Exponential Mixing and Smooth Classification of Commuting Expanding Maps

We show that genuinely higher rank expanding actions of abelian semi-groups on compact manifolds are $C^{\infty}$-conjugate to affine actions on infra-nilmanifolds. This is based on the classification of expanding diffeomorphisms up to \holder conjugacy by Gromov and Shub, and is similar to recent work on smooth classification of higher rank Anosov actions on tori and nilmanifolds. To prove regularity of the conjugacy in the higher rank setting, we establish exponential mixing of solenoid actions induced from semi-group actions by nilmanifold endomorphisms, a result of independent interest. We then proceed similar to the case of higher rank Anosov actions.

1. Introduction 1.1. Smooth classification of higher rank expanding actions. Let G be a connected and simply connected nilpotent Lie group. Let End(G) and Aut(G) denote the semi-group of endomorphisms of G and group of automorphisms of G respectively. Let Γ ⊂ G be a discrete subgroup such that the quotient space Γ \ G is compact. Then we call the compact manifold Γ \ G a nilmanifold. A compact manifold M is called an infra-nilmanifold if it admits a finite nilmanifold covering Γ \ G. If A ∈ G ⋊ End(G) satisfies that A(Γ) ⊂ Γ, then A induces a smooth map on Γ \ G, a so-called affine nilendomorphism of Γ \ G. The Aut(G) component of A is called the linear part of A. A map on M is called an affine infra-nilendomorphism if it lifts to an affine nilendomorphism of a finite nilmanifold covering Γ \ G.
Let M be a compact smooth manifold endowed with a Riemannian metric · . We say a smooth map τ : M → M is expanding if there exists a constant c > 1 such that for any v in the tangent bundle TM, we have It is natural to ask if one can classify all expanding maps, up to conjugacy. Based on the work of Shub [Shu70], Gromov [Gro81] found the best possible answer to this question: he proved that every expanding map on a compact manifold is topologically conjugate to an affine infra-nil endomorphism. In other words, for a finite cover M of M, there exists a compact nilmanifold Γ \ G, a homeomorphism φ : M → Γ \ G and an affine nilendomorphism A on Γ \ G such that the following diagram commutes: Moreover, φ is bi-Hölder. The linear part of A = φ • τ • φ −1 is given by the induced action of τ on the fundamental group Γ of M. We remark that any expanding map has at least one fixed point p ∈ M [Shu69, Theorem 1] so that the induced map on π 1 (M, p) is well defined. Finally, if M = Γ \ G then φ can be chosen to be homotopic to the identity Id. We remark that Dekimpe clarified some of the algebraic issues with the notions of affine nilendomorphisms and the proof of the Gromov and Shub classification result [Dek12].
We remark that in dimensions at least 5, passing to a finite cover M of M if necessary, we can further assume that M is actually diffeomorphic to a nilmanifold Γ \ G . Indeed, exotic differentiable structures on nilmanifolds always become standard on a finite cover -cf. the Appendix by J. Davis in [FKS13].
In general, an expanding map is not C 1 -conjugate to an affine infra-nilendomorphism. One can construct simple examples by perturbing a suitable affine example locally at a fixed point, changing the derivative at the fixed point. Furthermore, Farrell and Jones constructed expanding maps on tori with exotic differentiable structures [FJ78]. For higher rank Z k + actions (k ≥ 2), the situation changes dramatically. Note that higher rank is needed as one can always take product actions of individual non-algebraic expanding maps. We also have to avoid finite symmetries that disguise a product of rank one actions. Hence we make the following definition.
Definition 1.1. Let ρ be a C ∞ Z k + (k ≥ 2) action on a manifold M. We call ρ genuinely higher rank if for all finite index sub-semigroups Z of Z k , no continuous quotient of any finite extension of the Z-action factors through a finite extension of a Z + action.
It is easy to show that after passing to a finite index sub-semigroup and a finite cover, a Z k + action ρ with an expanding map is C 0 conjugate to an affine action ρ l on a nilmanifold via the conjugacy φ we get from a single expanding map, cf. Lemma 2.3. We will use genuine higher rank in this paper to show that the Hölder conjugacy φ is actually C ∞ .
We summarize the main result of this paper as follows: Theorem 1.2. Let ρ be a C ∞ Z k + (k ≥ 2) action on a compact manifold M. Suppose that ρ is genuinely higher rank and contains an expanding map ρ(a), for some a ∈ Z k + . Then M is diffeomorphic to an infra-nilmanifold M and ρ is C ∞ conjugate to a Z k + action ρ l on M by affine nil-endomorphisms.
1.2. Related results. Rigidity of higher rank actions on compact manifolds has been studied in different contexts. For Anosov actions, Rodriguez Hertz [RH07] classified Z k , k ≥ 2, actions on tori containing one Anosov element and satisfying certain additional conditions. His work required that the rank k of the action is comparable to the dimension of the torus. Kalinin, Fisher and Spatzier [FKS13] proved that if a Z k action α on a torus or nilmanifold is genuinely higher rank and contains "many" Anosov elements, then it is C ∞ conjugate to affine actions. Later Rodriguez Hertz and Wang [HW14] obtained the optimal result for Anosov actions on nilmanifolds by showing that existence of a single Anosov element implies existence of "many" Anosov elements. Here by saying "many" Anosov elements we mean that there is an Anosov element in each so-called Weyl chamber of Lyapunov exponents of α. We refer the reader to the introduction of [FKS11] and to [Spa16] for brief surveys of results and methods in the classification of higher rank Anosov actions.
1.3. Exponential mixing of nilendomorphisms, expanding maps and their solenoids. We will apply techniques similar to those in [FKS13] and [HW14] to show the conjugacy φ is C ∞ . The first difficulty here is that these actions are not invertible, so the Weyl chambers of the Lyapunov exponents of ρ are only indirectly defined, and not in terms of the dynamical behavior (slow exponential growth) of actual elements close to the respective Weyl chamber walls. To overcome this difficulty, we shall extend the action ρ to the solenoid S(M) of M, defined below in detail in §2.4. Basically one wants to invert a covering map from a space to itself by considering the space of all possible orbits on which one has a tautological inverse. As the future orbits are well defined this becomes the space of pasts. One can easily generalize this construction to semigroups generated by commuting covering maps. As discussed in [KS96], the solenoid has a completely algebraic description in terms of a p-adification of the space. As it turns out, we will actually never need the original notion of the space of pasts and will work directly with the algebraic definition.
The notion of solenoid was used by Williams [Wil74] to study expanding attractors, and also by Katok and Spatzier in [KS96] to prove measure rigidity statements. This makes ρ (and ρ l ) a partially hyperbolic Z k action on S(M). According to [KS96], the Lyapunov exponents and Weyl chambers of both ρ and ρ l are well defined. This allows us to proceed as in the first step of the proof of global rigidity for Anosov actions.
The crucial next step in the proof of Theorem 1.2 is to prove an exponential mixing result for the action of ρ on the solenoid S(M). This is the main novelty in this paper, and has independent interest. We actually prove such mixing quite generally for semigroups of endomorphisms of nilmanifolds. Indeed, passing to a finite cover, we may identify M as a nilmanifold Γ \ G.
By the theory of nilpotent Lie groups, there exists a nilpotent algebraic group N over Q such that the nilpotent Lie group G = N(R) and Γ = N(Z). We will see later that the solenoid S(M) of M can be identified with an S-adic nilmanifold N(Z) \ N(R) × p∈S N(Z p ) for a finite subset S of primes. Let µ denote the probability measure on S(M) induced by the Haar measure on N(R) × p∈S N(Z p ). Note that ρ l is a Z k action on S(M) by nilendomorphisms.
Our main exponential mixing result is the following: respectively. Let · θ denote the θ-Hölder norm. Let ρ l denote a Z k action on S(M) by nilendormorphisms. Suppose that ρ l (a) acts ergodically on S(M) for every a ∈ Z k . Then there exist constants a 1 > 0 and η ′ > 0 depending on θ, such that for any a ∈ Z k , any f ∈ C θ (M), regarded as a function on S(M), and any g ∈ C θ (S(M)), we have where · denotes the supremum norm on Z k .
In the case of Z k actions by ergodic automorphisms on nilmanifolds, exponential mixing was established by Gorodnik and Spatzier [GS14] and [GS15], based on the work of Green and Tao [GT12,GT14]. In our case, the structure of S(M) is essentially different from a real nilmanifold. Since S(M) is an S-adic nilmanifold, p-adic analysis will play an important role in the proof. We will prove the theorem in §3.
In §3, we will show that if the action ρ l is genuinely higher rank, then there exists a subgroup Σ ⊂ Z k isomorphic to Z 2 such that every element in Σ acts ergodically on S(M). Therefore, under the genuinely higher rank hypothesis, we can choose a subgroup Σ ∼ = Z 2 of Z k such that the above exponential mixing result holds for Σ. Note that ρ is conjugate to ρ l via a Hölder homeomorphism φ, if Theorem 1.3 holds for ρ l and µ, then it will also hold for ρ andμ := φ −1 * µ. Therefore, Theorem 1.3 implies the following corollary which is crucial to establish the smoothness of φ: ) and · θ be as above. Suppose the action ρ is genuinely higher rank, then there exists a subgroup Σ of Z k isomorphic to Z 2 and constants a 1 > 0 and η ′ > 0, such that the following holds: for any f ∈ C θ (M), regarded as functions on S(M), any g ∈ C θ (S(M)) and any a ∈ Σ, Let us emphasize that our exponential mixing results are different from exponential mixing for just the semi-group. Indeed we can go to infinity in the solenoid in a variety of ways, e.g. by going back far in the past and returning to the present. Exponential mixing for just the future of an expanding map follows from the standard techniques of Markov sections and transfer operators. These techniques however are not able to handle our case. In addition, we allow for quite general semi-groups of endomorphisms of nilmanifolds, not just expanding and hyperbolic ones. Quite generally, there are now several techniques available to prove exponential mixing: Fourier analysis, representation theory, Markov systems and transfer operators especially in combination with contact structures. However, for one reason or another, none of these work generally for semi-groups of nilmanifold endomorphisms.
