x is defined for all x ∈ Λ and measurable. We will show later that they are also suitably tempered in x.
Let R (n) x be the space of all homogeneous polynomial maps on E x of degree n, and let S (n) x and N (n) x be the subspaces of sub-resonance and non subresonance polynomials respectively. We seek H (n) x so that the right side of (4.11) is in S (n) x , and hence so is P (n) x when defined by this equation. Projecting (4.11) to the factor bundle R (n) /S (n) , our goal is to solve the equation whereH (n) andQ are the projections of H (n) and Q respectively.
We consider the bundle automorphism Φ : Since F preserves the splitting E = E 1 ⊕· · ·⊕E , it follows from the definition that the sub-bundles S (n) and N (n) are Φ-invariant. We denote byΦ the induced automorphism of R (n) /S (n) and conclude that (4.13) is equivalent tō Thus a solution of (4.13) is aΦ-invariant section of R (n) /S (n) . We will show thatΦ is a nonuniform contraction and that it has a unique measurable tempered invariant section. First, for polynomials of specific homogeneous type the exponent of Φ is determined by the exponents of F as follows.
Since R has homogeneous type s = (s 1 , . . . , s ) we obtain by (2.1) that Since this holds for any ε > 0, using (3.10) to compare the Lyapunov and standard norms, one can conclude that the Lyapunov exponent of Φ on R is For all non sub-resonance homogeneous types we have −χ i + s j χ j ≤ λ by the definition (4.2) of λ. Thus Lemma 4.2 yields the following lemma.  .14) is a nonuniform contraction over f −1 , and hence so isΦ : Proof. The statement aboutΦ follows since the linear partΦ ofΦ is given by Φ when R (n) /S (n) is naturally identified with N (n) . By the choice of ε, we have λ + (n + 1)ε < 0.
It follows from the previous remark that λ is the maximal Lyapunov exponent of Φ over f −1 on the space of non sub-resonant polynomials and that all Lyapunov exponents of Φ| S (n) are non-negative. Now we construct aΦ-invariant measurable section of B = R (n) /S (n) and study its properties. The construction is orbit-wise. We fix a point x ∈ Λ, consider its positive orbit {x k = f k x : k ≥ 0}, and define the Banach space and claim that it is in B x . For this we need to estimate the growth of the Lyapunov norm of (4.12) along the trajectory: for all x and k by (3.4). The exponential growth rate in k of F (n) x k x k+1 ←x k is at most 2ε. Indeed, using (3.10) and (3.6) we can obtain from (4.5) the corresponding estimate for C N,α norm with respect to the Lyapunov metric on E x k : Then using the inductive assumption (4.10) for the terms of order i , j < n, we can estimate the exponential growth rate of the two terms in the sum respectively as (i 2 + 2i )ε and ( j 2 + i 2 j )ε, which are at most ((n/2) 2 + i n)ε < n 2 ε. So the exponential growth rate of Q x k x k ←x k can be estimated by n 2 ε and thus Q < ∞.
Since λ+(n 2 +n+1)ε < 0 by the choice of ε (4.4),Φ x is a contraction and thus has a unique fixed point R x ∈ B x . We claim thatH (n) x = R x 0 is a measurable function which is a unique solution of (4.15) or equivalently (4.13). Measurability follows from the fact that the fixed point can be explicitly written as a series and since R x ∈ B x , the exponential growth rate of H (n) x k x k ←x k is at most n 2 ε. Now we can choose H (n) x as a lift ofH (n) x to R (n) x which is measurable in x and satisfies (4.10). Then we define P (n) x by equation (4.11). It also satisfies (4.10) as so do H and Q and as F x x← f x and F −1 x f x←x are uniformly bounded. This completes the inductive step and the construction of H (n) and P (n) , n = 1, . . . , N , satisfying (4.10).
Thus we have constructed the N th Taylor polynomial for the coordinate change and the polynomial map P x (t ) = d n=1 P (n) x (t ).

4.2.
Construction of the coordinate change H . We rewrite the conjugacy equa- We will find H in the form H = H N + R, where H N is given by (4.21). We denote and observe that T (H ) = H if and only ifT (R) = R. We will find R x using the fixed point of a contractionT x induced byT on a certain space C x of sequences of functions along the orbit of x. Now we define the space C x .