Finally, let us note three more corollaries of exponential mixing, similar to results in [GS14]. We refer there for a more extensive discussion of ideas and background. The proofs are identical, and we will not discuss them here in detail.
First consider a single ergodic nilendomorphism α on a nilmanifold X. For a function f : X → R, we set and for simplicity assume that X f dµ = 0.
One says that the sequence f • α n satisfies the central limit theorem if for some σ > 0, n −1/2 S n (f, ·) converges in distribution to the normal law with mean 0 and variance σ 2 . More generally, the sequence f • α n satisfies the central limit theorem for subsequences if there exists σ > 0 such that for every increasing sequence of measurable functions k n (x) taking values in N such that for almost all x, lim n→∞ variance σ 2 /c. We define S t (f, x) for all t ≥ 0 by linear interpolation of its values at integral points. The sequence f • α n satisfies the Donsker invariance principle if there exists σ > 0 such that the sequence of random functions (nσ 2 ) −1/2 S nt (f, ·) ∈ C([0, 1]) converges in distribution to the standard Brownian motion in C([0, 1]). The sequence f • α n satisfies the Strassen invariance principle if there exists σ > 0 such that for almost every x, the sequence of functions (2nσ 2 log log n) −1/2 S nt (f, x) is relatively compact in C([0, 1]) and its limit set is precisely the set of absolutely continuous functions g on [0, 1] such that g(0) = 0 and 1 0 g ′ (t) 2 dt ≤ 1. This is a strong version of the law of the iterated logarithm. Corollary 1.5. Let α be an ergodic endomorphism of a compact nilmanifold X, and let f be a Hölder function on X which has zero integral.
(1) If f is not a measurable coboundary. then the sequence {f • α n } satisfies the central limit theorem, the central limit theorem of subsequences, and the Donsker and Strassen invariance principles.
Livsic proved for Anosov diffeomorphisms that a measurable coboundary for a smooth function is automatically smooth. Veech discussed this issue for ergodic toral automorphisms in [Vee86] using sophisticated Fourier analysis. He also gave counter examples in the C 1category. Gorodnik and Spatzier proved the nilautomorphism version of this result in [GS14].
Corollary 1.6. Let M be as above, and let α : M → M be a nilendomorphism, ergodic w.r.t.
For the proof we just extend f and g to functions on the solenoid. Note that they are independent of the p-adic direction. Hence we can use the central limit theorem, Corollary 1.5, and exponential mixing as in [GS14].
In our last corollary, we consider genuinely higher rank actions. Again the proof is identical to [GS14].
1.4. Organization of the paper. Before §6, we always assume that dim M ≥ 5. In §2, we recall the result of Gromov and Shub on bi-Hölder conjugacies between expanding maps and their linearizations, reduce the main theorem to the case of actions Hölder conjugate to nilmanifold endomorphisms, recall the structures of solenoids, and discuss the Lyapunov exponents of the extended Z k -actions of ρ and ρ l on the solenoid S(M). In §3, we will prove Theorem 1.3 and Corollary 1.4. In §4, we apply Corollary 1.4 and the techniques developed in [HW14] to show that every coarse Lyapunov distribution of ρ admits a Hölder foliation with C ∞ leaves. This result is crucial for applying the techniques developed in [FKS13]. In §5, we combine the exponential mixing result, the result proved in §4 and the techniques developed in [FKS13] Acknowledgements. The first author thanks David Fisher and Boris Kalinin for early discussions on these matters. Both authors thank MSRI where they collaborated on this work in Spring 2015.

Preliminaries on expanding maps and solenoids
In this section we review and refine some background material on expanding maps and their conjugacies and centralizers. This allows us to reduce the main theorem to the case of actions Hölder conjugate to nilmanifold endomorphisms. We then define solenoid extensions and also Lyapunov exponents.
2.1. Gromov's conjugacy theorem on expanding maps. We recall the result of Shub [Shu70] and Gromov [Gro81]: Moreover, the covering mapτ l of τ l on G is the automorphism of G induced by the map τ * l : Γ → Γ. 2.2. Reduction to nilmanifolds. We discuss several basic properties of expanding maps and their commuting maps. We use these to reduce the proof of our main result to the case when the semi-group has a fixed point and when the linearization of the action of the semigroup action is on a nilmanifold rather than an infra-nilmanifold.
Proposition 2.2. Suppose the semigroup Z k + acts by ρ on a compact manifold M with an expanding map ρ(a). Then the action is bi-Hölder -conjugate to an action by affine infranilendomorphisms on an infra-nilmanifold M.
Proof. By Theorem 2.1, ρ(a) is C 0 -conjugate to an infra-nilendomorphism α by a homeomorphism φ. Any such conjugacy is bi-Hölder as is well-known. Lemma 2.3. Let ρ be a Z k + action on a compact manifold M with an expanding map ρ(a) which is Hölder conjugate to an affine action ρ l on an infra-nilmanifold M. Then there is a sub-semigroup Σ + of finite index in Z k + which acts on a finite cover M of M by nilendomorphisms covering the restriction of the original action to Σ + .
Proof. It suffices to prove that the linearization ρ l lifts since ρ and ρ l are C 0 -conjugate. By [Shu69, Theorem 1], ρ l (a) has a fixed point p ∈ M. Moreover, the set of fixed points of ρ l (a) is finite as ρ l (a) is expanding and M is compact. Hence there is a sub-semigroup Σ + of finite index in Z k + which also fixes p. As Σ + is finitely generated, we can find a finite cover M of M such that all elements of Σ + lift to M as affine nil-endomorphisms. Furthermore, as Σ + is finitely generated, we can pick lifts of generators of Σ + that all fix a given point p in the pre-image of p. Since these lifts are determined by their derivative action at p, we see that all the lifts of the generators of Σ + commute. Thus they define a lift of the action of Σ + to M which covers the Σ + action on M, as desired.
Under our higher rank assumptions on the semi-group actions, we will show that the covering map ρ of ρ is C ∞ -conjugate to ρ l by φ. This implies that ρ is C ∞ -conjugate to ρ l , as desired. Thus we can always work with the finite covers M and M and actions ρ and ρ l which have a common fixed point.
If the dimension dim(M) ≥ 5, we can make further reductions. Indeed, by Davis' work on exotic differentiable structures on nilmanifolds [FKS13, Theorem A.0.1, Appendix] and passing to a finite cover, we may assume the conjugacy is isotopic to a diffeomorphism ψ : M → Γ \ G. Then we can conjugate ρ by ψ to a smooth action on the nilmanifold Γ \ G. We will deal separately with the case dim(M) ≤ 4 in §6.
These reductions allows us to make the following hypotheses throughout except in §6.
2.3. Standing Assumption. Henceforth, M = Γ \ G will denote a compact nilmanifold and ρ will denote a C ∞ genuinely higher rank action of a semigroup Z k + with k ≥ 2 on M such that • ρ(a) is an expanding map for some a ∈ Z k + , • ρ is Hölder conjugate to an action of Z k + by affine nil-endomorphisms on M, • ρ has a common fixed point. Note that the linearization of ρ is given by the induced action on the fundamental group Γ, thanks to existence of a common fixed point.
2.4. Solenoids and extended actions. We will define the solenoid S(M) of M, extend ρ and ρ l to Z k actions on S(M), and define Lyapunov exponents of ρ and ρ l on S(M), following [KS96].
First recall Mal'cev's theorem from the theory of nilpotent Lie groups (see [Rag72,GT12,CG90], for example) that any lattice Γ of a nilpotent Lie group G must be arithmetic, i.e., there is a simply connected nilpotent algebraic group N over Q such that G = N(R) and Γ = N(Z). Then M = N(Z) \ N(R).
By [KS96], the abstract solenoid S(M, ρ) of M is naturally defined as follows: (2.1) S(M, ρ) := (z n ) ∈ M Z k ≤0 : z n+a = ρ(a)z n : for all a ∈ Z k + . In other words, we attach each point on M with all possible pasts with respect to all a ∈ Z k + . On this space, one can easily define a Z k action which extends the original Z k + action ρ (see [KS96,§3] for details). The disadvantage is that it is hard to do concrete analysis and calculation with this definition. Therefore, we will give another definition and stick with it throughout the paper. Given a ∈ Z k + , ρ(a) can be extended to a homeomorphism from N(R) to itself (cf. [Shu69] and [Shu70]). For a fixed z ∈ M, the preimage of z with respect to ρ(a) is {ρ −1 (a)(nz) : n ∈ N(Z)}. Therefore, to attach z with a past with respect to ρ(a) is the same as to attach z with an element n ∈ N(Z). Moreover, if ρ −1 l (a)(n −1 1 n 2 ) ∈ N(Z), then ρ −1 (a)(n 1 z) = ρ −1 (a)(n 2 z). Taking this congruence condition into account and passing to the inverse limit for all possible a ∈ Z k + , we will attach each z ∈ M with several p-adic components ξ p ∈ N(Z p ). This discussion brings us the new definition of the solenoid S(M) of M: Definition 2.4 (see [KS96,Appendix]). Let where N(Z) acts on N(R) × p prime N(Z p ) diagonally, and in the product, p runs over all primes. For each prime number p, we define where · p denotes the p-adic norm. Then S(M) is defined as follows: Equip S(M) with product structure.
(1) M p = N(Z p ) for all but finitely many p's. Therefore, there exists a finite set S of primes such that S(M) = N(Z) \ N(R) × p∈S N(Z p )/M p . (2) Generalizing the argument from [KS96, Lemma 8.2], we see that every quotient N(Z p )/M p is torsion free. (3) For each prime p, we denote by ν p the Haar measure on N(Z p ). By normalization, we assume that ν p (N(Z p )) = 1. Let ν denote the Haar measure on N(R) and also the induced measure on N(Z) \ N(R). By normalization, we assume that ν(N(Z) \ N(R)) = 1. Then the product measure ν × p∈S ν p induces a probability measure on the solenoid S(M), which we denote by µ. Defineν := φ −1 * (ν) andμ :=ν × p∈S ν p , thenν is absolutely continuous with respect to ν (since φ is Hölder ), andμ is preserved by the action of ρ. Moreover, for any a ∈ Z k , the action of ρ(a) is ergodic with respect toμ if and only if ρ l (a) is ergodic with respect to µ. (4) The definition above depends on the homotopy class of the action ρ as the M p 's do.
Since throughout this paper we fix the homotopy type, i.e., the induced action ρ * on N(Z), we may regard S(M) as a fixed space. (5) We note that that an infra-nilmanifold M can be regarded as a finite index factor of a homogeneous space N(Z) \ N(R) where N denotes a nilpotent Q-group. Once we define the solenoid S(N(Z) \ N(R)) of N(Z) \ N(R), the solenoid of M is just the quotient of S(N(Z) \ N(R)) by a finite group action. Thus solenoids for infranilmanifolds also have an explicit description.
For nilpotent algebraic group N, we have the following version of Chinese remainder theorem: Lemma 2.6. Given a finite subset of primes S, ξ p ∈ N(Z p ) for p ∈ S and l p ∈ Z + for p ∈ S, there exists n ∈ N(Z) such that Proof. We prove the statement by induction on the nilpotency degree of N.
If N is abelian, this is just the Chinese remainder theorem as the action n −1 ξ p is a linear expression.
Suppose the statement holds if the nilpotency degree is < d. Now we assume that the nilpotency degree of N is d. Take a nilpotent subgroup N ′ of N such that N ′ \ N is abelian and the nilpotency degree of N ′ is d−1. Then considering the image of ξ p on N ′ (Z p )\N(Z p ), and applying the Chinese remainder theorem, we have that there exists n 1 ∈ N(Z) such that , for all p ∈ S. Then n = n 1 n 2 satisfies our condition.
This proves the lemma.
The informal discussion before Definition 2.4 may help with the next result and its proof.
Proposition 2.7. ρ and ρ l can be extended to Z k actions on S(M).
Proof. For a ∈ Z k + , the action ρ(a) on S(M) can be naturally defined as follows: for z = (z, (ξ p ) p∈S ) ∈ S(M), ρ(a)(z) := (ρ(a)z, (ρ * (a)ξ p ) p∈S ). To extend ρ to a Z k action on S(M), it suffices to define the inverse of ρ(a) for each a ∈ Z k + . For a ∈ Z k + , ρ(a) can be extended to a homeomorphism of the universal covering N(R) of M to itself (cf. [Shu69] and [Shu70]). Therefore ρ −1 (a) is well defined on N(R). Recall that ρ * (a) agrees with ρ l (a) when restricted to N(Z). Then we define ρ −1 (a) as follows: for z = (z, (ξ p ) p∈S ) ∈ S(M), we may pick l p for each p ∈ S such that ρ −1 * (a)(N(p lp Z p )) ⊂ N(Z p ). By Lemma 2.6, we can find n ∈ N(Z) such that n −1 ξ p ≡ 0(mod p lp ). Let us write (z, (ξ p ) p∈S ) = (n −1 z, (n −1 ξ p ) p∈S ), then ρ −1 * (a) is well defined on each p-adic component. Thus, we can define . The same extension works for ρ l as well.
The conjugacy φ : M → M can be extended to a homeomorphism φ : S(M) → S(M) as follows: on the real component, it is φ, and on p-adic components, it is the identity map. It is easy to see that φ conjugates the extended actions ρ and ρ l on S(M).

Lyapunov exponents and coarse Lyapunov decomposition.
We need the following notation to define Lyapunov exponents of ρ and ρ l . The smoothness of a map defined on S(M) is defined as follows: Definition 2.9. We say a continuous map defined on S(M) is C ∞ if it is C ∞ when restricted to every manifold slice.
Remark 2.10. It is easy to see that for each a ∈ Z k , ρ(a) is C ∞ . Indeed, ρ(a) maps every manifold slice M(z) to another manifold slice M(z ′ ), and when restricted to the manifold slice, ρ(a) is smooth (because the map only depends on the real component). The same holds for ρ l .
Definition 2.11. Since ρ l acts on S(M) by affine nilendomorphisms, it naturally induces a Z k action on the nilpotent Lie group N(R) by automorphisms, which we still denote by ρ l . Let Dρ l denote the action on n(R) induced by ρ l . A character χ ∈ (R k ) * is called a real Lyapunov exponent of ρ l if the real Lyapunov subspace corresponding to χ defined as follows: is nontrivial. Let Dρ l denote the action on n(Q p ) induced by ρ l . Note that ρ induces the same action on p-adic components, so Dρ l is also the action induced by ρ. A character χ ∈ (R k ) * is called a p-adic Lyapunov exponent of ρ l (and also ρ) if the p-adic Lyapunov subspace corresponding to χ defined as follows: is nontrivial. For z ∈ S(M), let T z (M) denote the tangent space of the manifold slice passing through z based at z. A character χ ∈ (R k ) * is called a real Lyapunov exponent of ρ if forμ-a.e. z ∈ S(M), the real Lyapunov distribution corresponding to χ defined as follows: Notation 2.12. To distinguish the Lyapunov exponents of ρ and ρ l , later in this paper, we denote Lyapunov exponents of ρ by χ, χ 1 , χ 2 , . . . , and denote Lyapunov exponents of ρ l by χ l , χ l 1 , χ l 2 , . . . . Since p-adic Lyapunov exponents of ρ and ρ l coincide, we do not distinguish the above two notions in p-adic directions. We say a Lyapunov exponent χ (or χ l ) is of type K (K = R or Q p ) if the corresponding Lyapunov distribution (or Lyapunov subspace) is in n(K). For K = R or Q p , let T (K) denote the set of type K.
(1) One can prove that (see [HW14]) (2) Since ρ contains an expanding element, it is ergodic with respect toμ. By Multiplicative Ergodic Theorem, there exist finitely many Lyapunov exponents χ's of type R, a set of fullμ-measure set P ⊂ S(M), and a ρ-invariant measurable splitting of the bundle E(S(M)) : We refer to [KK01], [KS06], and [FKS13] for details.
Definition 2.14. For a Lyapunov exponent χ l of ρ l , we define the coarse Lyapunov subspace associated with χ l as follows: The corrosponding decomposition is called the coarse Lyapunov decomposition of the real component of S(M).
Similarly, for a Lyapunov exponent χ of ρ, we define the coarse Lyapunov distribution associated with χ as follows: Remark 2.15. One can prove that [σ χ l 1 , σ χ l 2 ] ⊂ σ χ l 1 +χ l 2 if χ l 1 and χ l 2 are of the same type, cf. [HW14, Lemma 2.3]. Therefore, each σ [χ l ] (χ l can be real or p-adic) is a Lie subalgebra (of n(R) or n(Q p )). Let V [χ l ] denote the corresponding Lie subgroup (of n(R) or n(Q p )), which will be called a coarse Lyapunov subgroup.
Definition 2.16. We define a Weyl chamber of ρ (respectively ρ l ) to be a connected component of R k \ χ kerχ (respectively R k \ χ l kerχ l ), and a real Weyl chamber to be a connected component of Thanks to the existence of the conjugacy φ, we have the following correspondence between coarse Lyapunov distributions of ρ and coarse Lyapunov subgroups of ρ l .  2.6. The cohomological equation I. Recall that ρ(a) and ρ l (a) are C ∞ for any a ∈ Z k (see Definition 2.9 and Remark 2.10).
Because the conjugacy φ : S(M) → S(M) is homotopic to Id, for any a ∈ Z k , ρ l (a) −1 ρ(a) is homotopic to Id. We write where Q a (z) ∈ N(R) × p∈S N(Z p ). It is easy to see that Q a is C ∞ since both ρ(a) and ρ l (a) are. Since ρ(a) and ρ l (a) are identical on p-adic components, Q a (z) is only nontrivial on the real component, so it can be regarded as a C ∞ map Q a : S(M) → N(R). In fact, for each p ∈ S, let l p (a) ≥ 0 be an integer such that ρ l (a)( . In other words, Q a can be treated as a C ∞ map defined on the finite cover N(nZ) \ N(R) of M where n = p∈S p lp(a) . Writing φ(z) = zh(z), then h(z) is also trivial on every p-adic component, and moreover, it only depends on the real component. Therefore, we can regard h as a map from M to N(R). Since φ conjugates ρ to ρ l , we have that for any a ∈ Z k and any z ∈ S(M), the following holds: In [HW14], the following lemma is proved. We follow its proof closely.
Proof. Let us first prove the following claim: If a map f : S(M) → N(R) is continuous on each manifold slice and satisfies that zf (z) = z for all z ∈ S(M), then there exists γ 0 ∈ N(Z) ∩ Z(N(R)) (where Z(N(R)) denotes the center of N(R)) such that f (z) = γ 0 for all z ∈ S(M).
In fact, f can be lifted to a function f : . This proves the claim.
From (2.2), we have that is continuous on each manifold slice. By the above claim, we conclude that ). This completes the proof.
Definition 2.19. Let F be a foliation of M with of C ∞ leaves. We consider derivatives of order k of functions or distributions f along F , which we denote by ∂ k F f . For θ ∈ (0, 1), let C ∞,θ, * F denote the space of distributions f on M such that all partial derivatives of f of any order along F exist as distributions on the space of θ-Hölder functions. Let C ∞,θ F denote the space of θ-Hölder functions f on M such that all partial derivatives of f of any order along F are θ-Hölder .