By the construction of H N and P , H N and T (H N ) have the same derivatives at the zero section up to order N , so we consider functions with vanishing derivatives at the zero section up to order N . First we describe the space of functions at each regular point x. For any x ∈ Λ we denote by B x,r the ball centered at 0 in E x of radius r < ρ(x) < 1 in the Lyapunov norm · x . We define . . , N }. Throughout this section we use the C N,α norms with respect to the Lyapunov metric on E x . They are estimated through the norms for the standard metric (2.5) in (4.19). In particular, we use the α-Hölder constant (2.4) of D (N ) R at 0 with respect to the Lyapunov metric, which for any R ∈ C x,r is given by For any R ∈ C x,r lower derivatives can be estimated by the mean value theorem as so using the above Hölder constant we obtain that for any 0 ≤ n < N and t ∈ B x,r , Thus the norms of all derivatives are dominated by the Hölder constant and hence It follows that C x,r equipped with the norm D (N ) R x,α is a Banach space.
We will choose a small r = r (x) < ρ(x) satisfying (4.40) and set where L is given by (4.1). We define C x as the following Banach space of sequences of functions along the orbit x k = f k x.
with the norm · x k ,α defined as in (4.25) and satisfying (4.28). We consider the operatorT x induced byT on C x : by the construction, and the latter satisfies F x k x k+1 ←x k ≤ e χ +ε and F −1 x k x k ←x k+1 ≤ e −χ 1 +ε .
Also, for 2 ≤ n ≤ d , the Lyapunov norms of D (n) 0 (P x k ) = P (n) x k and D (n) 0 (H x k ) = H (n) x k grow at most at the exponential growth rate n 2 ε in k by (4.10).
Recall that the inverse of P x k is also a sub-resonance polynomial P −1 . We now show that the Lyapunov norms of (P −1 x k ) (n) also grows at most at the exponential rate n 2 ε in k. First, its linear term (P −1 x k has bounded Lyapunov norm. Inductively, we consider terms of order n > 1 in the equation P • P −1 = Id and obtain The terms in the sum can be estimated as P (i ) x k ) ( j ) i and hence are (i 2 + j 2 i )ε-tempered by the inductive assumption. Since i , j ≤ n 2 we obtain i 2 + j 2 i = i 2 + ni ≤ n 2 4 + n 2 2 < n. Multiplying the equation by bounded P (1) x k −1 we conclude that P −1 x k (n) is n 2 ε-tempered, completing the induction. can be inductively estimated similarly to (4.26), x x←x yielding the same estimate of the exponential rate as for H (N ) x , Similarly for P −1 x k the derivative of order d ≤ N is constant on E x k , higher derivatives are zero, and the lower derivatives can be estimated as for H , so we obtain To obtain estimates for (T (H N )) x = P −1 x •H N f x •F x we use the following lemma.

LEMMA 4.5. If Q is a polynomial of degree at most N and F is C N,α then Q •F is C N,α and Q
First we estimate α-Hölder constant at 0 of the first term. As DQ is linear, we get So the α-Hölder constant at 0 of D F (t ) Q•D (N ) t F is estimated by 2 Q C N F 2 C N,α . The other terms in the sum are C 1 and hence are Lipschitz with constant bounded by supremum norms of their derivatives. These norms, along with the norms of lower derivatives of Q • F can be estimated as a sum of termss of the type Later we will also need a similar result for the case when Q is not a polynomial.
Proof. The proof is the same as in Lemma 4.5 except that, since D (N ) Q is only Hölder, we also need to estimate the α-Hölder constant at 0 of the term D (N ) We consider Here we estimated the Lipschitz constant Lip D (N ) 0 Q of the homogeneous polynomial N -form D (N ) 0 Q on a ball of radius R = F C 1 by the supremum of its derivative on that ball, which is a homogeneous polynomial (N −1)-form whose norm can be estimated by D (N ) 0 Q with some constant c N depending on N only.