Notation 2.20. Throughout this paper, we will denote coarse Lyapunov subgroups cor- Outline of the proof. We briefly describe the basic idea (developed in [FKS13] and [HW14]) to establish the smoothness of h.
We first establish the smoothness when M is a torus.
To this end we first show that for every coarse Lyapunov exponent [χ] of ρ, the corresponding coarse Lyapunov distribution E [χ] admits a Hölder foliation consisting of C ∞ leaves. In order to show this, we make use of our exponential mixing result for solenoids (Corollary 1.4) proved in §3 and the techniques and results developed by Rodriguez Hertz and Wang [HW14].
Let V be a coarse Lyapunov subgroup corresponding to [χ l ]. We define h V (z) to be the projection of h(z) on V . We want to show that for any coarse Lyapunov subgroup V , From (2.2) we get the corresponding equation for h V (z): By iterating (2.3), one finds the solution by the following formal series: We will apply the exponential mixing result again to prove that for any coarse Lyapunov foliation V 1 , h V ∈ C ∞,θ, * V 1 . In the case of Z k actions, exponential mixing was established by Fisher, Kalinin and Spatzier [FKS13] for tori and by Gorodnik and Spatzier [GS14], [GS15] for nilmanifolds.
By variations of results of Rauch and Taylor in [RT05] proved by Fisher, Kalinin and Spatzier [FKS13], and by Rodriguez Hertz and Wang [HW14], h V ∈ C ∞,θ, * V 1 for all possible coarse Lyapunov foliations V 1 will imply that h V is C ∞ . Then it follows that h is C ∞ since h can be written as the sum of h V 's for all coarse Lyapunov subgroups V . This will prove the smoothness when M is a torus.
For the general case, we follow the approach of Margulis and Qian [MQ01]. We consider the derived series of N: and prove the smoothness of h by induction on k.
3. Exponential mixing for extended Z k -actions on solenoids In this section, we will prove Theorem 1.3 and Corollary 1.4. As we discussed in §2, Corollary 1.4 is crucial to establish the smoothness of φ.
3.1. Preparation for the proof. We need some preparation before proving the theorem.
Then M 0 is a torus and M is a bundle over M 0 with M ′ fibers. We call M 0 the maximal torus factor of M.
We will need the following effective equidistribution result for box maps on nilmanifolds.
Theorem 3.2 (See [GS14, Theorem 2.1]). Let w 1 , w 2 , . . . , w r be r linearly independent vectors in the Lie algebra n(R). Let v ∈ n(R) be a fixed vector. We define the box map as follows: There exist constants L 1 , L 2 > 0 such that for every δ ∈ (0, 1/2), every u ∈ n(R), every and every box map ι : B → n(R), one of the following holds: Another important result we will need is the following effective equidistribution result for polynomial orbits on nilmanifolds, proved by Green and Tao [GT12].
be a polynomial map. For i = 1, 2, . . . , r, let e i ∈ [N] denote the vector with 1 on the ith component and 0 on other components, and let ∂ i π(p(n)) := π(p(n)) − π(p(n − e i )). Then there exist constants L 1 , L 2 > 0 such that for every δ > 0 and every f ∈ C θ (M), one of the following holds: Remark 3.4.
(1) The statement in [GT12] is different from the above theorem. For example, the function is assumed to be Lipschitz continuous, and case AA.2 above is stated differently. The statement above follows the one stated and applied in [GS14]. To get the above modified version from the statement in [GT12] , one needs to approximate Hölder functions by Lipschitz functions, keeping control of all the desired estimates. We refer to [GS14] for details.
(2) The non-effective version of the equidistribution of polynomial orbits in nilmanifolds is proved by Leibman [Lei05].
We introduce some notation.
Lemma 3.7 (see [GS14, Lemma 3.3]). Let V ⊂ R l be a subspace defined over Q ∩ R such that V is not contained in any proper subspace defined over Q. Then there exists w ∈ V ∩ Q l whose coordinates are real numbers linearly independent over Q.
For p-adic vector spaces, we have the following similar result: Lemma 3.8. For any prime p, let Q al p ⊂ Q p denote the field of p-adic algebraic numbers. Let V ⊂ Q l p be a subspace defined over Q al p such that V is not contained in any proper subspace defined over Q. Then there exists w ∈ V ∩ (Q al p ) l whose coordinates are linearly independent over Q.
Proof. The proof is the same as that of Lemma 3.7 (see [FKS13,Lemma 3.3]). We will include the proof for completeness.
Since V is defined over Q al p , we can choose a basis {u i : i = 1, 2, . . . , s 1 } of V with coordinates in Q al p . Let K ⊂ Q al p denote the field generated by their coordinates. Then K is a finite extension of Q in Q p . Then we may choose α 1 , α 2 , . . . , α s 1 ∈ Q al p which are linearly independent over K. Let w := s 1 i=1 α i u i . Let us denote u i = (u i1 , u i2 , . . . , u il ) for i = 1, 2, . . . , s 1 . Then for any z = (z 1 , . . . z l ) ∈ Q l , we have that Since {α 1 , . . . , α s 1 } are linearly independent over K, we have that z, w = 0 unless l j=1 z j u ij = 0, for all i = 1, 2, . . . , s 1 , i.e., z, W = 0. Since we assume that V is not defined over Q, we have proved that the coordinates of w are linearly independent over Q.
This completes the proof.
By [BG06, Theorem 7.3.2], a vector w ∈ R l whose coordinates are algebraic numbers that are linearly independent over Q is (c, L)-Diophantine for some c, L > 0.
For p-adic vectors, we have the same result: Lemma 3.9. Let w ∈ (Q al p ) l such that its coordinates are linearly independent over Q. Then w is (c, L)-Diophantine for some constants c, L > 0.
Therefore, the vector w we get from Lemma 3.7 (or Lemma 3.8) is (c, L)-Diophantine for some constants c, L > 0.
3.2. Ergodicity of the action. Before proving Theorem 1.3, let us first study ergodicity of the Z k action ρ l on the solenoid S(M). It is proved in [Sta99] and [HW14] that a Z k action on a nilmanifold M by automorphisms is genuinely higher rank if and only if there exists a Z 2 subgroup of Z k all of whose nontrivial elements act ergodically. We will prove a similar result for Z k actions on solenoids. Then we will prove Corollary 1.4 assuming Theorem 1.3.  We next claim that every nontrivial ρ l (a) does not admit any nontrivial proper invariant subtori of M. Otherwise, take a nontrivial proper ρ l (a) invariant subtorus with minimal dimension, say M ′ . Then M ′ corresponds to a minimal ρ l (a)-invariant rational subspace V ′ of R dim M . For any b ∈ Z k , it is easy to see that V ′ ∩ ρ l (b)V ′ is also a ρ l (a)-invariant subspace defined over Q. Since V ′ is assumed to be minimal, we have Thus there are only finitely many possible Lie algebras V " that ρ l (b)V ′ can be. This implies that there exists a subgroup Γ of Z k of finite index such that V ′ is ρ l (Γ)-invariant. This contradicts the assumption on irreducibility. Therefore, no ρ l (a) leaves any nontrivial proper subtori invariant.
Note that the dual space of S( Lemma 3.12. Let ρ l : Z k S(M) be a genuinely higher rank action as above. Then there exists a ∈ Z k such that ρ l (a) is ergodic.
Proof. For the same reason as above, we may assume that M is a torus Z dim M \ R dim M . By [Sta99, Corollary 6], we may assume that every ρ l (a) ∈ GL(dim M, Q) is semisimple.
By passing to a subgroup of Z k of finite index, we can decompose M into almost direct product of ρ l -invariant irreducible subtori: For each i = 1, . . . , s, ρ l (a) acts either ergodically or trivially on M i . Our goal is to find a ∈ Z k such that ρ l (a) is not trivial on each M i .
For contradiction, we suppose that every ρ l (a) acts trivially on some M i . Define then every F i is a subgroup of Z k and Z k = s i=1 F i . This implies that some F i is a subgroup of Z k of finite index, which contradicts our higher rank assumption (since the restriction of ρ l on M i is essentially trivial).
Definition 3.13. We call an integer triple (k, l, m) ∈ Z 3 primitive if k, l, m do not have nontrivial divisor.
Lemma 3.14. Let semisimple elements A, B, C ∈ GL(d, Q) commute and act on S(Z d \R d ).
Suppose for any i, j ∈ Z, A i B j is ergodic unless i = j = 0. Then there exist at most finitely many primitive triples (k, l, m) ∈ Z 3 such that A k B l C m is not ergodic.
Let S = A k B l C m be a non-ergodic element. Then for some r ∈ Z, Fix(S r ) is nontrivial. Let V = Fix(S r ). Then V is rational and invariant under the action of A, B and C.
If V = R d , then there exists only one primitive non-ergodic triple. In fact, for any primitive non-ergodic triple (k 1 , l 1 , m 1 ), there exist r 1 ∈ Z and v 1 ∈ R d \ {0}, such that A r 1 k 1 B r 1 l 1 C r 1 m 1 v 1 = v 1 . Since A rk B rl C rm v 1 = v 1 and A i B j is ergodic for any (i, j) = (0, 0), we have that k 1 /k = l 1 /l = m 1 /m.
If V = R d , then there exists a rational nontrivial A, B, C-invariant subspace V ′ such that R d = V ⊕ V ′ . V and V ′ correspond to nontrivial subtori T and T ′ , respectively, such Z d \ R d = T ⊕ T ′ and S(T ) and S(T ′ ) are both A, B, C-invariant. Since A i B j is ergodic on S(T ) and S(T ′ ) for every (i, j) = (0, 0) and A k B l C m is ergodic on S(Z d \ R d ) if and only if it is ergodic on both S(T ) and S(T ′ ), we can complete the proof by induction.
Proposition 3.15. Let ρ l be a Z k action on the solenoid S(M) of a nilmanifold M, extended from a Z k + action on M by affine endomorphisms. If ρ l is genuinely higher rank, then there exists a subgroup Σ of Z k isomorphic to Z 2 consisting of ergodic elements.
Proof. We will prove the following stronger statement: for any ergodic element ρ l (a), there exists a subgroup Σ ∼ = Z 2 containing a which consists of ergodic elements.
For the same reason as above, throughout this proof, we will assume that M is a torus Z dim M \ R dim M and every ρ l (a) ∈ GL(dim M, Q) is semisimple.
By passing to a subgroup of Z k of finite index, we can decompose M into almost direct product of ρ l -invariant irreducible subtori:

Let us prove the statement by induction on s.
When s = 1, the action ρ l is irreducible. Then the statement follows from Lemma 3.11. In fact, every nontrivial ρ l (a) is ergodic.
Suppose the statement holds for s − 1, we shall prove the statement for s. By Lemma 3.12, there exist ergodic elements. Take an ergodic element a ∈ Z k , we want to show that there exists a subgroup Σ ∼ = Z 2 containing a which consists of ergodic elements. By inductive assumption, there exist b 1 , b 2 ∈ Z k such that the restriction of ρ l (a) and ρ l (b 1 ) to S(M 1 ) generate a Z 2 action consisting of ergodic elements, and the restriction of ρ l (a) and ρ l (b 2 ) to S(M ′ 1 ) generate a Z 2 action consisting of ergodic elements. By Lemma 3.14 applied to M 1 and M ′ 1 , there are at most finitely many primitive triples (k, l, m) ∈ Z 3 such that the restriction of ρ l (ka + lb 1 + mb 2 ) onto S(M 1 ) or S(M ′ 1 ) is not ergodic. This implies that for all but finitely many primitive triples (k, l, m) ∈ Z 3 , ρ l (ka + lb 1 + mb 2 ) is ergodic on S(M). This implies that there exists a Z 2 subgroup of Z k containing a which consists of ergodic elements.
This completes the proof.
Proof of Corollary 1.4 assuming Theorem 1.3. By our assumption, ρ l is genuinely higher rank, then by Proposition 3.15, there exists a subgroup Σ ∼ = Z 2 of Z k such that for every a ∈ Σ, ρ l (a) is ergodic. Then by Theorem 1.3, there exist constant a 1 > 0 and η ′ > 0, such that for every a ∈ Σ, any f ∈ C θ (M), considered as a function on S(M), and any g ∈ C θ (S(M)), (1.1) holds. Recall that ρ l and ρ are conjugate via the bi-Hölder conjugacy φ. Thus if the exponential mixing holds for ρ l and µ, it also holds for ρ andμ = φ −1 * (µ). This proves Corollary 1.4.
3.3. Maximal expanding factor. Assuming ρ l (a) is ergodic for every a ∈ Z k , we will study the maximal expanding factor of every ρ l (a). Proof. By passing from the action ρ l to its maximal torus factor M 0 , we can reduce the proof to the case that M is a torus Z dim M \ R dim M , again using Parry's theorem [Par69] that a nilmanifold endomorphism is ergodic precisely when its projection to the maximal toral factor is ergodic. Then every ρ l (a), a ∈ Z k , can be expressed as an element in GL(dim M, Q). We first prove the statement assuming every ρ l (a), a ∈ Z k , is semisimple. Let K denote R or Q p . Then For a ∈ Z k , since ρ l (a) is semisimple, σ χ l is a generalized eigenspace of ρ l (a) with generalized eigenvalue χ l (a). For contradiction, suppose that there exists a sequence {a r : r ∈ N} such that a r → ∞ as r → ∞, and S(a r ) → 0 as r → ∞. Let K ⊂ Q be a finite field extension of Q such that every ρ l (a r ) is diagonalizable in GL(dim M, K). So we may assume that every ρ l (a r ) is a diagonal matrix in GL(dim M, K), denoted by diag{a r (i, i) : 1 ≤ i ≤ dim M}. Then S(a r ) → 0 implies that for any prime ideal p of the ring of algebraic integers O K , |a r (i, i)| p → 1 as r → ∞, for all i = 1, 2, . . . , dim M. Moreover, |a r (i, i)| → 1 as r → ∞ for all i = 1, 2, . . . , dim M. There are only finitely many units in K with absolute value equal to 1, thus there exists an element u ∈ GL(dim M, K) with some power of u is identity, such that passing to a subsequence, ρ l (a r ) → u, as r → ∞ in GL(dim M, C) and also in GL(dim M, K p ) for any prime ideal p in O K .
Thus ρ l (a r ) → u as l → ∞ in GL(dim M, A K ) where A K denotes the adeles of K.
Since GL(dim M, K) is discrete in GL(dim M, A K ), we will have ρ l (a r ) = u for r large enough. This implies that for r large enough ρ l (a r ) = u ∈ GL(dim M, Q) and some power of u is identity. This contradicts the assumption that every ρ l (a r ) is ergodic since some power of ρ l (a r ) is identity. This shows the statement assuming every ρ l (a r ) is semisimple. Now we prove the statement in general. Suppose Z k is generated by a 1 , a 2 , . . . , a k . Consider the Jordan decomposition of ρ l (a 1 ): ρ l (a 1 ) = b 1 c 1 with b 1 semisimple and c 1 unipotent. Since c 1 ∈ GL(dim M, Q), the eigenspace of c 1 with eigenvalue 1, which we denote by W 1 , is nontrivial and defined over Q. Also, W 1 is ρ l (Z k )-invariant, and the restriction of ρ l (a 1 ) onto W 1 is semisimple. Repeating this argument, we can find a sequence of rational ρ l (Z k )invariant subspaces W k ⊂ · · · ⊂ W 2 ⊂ W 1 such that for i = 1, 2, . . . , k, the restriction of ρ l (a i ) on W i is semisimple. Then the restriction of ρ l (Z k ) on W k is semisimple. By the special case above, inf a∈Z k \{0} {S(a| W k )} > 0. Then the statement follows since Proof. Let us first prove S(a) > 0 for every a ∈ R k \ {0}. Suppose S(a) = 0 for some a ∈ R k \ {0}. Since the line {ta : t ∈ R} comes arbitrarily close to integer points in Z k , we can find t l ∈ R and a l ∈ Z k with a l − t l a → 0 as l → ∞. Since S(t l a) = 0, we have S(a l ) → 0 as l → ∞. This contradicts Lemma 3.16. This proves that S(a) > 0. Then the statement follows as S is continuous.
Remark 3.18. Note that there are only finitely many Lyapunov exponents for ρ l and for any a ∈ Z k , χ l χ l (a) = 0. Thus, the above lemma implies that there exists a constant L ′ > 0 such that for any a ∈ Z k , there exists a Lyapunov exponent χ l such that χ l (a) ≥ L ′ a .
3.4. Proof of Theorem 1.3. Now we are ready to prove Theorem 1.3.
We first prove the theorem for irreducible actions, and then deal with the general case.
Irreducible Case. In this case we assume that the action ρ l : Z k S(M) is irreducible. We deal with the following two cases separately: Case 1 Let L ′ > 0 be the constant given in Remark 3.18. For all a ∈ Z k , there exists a real Lyapunov exponent χ l such that χ l (a) ≥ L ′ a . Case 2 Case 1 fails.
Proof for Case 1. For this case, the proof is more or less the same as that in [GS14].
Suppose χ l 1 (a) = max{χ l (a) : χ l ∈ T (R)}. Then by our asssumption, χ l 1 (a) ≥ L ′ a . Then σ χ l 1 ⊂ n(R) is the generalized eigenspace of Dρ l (a) with generalized eigenvalue e χ l 1 (a) . It is easy to see that σ χ l 1 is defined over Q∩R. Thus W ′ := Dπ(σ χ l 1 ) ⊂ R l is also defined over 20 Q ∩ R. Then by Lemma 3.7, there exists w ∈ W ′ whose coordinates are algebraic numbers that are linearly independent over Q. As discussed after Lemma 3.9, there exist constants c, L > 0 such that | w, z | ≥ c z −L , for all z ∈ Z l \ {0}. Let us fix a small constant ǫ > 0. For each p ∈ S, we pick an integer l p (a) ≥ 0 such that p −lp(a) ≤ ǫ, and ρ l (a)(N(p lp(a) Z p )) ⊂ N(Z p ). We cut S(M) into small pieces along p-adic directions: for a fixed fundamental domain F ⊂ N(R) of N(Z) \ N(R) and some ξ j (p) ∈ N(Z p ). We fix a basis {w 1 , . . . , w s 1 } of σ χ l 1 and extend it to a fixed basis {w 1 , . . . , w s 1 , v 1 , . . . , v s 2 } of n(R). For ǫ > 0, we define We then cut each B i into small pieces along the real component. In other words, we write It is easy to see that where z i,j = (x i , (ξ j (p)) p∈S ) ∈ B i,j . By Lemma 2.6, for each j, we may choose n j ∈ N(Z) such that n −1 j ξ j (p) ∈ N(p lp(a) Z p ) for all p ∈ S. Therefore, Since ρ l (a)(N(p lp(a) Z p )) ⊂ N(Z p ) and since the value of the function f only depends on its projection on M, we have that where V = p∈S ν p (N(p lp(a) Z p )). Let y i,j := n −1 j x i , then we have that f (y i,j exp(Dρ l (a)v + Dρ l (a)w))dwdv.
We want to show that for all v ∈ n(R), the integral Here Vol(·) denotes the volume with respect to the normalised Lebesgue measure dw.