First we check that the compositions in T (R) k are well-defined. We take t ∈ B x k ,r k and show that t = F x k (t ) is in B x k+1 ,r k+1 . Since by (4.37) t is in the ball B x k ,ρ(x k ) in standard metric, the estimates in Lemma 4.1 hold for any k. In particular, by (2),(5) Since t x k ≤ r k = r e −2N Lkε , this yields where we have estimates for (P x k ) −1 . Using the mean value theorem, (4.38), and the inequality γ < γ we obtain as 2N − 1 ≥ 1. Using (4.28) we obtain similarly that for anyR ∈ B x (γ), Since ρ(x k ) ≥ ρ(x)e −κεk ≥ ρ(x)e −Lεk and 2γr < ρ(x) by (4.40), we obtain (4.43) Now we show thatT x is a contraction on B x (γ) by estimating its differential. For anyR,S ∈ B x (γ) we can write Differentiating (P x k ) −1 and denoting we obtain where E is a polynomial with terms of degree at least two. Thus E (z(t )) C x = O( S 2 C x ) and so the differential ofT x is given by where A k (s) = D s (P x k ) −1 . To estimate the norm we consider the derivative of order N . Since A k (y(t )) is a linear operator on z, the product rule yields (4.44) where m + l = N and l < N for all terms in the sum. Differentiating z(t ) we get Only the first term in (4.44) contains D (N ) S k+1 so where J k consists of a fixed number of terms of the type
Finally we estimate (4.45) combining (4.46) and (4.48). For anyR ∈ B x (γ) we have Since r k = r e −2N Lkε and 2N L ≥ M , as L ≥ N 3 + 3N 2 + 1, we see that for all k ≥ 0 as r ≤ θ/(c 5 (x)γ N e (M +L)ε ) by (4.40). Then for allR ∈ B x (γ) we obtain and so Thus . ThusT x has a unique fixed point R x ∈ B x (γ) which depends measurably on x. As in the construction of Taylor coefficients, the uniqueness implies that R x := (R x ) 0 is Lε-tempered and solves the equationT (R) = R whereT is given by (4.24). We conclude that the measurable family of C N,α maps H x = H N x +R x , is Lε-tempered and satisfies (4.23), i.e., which has the same Taylor terms at 0 as H up to order N . In fact, for n > d we have S (n) = 0 and hence ∆ (n) = 0, so that H N = H d . Now we show that H = H N , which also proves the last statement in part (2) of the theorem. The equality follows from the uniqueness in the final step of the construction. Indeed, for H N given by (4.21), both differences R = H −H N and R = H N −H N are fixed points of operatorT given by (4.24). Hence R = R by uniqueness of the fixed point in the appropriate space C r,x on whichT induces a contraction.
To ensure that the sequence (R x k ) is in C r,x we need to estimate temperedness of α-Hölder constant at 0 for H (N ) N . As above one can see that all terms in the polynomial G x are N 2 ε-tempered. Then using Lemma 4.5 and the assumption onH we obtain that H N ,x C N,α isLε-tempered forL = (N 2 + N L) < (N + 1)L and hence (R x k ) is in C r,x withL in place of L. Since the proof of part (1) is for any L ≥ L(N , α), we conclude thatT induces a contraction in such C r,x provided that ε < ε 1 = ε 0 /(N + 1), which is less than ε 0 withL in pace of L in (4.4). Thus First we prove that the derivative of G at zero section, Γ x = D 0 G x , is sub-resonance. Since Γ x is linear, this is equivalent to the fact that Γ x preserves the fast flag associated with the Lyapunov splitting Suppose to the contrary that for some x ∈ Λ and some i < j we have a vector t in E i x such that t = Γ x (t ) has nonzero component t j in E j g x . Then and on the other hand which is impossible for large n since ε is small. Here we used the fact that the C N,α norm G x C N,α ,x on E x is 2ε-tempered with respect to the Lyapunov metric (3.2) for F . This follows as in (4.19) since G x C N,α in standard norm is ε-tempered by assumption.
We conclude that Γ x is sub-resonance for each x ∈ Λ. Now we consider a new family of coordinate changesH which also satisfiesH x (0) = 0 and D 0H x = Id. A direct calculation shows that whereP x is a sub-resonance polynomial as a composition of sub-resonance polynomials. Now we would like to to apply the uniqueness part of the theorem, which would giveH x = G x H x for some tempered function G x ∈ G χ . Then it follows from the definition ofH x that which is again a sub-resonance polynomial, as claimed.