Let v ∈ n(R) be fixed. We consider the box map Then the integral (3.2) is the integral of f along the box map ι and based at the point y i,j . For contradiction, suppose (3.2) does not estimate M f dν with error O(e −η a f θ ). Then by Theorem 3.2, there exists z ∈ Z l \ {0} such that z ≪ e L 1 η a and | z, Dπ(w ′ i ) | ≪ e L 2 η a /ǫ. Since w ′ i = Dρ l (a)w i ≍ e χ l 1 (a) w i ≫ e L ′ a and σ χ l 1 is spanned by {w ′ 1 , . . . , w ′ s 1 }, we have that for all w ∈ σ χ l 1 with w ≍ 1, On the other hand, by Lemma 3.7, there exists w ∈ W such that Let ǫ = e −L 3 a such that 0 < L 3 < L ′ /2. Then (3.3) and (3.4) will lead to a contradiction if L 3 + L 2 η − L ′ < −L 1 Lη. This shows that there exists constant η > 0 such that, for all v ∈ n(R), integral (3.2) estimates M f dν with error O(e −η a f θ ). This implies that Therefore, a f θ )).
Proof for Case 2. By Remark 3.18, there exists a Lyapunov exponent χ l 2 ∈ T (Q p ) for some p ∈ S such that χ l 2 (a) ≥ L ′ a . We may further assume that χ l 2 (a) = max χ l {χ l (a)}. Let σ χ l 2 ⊂ n(Q p ) denote the generalized eigenspace of Dρ l (a) with generalized eigenvalue e χ l 2 (a) . Let us fix a basis {w 1 , . . . , w s 1 } of σ χ l 2 and extend {w 1 , . . . , w s 1 } to a basis {w 1 , . . . , w s 1 , v 1 , . . . , v s 2 } of n(Q p ). Without loss of generality, we may assume that w i p = 1 for i = 1, . . . , s 1 . Then {Dρ l (a)w 1 , . . . , Dρ l (a)w s 1 } is also a basis of σ χ l 2 and Dρ l (a)w i p = e χ l 2 (a) for i = 1, 2, . . . , s 1 . Denote e χ l 2 (a) = p h , and Dρ l (a)w i = p −h u i for i = 1, . . . , s 1 . Then {u 1 , . . . , u s 1 } is a basis of σ χ l 2 and u i p = 1 for i = 1, . . . , s 1 . Pick δ > 0 small enough such that δ < e −L ′ a and the diameter of ρ l (a)(U(x, δ)) is less than e −L ′ a for all x ∈ M. Let ǫ = e −L ′ a /2 . For S ∋ q = p, let l q (a) > 0 denote the smallest integer such that q −lq(a) ≤ ǫ and ρ l (a)(N(q lq(a) Z q )) ⊂ N(Z q ). Let l p > 0 denote the smallest integer such that p −lp ≤ ǫ. Let us cut S(M) into small pieces as follows: for some x j ∈ N(R), ξ j (p) ∈ N(Z p ) and ξ j (q) ∈ N(Z q ). Since e χ l 2 (a) = p h and since χ l 2 (a) is maximal among all χ l (a), we have that ρ l (a)N(p h Z p ) ⊂ N(Z p ). For a positive integer h ′ , define Let us further cut each B j along the p-adic direction: . By Lemma 2.6, there exists n i,j ∈ N(Z) such that n −1 i,j ξ i,j (p) ∈ U p (h) and n −1 i,j ξ j (q) ∈ N(q lq(a) Z q ) for q = p. Then By the argument in the proof for Case 1, to prove the exponential mixing result it suffices to show that for a constant η > 0. To estimate the above integral, we cut U p (h) further as follows: where κ l (p) ∈ U p (h). According to this we can cut B i,j into small pieces: Now let us look at B i,j,l f (ρ l (a)z)dµ(z) more carefully. First note that we can choose κ l (p) to run over elements in ∆(w 1 , . . . , w s 1 ) := {exp(t 1 w 1 + · · · + t s 1 w s 1 ) : t i = 0, p lp · 1, p lp · 2, . . . , p lp (p h−lp − 1)}.
By Lemma 2.6, for each w i , there exists w i (h) ∈ n(Z) such that w i (h) ≡ w i (mod p h ) and w i (h) ≡ 0(mod q lq(a) ) for q = p. Suppose κ l (p) = exp(t 1 w 1 + · · · + t s 1 w s 1 ), then direct calculation shows that where V := ν p (N(p h Z p )) × S∋q =p ν q (N(q lq(a) Z q )) and n l := exp(t 1 w 1 (h) + · · · + t s 1 w s 1 (h)). Note that the diameter of ρ l (a)U(y i,j , ǫ) is less than e −L ′ a , we have that ′ θ a f θ )). Therefore, to show the exponential mixing result, it suffices to show that the following summation 1 p (h−lp)s 1 t 1 ,...ts 1 f (ρ l (a)(exp(−(t 1 p lp w 1 (h) + · · · + t s 1 p lp w s 1 (h)))y i,j )) where z i,j := ρ l (a)y i,j . By our previous discussion Dρ l (a)w j = p −h u j , for j = 1, . . . , s 1 . Let u j (h) ∈ n(Z) be such that u j (h) ≡ u j (mod p h ) and u j (h) ≡ 0(mod q lq(a) ) for S ∋ q = p. Then the difference between exp(− s 1 j=1 t j p lp Dρ l (a)w j (h))z i,j and exp Therefore we reduce our task to proving that 1 p hs 1 t 1 ,...,ts 1 For t = (t 1 , . . . , t s 1 ) ∈ [0, p h−lp − 1] s 1 , let us denote then it is easy to see that P : [0, p h−lp −1] s 1 → N(R) is a polynomial map. Then by Theorem 3.3, either 1 which is case AA.1 in Theorem 3.3, or there exists z ∈ Z l \ {0} such that z ≪ e L 1 η a and dist(| z, ∂ i π(P (t)) |, Z) ≪ e L 2 η a /p h−lp for all i = 1, . . . , s 1 and t ∈ [0, p h−lp − 1] s 1 , which is case AA.2. Suppose the latter holds. Since N ′ (Q p )\N(Q p ) ∼ = Q l p is abelian, we may identify Q l p with its Lie algebra. For v ∈ n(Q p ), letṽ ∈ Q l p denote the projection of v onto Q l p under Dπ. Then it is easy to see that ∂ i π(P (t)) = p −h+lpũ i (h) for i = 1, . . . , s 1 . Then for i = 1, . . . , s 1 . In other words, in (mod p h−lp ) sense, | z,ũ j (h) | ≪ e L 2 η a . Let k denote a positive integer with (k, p) = 1, In the construction above, we can replace the basis w 1 , . . . w s 1 by kw 1 , . . . kw s 1 as the generalized eigenspace σ χ l 2 is invariant under multiplication by k. Note that multiplication by k does not change the p-adic norm of any vector since (k, p) = 1. Tracking the argument, we eventually replace the polynomial P (t) by P k (t) := exp(−p −h+lp s 1 j=1 t j ku j (h))z i,j ,. By the same argument as for P , either the sum η a f θ ), and we are done, or there exists z(k) ∈ Z l \ {0} with z(k) ≪ e L 1 η a and in (mod p h−lp ) sense, | z(k), kũ i (h) | ≪ e L 2 η a for i = 1, . . . , s 1 . We choose η small enough such that (e L 1 η a ) l × (e L 2 η a ) s 1 ≤ (p h−lp ) 1/8 .
Let W ⊂ N ′ (Q p ) \ N(Q p ) = Q l p denote the projection of σ χ l 2 on Q l p under π. Apparently W is defined over Q al p and not defined over Q. Let us choose a basis {ũ i : i = 1, 2, . . . , s 1 } of W with coordinates in Q al p . By Lemma 3.8, there existsũ = s 1 i=1 α iũi ∈ W ∩ (Q al p ) l whose coordinates are linearly indepedent over Q. Moreover, multiplying by suitable powers of p, we can choose α i ∈ Z p for i = 1, 2, . . . s 1 . Then we have for any z ′ ∈ Z l , z ′ ,ũ i = 0.
On the other hand, by Lemma 3.9, there exists a constant L > 0 such that for all z ′ ∈ Z l \ {0}, z ′ ,ũ p ≫ z ′ −L ≫ e −LL 1 η a . The above two inequalities will lead to contradiction if LL 1 η < 3L ′ /8. This shows that for a constant η > 0. By the argument in the proof for Case 1, this finishes the proof of exponential mixing for Case 2.
This completes the proof of irreducible part of Theorem 1.3.
In fact, using the same argument, one can prove the following slightly stronger result: Proposition 3.19. Under the irreducible assumption as above, let β be an automorphism of N defined over Q such that β = Id on N ′ \ N, then for the same constant η ′ > 0 as above, for any f ∈ C θ (M) and any g ∈ C θ (S(M)).
Proof. From the proof above, we see that the basic scheme of the argument goes as follows: We first cut the whole space S(M) into small pieces along real and p-adic directions, and then apply Theorem 3.2 or Theorem 3.3 prove each small piece estimates the integral of the whole space with exponentially small error. The obstruction of the effective equidistribution is that Dβ(ρ l (a)(Dπ(σ χ l ))) lies in a rational linear subspace. Since β acts trivially on N ′ \N, the proposition follows from the argument above.
General Case. We will prove the statement in general using induction on the dimension of M. By [GS14, Lemma 3.5], if the action ρ l on M is not irreducible, then there exists a ρ l -invariant normal subgroup N 1 of N defined over Q satisfying the following: (1) The restriction of ρ l on N 1 is irreducible. N 1 ]. Then Y := N(Z)N 1 (R) \ N(R) and Z := N 1 (Z) \ N 1 (R) are both compact nilmanifolds, and moreover, M fibers over Y with fibers isomorphic to Z. Let µ Y and µ Z denote the normalized measures on the solenoids S(Y ) and S(Z) respectively, defined as the measure µ on S(M). Then for any continuous function f defined on S(M), we have the following disintegration formula: Since N 1 is ρ l -invariant, ρ l defines transformations of Y and Z. Then where F ⊂ N(R)× p∈S N(Z p ) is a bounded fundamental domain for S(Y ), and m F denotes the measure on F induced by µ Y .