To complete the proof it remains to show thatH x is suitably tempered to obtain uniqueness. The n th Taylor term at 0,H (n) x , is the sum of the terms of the form Γ −1 x with n = k j , whose Lyapunov norms as we can estimate as before Thus we obtain thatH (n) x is mε-tempered with m ≤ 2 + k 2 + 2k < 3n 2 for n ≥ 2. Since H C N,α is Lε-tempered, using Lemma 4.6 with Q = H and F = G we obtain that H • G C N,α is (L + 2(N + α))ε-tempered. Then Lemma 4.5 implies that H C N,α is (2 + L + 2(N + α))ε-tempered and hence 3Lε-tempered since L ≥ N + 2. So the uniqueness result in part (2) of the theorem applies for ε < ε * = ε 1 /3 = ε 0 /3(N + 1). This completes the proof of Theorem 2.3.
5. PROOF OF THEOREM 2.5 5.1. Proof of (i), (ii), (iii), (v). We will apply Theorem 2.3. First we note that the integrability condition for the derivative in Theorem 2.3 was used in the proof only to obtain the Lyapunov splitting and the Lyapunov metric. So while the restriction D f | E may not satisfy this integrability condition, the Lyapunov splitting and the Lyapunov metric are obtained in this case from the results for the full differential D f . The centralizer part (v) will follow directly from part (3) of Theorem 2.3 since X = n∈Z g n (X ) is the desired invariant set of full measure as g preserves the measure class of µ. Moreover, g (W x ) = W g x since g is a diffeomorphism commuting with f , so that X is also saturated by the stable manifolds.
Parts (i), (ii), (iii) essentially follow from Theorem 2.3, which is formulated so as to apply to this setting. First we consider the regular set Λ for (D f , µ). We fix a family of local (strong) stable manifolds W x,r (x) for x ∈ Λ of sufficiently small size r (x). Identifying W x,r (x) by an exponential map with a neighborhood of 0 in E x we obtain the extension F = {F x } of f . Then the properties of local stable manifolds ensure that F satisfies the assumptions of Theorem 2.3. Indeed, they are given by C N,α embeddings so that the C N,α norm and 1/r (x) are ε-tempered for any ε > 0 (see [3] for a general reference and [16,Theorem 5] for a convenient statement of the stable manifold theorem). Hence Theorem 2.3 yields existence of the desired family of local diffeomorphisms H x , x ∈ Λ, which can be uniquely extended by invariance to the global stable leaf W x , which consists of those t ∈ M for which f k t is in the local stable leaf of f k x for some k. Now we define X = x∈Λ W x and explain the construction of H y for any y ∈ X . By iterating it forward we may assume that y ∈ W x,r (x) . While the individual Lyapunov spaces E i may not be defined for all points y ∈ W x,r (x) , the flag V of fast subspaces (4.49) is defined for each E y = T y W x,r (x) ; moreover, the subspaces V i y depend Hölder continuously, and in fact C N −1,α , on y along W x [24,Theorem 6.3].