Then we have φ 0 θ ≪ f θ and φ 1 θ ≪ g θ , and S(Z) Since every ρ l (a) is still ergodic when restricted to S(Z), we can apply Proposition 3.19 to conclude that there exists a constant η ′ > 0 such that uniformly over h ∈ F . This proves the claim.
Let us define f (y) := S(Z) f (zy)dµ Z (z) and g(y) := S(Z) g(zy)dµ Z (z) for y ∈ S(Y ). Then by the above claim, we conclude that Since dim Y < dim M, the statement follows by induction.

Coarse Lyapunov foliations and smooth leaves
In this section, we will prove that for every coarse Lyapunov exponent [χ], the corresponding coarse Lyapunov distribution E [χ] := χ∈[χ] E χ admits a Hölder foliation with C ∞ leaves. We follow the argument developed by Rodriguez-Hertz and Wang [HW14] with minor modification for our setup.
We first make the following definition: Remark 4.2. When ρ(a) is uniformly hyperbolic, both E s a and E u a admit Hölder foliations with C ∞ leaves. We denote the corresponding foliations by W s a and W u a , respectively. For z ∈ S(M) and = s or u, we denote by W a (z) the leaf of W a passing through z.
The basic idea to establish the smoothness of leaves goes as follows: we start with a real Weyl chamber C 0 ⊂ R k with an element a ∈ Z k ∩C 0 such that ρ(a) is uniformly hyperbolic. We shall prove that any real Weyl chamber C adjacent to C 0 also contains a uniformly hyperbolic element a ′ ∈ Z k . Therefore every real Weyl chamber of the action ρ contains a uniformly hyperbolic element. For every coarse Lyapunov exponent [χ], we choose two adjacent Weyl chambers C 1 and C 2 such that kerχ is the only Weyl chamber wall separating C 1 and C 2 . Take a i ∈ Z k ∩ C i such that ρ(a i ) is uniformly hyperbolic. Without loss of generality, we may assume that χ(a 1 ) < 0 and χ(a 2 ) > 0. Then the intersection W s a 1 ∩ W u a 2 defines a Hölder foliation, and passing through every z ∈ S(M), the intersection W s a 1 (z) ∩ W u a 2 (z) is C ∞ . From our assumption it is easily seen that W s a 1 ∩ W u a 2 corresponds to the distribution E [χ] . This proves that E [χ] admits a Hölder foliation with C ∞ leaves. To show that C contains a uniformly hyperbolic element, we basically follow the argument by Rodriguez Hertz and Wang [HW14] with minor modifications. 4.1. Correspondence between foliations of ρ and ρ l . Let us fix a real Weyl chamber C 0 and choose a ∈ Z k ∩ C 0 such that ρ(a) is uniformly hyperbolic.
Recall that for each z ∈ S(M), M(z) denotes a manifold slice passing through z. Then the leaves W s a (z) and W u a (z) are given by Now, let us turn our attention to the affine action ρ l (a). Let g s a (R) (and g u a (R), respectively) denote the direct sum of σ χ l such that χ l ∈ T (R), and χ l (a) < 0 (and χ l (a) > 0, respectively). From Remark 2.15, we can see that g s a (R) and g u a (R) are both Lie subalgebras of n(R). Let G s a and G u a denote the corresponding Lie subgroups. Then it is easy to see that g s a (R) and g u a (R) correspond to the stable and unstable foliations of M with respect to the action ρ l (a), and moreover, the stable (and unstable) leaf passing through any point z ∈ S(M) is zG s a (and zG u a respectively). Because φ conjugates ρ to ρ l , we have that φ(W s a (z)) = φ(z)G s a , and φ(W u a (z)) = φ(z)G u a . In particular, we have that dim E s a = dim g s a and dim E u a = dim g u a . Therefore, for every real Lyapunov exponent χ l for ρ l , χ l (a) = 0, and n(R) = g s a (R) ⊕ g u a (R). Let C be a real Weyl chamber adjacent to C 0 . Let kerχ l denote the real Weyl chamber wall separating C 0 and C. Then by Remark 2.15, σ := σ [χ l ] is a Lie subalgebra of n(R). Let V := V [χ l ] = exp(σ [χ l ] ) denote the corresponding Lie subgroup.
Without loss of generality, we may assume that χ l (a) < 0, then V ⊂ G s a . Let us define the strong stable subspace of ρ l (a) by Lemma 4.3 (see [HW14,Lemma 3.1]). 1. g ss a is a Lie subalgebra. 2. σ ⊕ g u is a Lie subalgebra. 3. [σ, g ss a ] ⊂ g ss a . Let G ss a := exp g ss a be the Lie subgroup corresponding to g ss a . On the level of Lie algebra, we have the following decomposition: On the level of Lie group, the following lemma is proved in [HW14]: 1. The multiplication map Remark 4.5. In [HW14], the order of the multiplication is G ss a × V × G u a → N(R). The same proof works for our order here. We make this change in this paper because here M = N(Z) \ N(R) while in [HW14] M = N(R)/N(Z). Also, later in this section, we will make similar changes due to this reason.
From the above lemma, any g ∈ N(R) can be uniquely written as g u g V g ss . Define g s := g V g ss . We call g ss (g V , g u and g s respectively) the projection of g onto G ss a (V , G u a and G s a respectively). Rodriguez Hertz and Wang [HW14] proved several results on the G ss a × V × G u a coordinate of N(R). We sum up them in the following proposition. We refer to [HW14] for proofs.
Let us recall an argument from [FKS13] and [HW14] concerning the choice of a ∈ Z k and estimate of Lyapunov exponents.
Recall that a ∈ C 0 where C 0 denotes a Weyl chamber of the action ρ l , C denotes a Weyl chamber adjacent to C 0 , and [χ l ] denotes the coarse Lyapunov exponent such that kerχ separates C 0 and C.
Recall that Σ ∼ = Z 2 denotes the subgroup of Z k given in Corollary 1.4. For any ξ > 0, we may choose b ∈ Σ ∩ C such that |χ l 1 (b)| < ξ b for all χ l 1 ∈ [χ l ] or [−χ l ]. Then the restriction of ρ −1 l (b) on V is contracting, and for any v ∈ σ and n ∈ N, Dρ l (nb)v = O(e nξ b ). For ξ > 0 small enough, we have that for any [χ l 2 ] = [χ l ] or [−χ l ], χ l 2 (a) and χ l 2 (b) have the same sign. Therefore, if σ [χ l 2 ] ⊂ g ss a , then χ l 2 (b) < 0. Combining this with the fact that |χ l 1 (b)| < ξ b for all χ l 1 ∈ [χ l ] or [−χ l ], we conclude that for any v ∈ g s a and any n ∈ N, Dρ l (nb)v = O(e nξ b ). Since the foliation W s a corresponds to G s a via φ, and φ is θ-Hölder, we conclude that for any w ∈ E s a , and any n ∈ N, Dρ(nb)w = O(e nξ ′ b ) where ξ ′ = ξ/θ. Summing up the argument above, we have the following proposition: . By Proposition 4.6, (g 1 g 2 ) V = (g 1 (g 2 ) u ) V (g 2 ) V . Projecting both sides of the equation above to V , we have . Notation 4.8. We borrow the following notation from [HW14].
For x ∈ S(M)( or N(R)) and ǫ > 0, let B ǫ (x) denote the ball inside M(x)( or N(R)) centered at x with radius ǫ. For a Lie subgroup F of N(R), f ∈ F and ǫ > 0, we denote by B F ǫ (f) the ball inside F centered at f with radius ǫ. Let us fix a constant δ > 0 such that for x, y ∈ S(M) belonging to the same manifold slice and dist(x, y) ≤ δ, there exists a unique element p(x, y) ∈ B δ (e) with y = xp(x, y). It is easy to see that the map p : {(x, y) ∈ S(M)×S(M) : x and y belong to the same manifold slice, and dist(x, y) ≤ δ} → N(R) For x, y ∈ S(M) belonging to the same manifold slice and dist(x, y) ≤ δ, we write φ(y) = φ(x)H x (y) where H x (y) := h −1 (x)p(x, y)h(y). It is easy to see that the map is θ-Hölder in the pair (x, y).
Proof. The proof we present here follows the proof of [HW14, Lemma 3.13, Corollary 3.14] with minor modification.
We write p −1 (x, y)h(x) = (p −1 (x, y)h(x)) u (p −1 (x, y)h(x)) s and h(y) = h u (y)h s (y). Then a if and only if (p −1 (x, y)h(x)) −1 u h u (y) = e, i.e., h u (y) = (p −1 (x, y)h(x)) u . Since G s a corresponds to the foliation W s a via the conjugacy φ. Therefore, near x the leaf W s a (x) is defined by Since on W s a (s), p(x, y) is C ∞ and h(x) is constant, we have that h u (y) is C ∞ along W s a (x). It remains to check that the partial derivatives along W s a vary Hölder continuously. We fix a neighborhood Ω of G s a ǫ (e). Then every y ∈ Ω can be projected to some x = x(y) ∈ Ω ∩ W u a (x 0 ) along W s a . Since W s a is a Hölder foliation, we have that the map y → x(y) is Hölder in y. Thus the map y → h(x(y)) is also Hölder . By (4.2), partial derivatives of h u (y) along W s a C ∞ depend on x and h(x), and thus are Hölder continuous in y.
This completes the proof. Now let us get back to the cohomological equation (4.1). Leth V := log h V and Ψ := log(Q ′ b (z)ρ −1 l (b)h u (ρ(b)z)) V . By Proposition 4.9, Ψ ∈ C ∞,θ W s a . Then (4.1) can be rewritten as ). By Baker-Campbell-Hausdorff formula, we have that where there are only finitely many terms on the right hand side since both sides of the equation belong to the Lie algebra σ of V which is nilpotent. Consider the derived series of σ: σ = σ 0 ⊃ σ 1 ⊃ · · · ⊃ σ l = {0}. For i = 1, 2, . . . , l, let π i : σ → σ i \ σ denote the canonical projection. Leth i := π i •h V and Ψ i := π i • Ψ. Projecting the equation to σ 1 \ σ, we will get the following linearized equation: (4.4)h 1 = ρ −1 l (b)h 1 • ρ(b) + Ψ 1 . We first prove the following lemma on linearized cohomological equations: then there exists ξ 0 > 0 such that for 0 < ξ ≤ ξ 0 , f ∈ C ∞,θ W s a . Proof. We first claim that in order to show the lemma, it suffices to show the lemma assuming that the integral S(M ) f dμ = 0. In fact, let f := S(M ) f dμ and let f 1 := f − f . Since f is a constant function, to show f ∈ C ∞,θ W s a , it suffices to show that f 1 ∈ C ∞,θ W s a . We have that S(M ) f 1 dμ = 0 and f 1 satisfies the following equation: This proves the claim. Therefore we may assume that S(M ) f dμ = 0.
By iterating (4.5), we have that We claim that in the sense of distributions. To show this, it suffices to show that for any g ∈ C ∞ (M), In fact, sinceμ is ρ-invariant and f has zero average with respect toμ, we have ψdμ, we conclude that S(M ) ψdμ = 0. Therefore, by Corollary 1.4, there exist constants C > 0 and η ′ > 0 such that for all g ∈ C ∞ (M), a is a Hölder continuous function, the term Fix a compactly supported positive C ∞ bump function δ on n(R) supported on a neighborhood around 0. For small ǫ > 0, define on N(R) a function where c ǫ > 0 is chosen such that N (R) δ ǫ (g)dg = 1. Let φ ǫ := φ * δ ǫ . By standard facts on convolutions, we have the following hold: where a 1 > 0 and η ′ > 0 are constants from Corollary 1.4, and C 1 := a 1 c k ψ θ .
We also need to estimate | ∂ k W s a (ψ • ρ(ia)), φ − φ ǫ |: (4.7) . Therefore, we have for a constant a 2 > 0 depending on k, b, ψ and dim W s a . Let For ξ small enough, we will have By the estimates above, we have that , φ | ≤ a 3 e −η 1 i b φ θ for a constant a 3 > 0. Since by assumption there exists a constant a 4 > 0 such that B −i ≤ a 4 for all i ∈ N, we will have that (4.9) where a 5 = a 3 a 4 1 1−e −η 1 b . This proves that ∂ k W s a f ∈ (C θ (M)) * . By our previous discussion, this completes the proof.
whereΨ i = Ψ i + [ higher order terms ] is in C ∞,θ W s a . Let B := ρ l (b). Since B −1 is contracting on σ, B −i is uniformly bounded when restricted to σ i \σ. By Proposition 4.10, we conclude thath i ∈ C ∞,θ W s a . For x ∈ S(M), let W ss a (x) denote the topological submanifold of M(x) passing through x defined by W ss a (x) := φ −1 (φ(x)G ss a ). Obviously every W ss a (x) is contained in a W s a leaf. Recall that H x (y) = h −1 (x)p(x, y)h(y).
The following proved in [HW14] gives the local description of W ss a (x): Lemma 4.12 (see [HW14,Lemma 4.1]). Inside W s a (x), W ss a (x) is locally defined by the equation (H x (y)) V = e.
Note that by Proposition 4.6, (H x (y)) V = h V (y)((h(x)p(x, y) −1 ) −1 s ) V . By Proposition 4.11, h V (y) is C ∞ when restricted to W s a (x). Combined with ((h(x)p(x, y) −1 ) −1 s ) V is C ∞ in y, this implies that y → H x (y) is C ∞ in small neighborhoods of x in W s a (x). Moreover, since partial derivatives of (H x (y)) V along W s a (x) are polynomial combinations of ∂ k W s a h V (y) and ∂ k W s a ((h(x)p(x, y) −1 ) −1 s ) V , we conclude that all partial derivatives ∂ k W s a | y=x (H x (y)) V are θ-Hölder continuous in x.
Our aim is to show that W ss a defines a Hölder foliation with C ∞ leaves. By [HW14,Corollary 4.3], to show the smoothness of every W ss a (x), it suffices to show that for any x ∈ S(M), the map (H x (y)) V is regular in y at y = x. We will modify the argument by Rodriguez-Hertz and Wang to prove the result.
Let A be the set of points x ∈ S(M) where (H x (y)) V is singular at x. We want to show that A is empty. Proof. Since D W s a | y=x (H x (y)) V depends continuously on x, and since being singular is a closed condition, we conclude that A is closed.
Since a ′ is fixed and both H x (y) and H ρ(a ′ )x (ρ(a ′ )y) are close to e, we conclude that H ρ(a ′ )x (ρ(a ′ )y) = ρ l (a ′ )H x (y).
Since W s a is ρ-invariant, we have that D W s a | y=ρ(a ′ )x (H ρ(a ′ )x (y)) V = Dρ l (a ′ )| σ (D W s a | y=x (H x (y)) V )(D ρ(a ′ )x ρ(a ′ )| E s a ) −1 . Since Dρ l (a ′ )| σ and D ρ(a ′ )x ρ(a ′ )| E s a are both regular, we conclude that D W s a | y=ρ(a ′ )x (H ρ(a ′ )x (y)) V is singular if and only if D W s a | y=x (H x (y)) V is so. In other words, x ∈ A if and only if ρ(a ′ )x ∈ A. Since a ′ ∈ Z k is chosen arbitrarily, we conclude that A is ρ-invariant.
such that for all y ∈ W s a (z) ∩ Φ z (B R N (τ l −1 (z))), π z maps the graph of g z,y to a piece of a hyperplane parallel to R N ss . Put P z := Φ z •π −1 z , then (H z ) V •P z is constant along hyperplanes parallel to R N ss . Now we are ready to give the contradiction, cf. [ where Vol φ(z)G s a and Vol W s a (z) denote the volume forms of the induced Riemannian metrics on φ(z)G s a and W s a (z) respectively. In fact, we may choose δ 0 > 0 and a neighborhood B z ⊂ W s a (z) such that B R N ss (δ 0 ) × B R N V (δ 0 ) ⊂ P −1 z (B z ). Fix a decreasing sequence {δ k > 0 : k ∈ N} approaching 0 as k → ∞. For each k ∈ N, define B k,z := P z B R N ss (δ 0 ) × B R N V (δ k ) .
Since P z is bi-Lipschitz, to show the claim, it suffices to show that The denominator is of order O(δ N V k ). Let us analyze the numerator. Our aim is to show that the numerator is of order o(δ N V k ) (cf. Notation 1.8). Since G s is decomposed as G ss · V and since V normalizes G ss , we have that dVol G s = dVol G ss · dVol V . It is easy to see that the G ss -projection of φ • P z (B R N ss (δ 0 ) × B R N V (δ k )) is uniformly bounded, so to show that Vol φ(z)G s a (φ • P z (B R N ss (δ 0 ) × B R N V (δ k ))) = o(δ N V k ), it suffices to show that Vol V (H z ) V • P z (B R N ss (δ 0 ) × B R N V (δ k )) = o(δ N V k ). We have seen that (H z ) V • P z only depends on the second coordinate, so (H z ) V • P z (B R N ss (δ 0 ) × B R N V (δ k )) = (H z ) V • P z (B R N ss (δ k ) × B R N V (δ k )).
Note that P z is Lipschitz, we have that P z (B R N ss (δ k ) × B R N V (δ k )) ⊂ B W s a (z) (z, Cδ k ) for a constant C = C(z), here B W s a (z) (z, r) denotes the ball in W s a (z) centered at z of radius r. Then to prove the claim, it is enough to show that Vol V (H z ) V (B W s a (z) (z, δ)) = o(δ N V ) as δ → 0. This is true since by our hypothesis, for z ∈ A ′′ ⊂ A, D W s a | y=z (H z (y)) V : E s a (z) → σ has rank less than N V = dim σ.
On the other hand, we claim that for every z ∈ S(M), there exists a positive continuous function J z such that φ * dVol W s a (z) = J z dVol φ(z)G s a . Note that this claim contradicts the previous one. Therefore, to complete the proof, it suffices to prove this claim.

Regularity of the conjugacy
In this section we will prove Theorem 1.2 when dim M ≥ 5. As we discussed in §2, we write φ(z) = zh(z). Then it suffices to show that h is C ∞ .
We follow the process described in §2.7. Recall that we follow Notation 2.12 and Notation 2.20 to denote coarse Lyapunov exponents and coarse Lyapunov subgroups. By Theorem 4.19, every coarse Lyapunov exponent [χ] of ρ admits a Hölder foliation with C ∞ leaves. Let us call it the coarse Lyapunov foliation associated with [χ]. 5.1. Case of tori. In this subsection, we assume that M is a torus. By our discussion in §2.7, it is enough to show that Proof of Theorem 1.2 for the case of tori. By [FKS13, Corollary 8.4], Proposition 5.1 implies that h V ∈ C ∞ (M). This proves that h is C ∞ since h is the sum of h V 's where V runs over all coarse Lyapunov subgroups of ρ l . 5.2. General case. Now we deal with the general case. As described in §2.7, we consider the derived series of N: Note that ρ l preserves N i for each i = 0, 1, . . . , r.
We will prove Theorem 1.2 by showing the following stronger statement.
Proposition 5.2. For i = 0, 1, . . . , r, let h : M → N i (R) (regarded as a map defined on S(M)) be a θ-Hölder map. Suppose for all a ∈ Z k , there exists a C ∞ map Q a : S(M) → N i (R), such that h(z) = Q a (z)ρ −1 l (a)h • ρ(a)(z). Then h is C ∞ .