The key observation is that the notion of sub-resonance polynomial depends only on the fast flag V [20, Proposition 3.2], not on the individual Lyapunov spaces E i , and thus is well-defined for E y . Then the sub-bundle S (n) of subresonance polynomials of degree n is well-defined, invariant under D f , and Hölder continuous in y along W , and hence so is the factor bundle R (n) /S (n) . Then for each y ∈ W x,r (x) we can define H y using the construction in Theorem 2.3. Indeed, first we constructed the Taylor term of degree n using the contractionΦ on the bundle R (n) /S (n) from Lemma 4.4 with linear part estimated as Φ x (R) ε,x ≤ e λ+(n+1)ε · R ε, f x . Then Φ y , the corresponding map at y, is Hölder close to Φ x . We note that since W x,r (x) are C N,α embedded, the derivatives F (n) y = D (n) 0 F y of all orders n ≤ N depend α-Hölder continuously on y in W x,r (x) . In fact, the linear operator Φ y depends only on the first derivative. Using the Lyapunov norm at x as the reference norm, we obtain that Φ y is also a contraction with similar estimate for all y ∈ W x,r (x) provided that r (x) is sufficiently small. Since f k y ∈ W f k x,r ( f k x) by the contraction property of W x,r (x) , the closeness persists along the forward trajectory. This argument is similar to the proof of Lemma 4.1. Then we obtain that the operatorΦ y on the sequence space is also a contraction. Thus we can defineH (n) y as before using the unique fixed point in the space of sequences. The last step of the construction can be carried out similarly as it involves only the estimates of the derivatives on the full space E and does not depend on the splitting. This completes the proof of (i), (ii), (iii). REMARK 5.1. Any measurable choice of transversalsẼ i to V i −1 inside V i , with i = 2, . . . , , yields a transversalÑ (n) to S (n) inside R (n) . The latter gives a preferred choice of the lift. The fixed point of the contractionH (n) y depends Hölder continuously, and even smoothly by appropriate C r section theorem as in [20], on y along W x,r (x) if the same holds for the dataQ obtained in the previous step of the construction. To complete the inductive step we need a Hölder lift H (n) y to R (n) . If there is a consistent choice which is Hölder on the full leaves of W , then we can obtain a family {H x } which is Hölder along the leaves of W . In contrast to the uniform setting of [20] It is easy to see that all derivatives of a sub-resonance polynomial are subresonance polynomials. In particular, the derivative D y P x at any point y ∈ E x is sub-resonance and hence is block triangular. Thus it maps subspaces parallel to V k x to subspaces parallel to V k f x . Hence the foliation of E by subspaces parallel to V k x in E x is invariant under the extension P and hence coincides withW k by uniqueness of the fast foliation. It follows that for any x ∈ X and any y ∈ W x the diffeomorphism maps the fast flag of linear foliations of E x to that of E y . In particular, the same holds for its derivative D 0 G x,y = D x H y : E x → E y and we conclude that D 0 G x,y is block triangular and thus is a sub-resonance linear map. 5.3. Proof of (iv): Consistency of normal form coordinates. We need to show that the map G x,y in (5.1) is a sub-resonance polynomial map for all x ∈ X and y ∈ W x . It suffices to consider x ∈ Λ and, using invariance, we may assume that y ∈ W x is sufficiently close to x. First we note that G x,y (0) = H y (x) =:x ∈ E y and D 0 G x,y = D x H y . Since and hence G f n x, f n y • P n x = P n y • G x,y . (5.2) Now we consider the Taylor polynomial for G x,y : x,y (t ).
Our first goal is to show that all its terms are sub-resonance polynomials. We proved in Section 5.2 that the first derivative G (1) x,y = D x H y is a sub-resonance linear map.
Inductively, we assume that G (m) x,y has only sub-resonance terms for m = 1, . . ., k − 1 and show that the same holds for G (k) x,y . Suppose for the contrary that G (k) x,y is not a sub-resonance polynomial and consider order k terms in the Taylor polynomial at 0 ∈ E x for (5.2). The Taylor polynomial for P n x at 0 coincides with itself, P n x (t ) = d m=1 P (m) x (t ). We also consider the Taylor polynomial for P n y at G x,y (0) =x ∈ E y : All terms Q (m) are sub-resonance as the derivatives of a sub-resonance polynomial. Consider the Taylor polynomial for f n x, f n y (t ).
Now we obtain from (5.2) the coincidence of the terms up to degree N in x,y (t ) .
Since any composition of sub-resonance polynomials is again sub-resonance, the inductive assumption gives that all terms of order k in the above equation must be sub-resonance polynomials except for G (k) f n x, f n y P (1) x (t ) and Q (1) y G (k) x,y (t ) .
Multiplying these terms on the left by sub-resonance linear map D 0 G f n x, f n y −1 = D f n x H f n y −1 and using the fact that P (1) x = F n x = D f n | E x and Q (1) y = Dx P n y = D f n x H f n y • F n x • (D x H y ) −1 we obtain that the following maps from E x to E f n x agree modulo sub-resonance terms: x,y mod S x, f n x .
Since x, f n x ∈ Λ and thus the spaces E x and E f n x have Lyapunov splittings, we can decompose these polynomial maps into sub-resonance and non subresonance terms. Taking non sub-resonance terms on both sides we obtain the equality N f n x • F n x = F n x • N x (5.3) where N f n x and N x denote the non sub-resonance terms in D f n x H f n y −1 • G (k) f n x, f n y and (D x H y ) −1 •G (k) x,y respectively. If the latter had only sub-resonance terms then so would G (k) x,y , contradicting the assumption. Hence N x = 0. We decompose N x into components N x = (N 1 x , . . . , N x ) and let i be the largest index so that N i x = 0, i.e., there exists t ∈ E x so that z = N (t ) has non-zero component in E i y , which we denote by z i . Then by (3.3) we obtain For non sub-resonance N s we have χ i > s j χ j and hence (5.5) decays faster than (5.4). Since there are no sub-resonance terms in N i f n x , this contradicts (5.3) for large n if ε is sufficiently small since N f n x f n x is tempered. The latter follows from temperedness of G (k) f n x, f n y and the fact that D f n x H f n y is Hölder close to the identity and so the norm of its inverse is bounded in Lyapunov metric.
We conclude that for all x ∈ X and y ∈ W x the Taylor polynomial G x,y of G x,y contains only sub-resonance terms. Now we will show that G y,x coincides with its Taylor polynomial. Again it suffices to consider x ∈ Λ and y ∈ W x which is sufficiently close to x. In addition to (5.2) we have the same relation for their Taylor polynomials G f n y, f n x • P n y = P n x • G y,x . (5.6) Indeed, the two sides must have the same terms up to order N , but these are sub-resonance polynomials and thus have no terms of degree higher than d ≤ N .
Denoting ∆ n = G f n y, f n x −G f n y, f n x we obtain from (5.2) and (5.6) that ∆ n • P n y = P n x • G y,x − P n x • G y,x . (5.7) We denote ∆ = G y,x −G y,x : E y → E x and suppose that ∆ = 0. Let i be the largest index for which the i component of ∆ is nonzero. Then there exist arbitrarily small t ∈ E y such that the i component z i of z = ∆(t ) is nonzero. Since P n x is a sub-resonance polynomial, the nonlinear terms in its i component can depend only on j components of the input with j > i , which are the same for G y,x and G y,x by the choice of i . Thus the i component of the right side of (5.7) is F n x (z i ) since the linear part of P n x is F n x and it preserves the Lyapunov splitting. So by (3.3) we can estimate the right side of (5.7) P n x • G y,x − P n x • G y,x (t ) f n x ≥ F n x (z i ) f n x ≥ e n(χ i −ε) z i x ≥ e n(χ 1 −ε) z i x . (5.8) Now we estimate the left side of (5.7). Since G f n y, f n x is C N,α there exists C n (x) determined by G f n x, f n y C N,α such that ∆ n (t ) ≤ C n (x) · t N +α (5.9) for all sufficiently small t ∈ E f n x . To estimate P n y we note that D 0 P n y = F n y = D f n | E y and its norm for y close to x can be estimated using Lemma 4.1(3). Then P n y itself can be estimated as in that lemma: P n y (t ) ≤ K e n(χ +3ε) t for all sufficiently small t ∈ E y . Combining this with (5.9) we obtain ∆ n • P n y (t ) ≤ C n (x) · P n y (t ) N +α ≤ C n (x) · (K t ) N +α e n(N +α)(χ +3ε) . This contradicts (5.7) and (5.8) for large n if ε is sufficiently small. Indeed (N + α)χ < χ 1 while C n (x) is tempered and the Lyapunov norm satisfies u ≥ K (x)e −nε u f n x . Thus, ∆ = 0, i.e., the map G y,x coincides with its Taylor polynomial.
This completes the proof of Theorem 2.5.

Proof of Corollary 2.6.
If d = 1 then all sub-resonance polynomials are linear, the maps H y • H −1 x : E x → E y are affine, and the family {H x } x∈X is unique by part (2) of Theorem 2.3. If we identify W x with E x by H x , then H y for y ∈ W x becomes an affine map E x → T y E x with identity differential and H y (y) = 0. Thus it depends C N on y as the coordinate system H x is C N .