Arithmeticity and topology of smooth actions of higher rank abelian groups

We prove that any smooth action of $\mathbb Z^{m-1}, m\ge 3$ on an $m$-dimensional manifold that preserves a measure such that all non-identity elements of the suspension have positive entropy is essentially algebraic, i.e. isomorphic up to a finite permutation to an affine action on the torus or its factor by $\pm\Id$. Furthermore this isomorphism has nice geometric properties, in particular, it is smooth in the sense of Whitney on a set whose complement has arbitrary small measure. We further derive restrictions on topology of manifolds that may admit such actions, for example, excluding spheres and obtaining below estimate on the first Betti number in the odd-dimensional case.


Introduction
Let α be a smooth action of of Z m−1 , m ≥ 3 on a compact mdimensional manifold M, not necessarily compact. We assume that α is uniformly C 1+θ , θ > 0 with respect to a certain smooth Riemanninan metric on M, i.e. the generators of the action and their inverses have uniformly bounded derivatives satisfying Hoelder condition with exponent θ and a fixed Hoelder constant. Naturally, if M is compact this condition does not depend on the choice of the Riemannian metric. This regularity assumption allows to apply standard results of smooth ergodic theory to any invariant measure of the action.
Following [10] we assume that α has an invariant probability ergodic measure µ such that (1) Lyapunov characteristic exponents are non-zero and are in general position, i.e. the dimension of the intersection of any l of their kernels is the minimal possible, i.e. is equal, to max{k − l, 0}, (2) at least one element in Z m−1 has positive entropy with respect to µ. We will call such a pair (α, µ) a maximal rank positive entropy action. 1 ) Based on research supported by NSF grant DMS 1002554. 2 ) Based on research supported by NSF grant DMS 1201326.
The main result of [10] is absolute continuity of the measure µ for a maximal rank positive entropy action. In [10,Sections 8.1,8.2] a program of further study of such actions has been formulated.
In the present paper we mostly complete this program. Firstly we extend the description of maximal rank actions on the torus with Cartan homotopy data from [15], where a positive entropy hyperbolic measure always exists, to maximal rank positive entropy actions on arbitrary manifolds. Secondly we obtain substantial information on topology of manifolds that may admit such actions., in particularly excluding spheres and many other standard manifolds.
Let us call an infratorus a factor of R m by a group E of affine transformations that contains a lattice L of translations as a finite index subgroup. Thus an infratorus is the factor of the torus R m /L by a finite group G of affine transformations. In this definition infratorus is a varifold and not necessarily a smooth manifold since the group G may not act freely; in particular it may have fixed points. In fact, the only examples of infratori that admit maximal rank abelian actions by affine transformations and that are not tori, are of that kind: such an infratorus is obtained by factorizing T m by the involution Ix = −x that has 2 m fixed points. Let us denote such an infratorus T m ± .
Remark 1. By blowing up the singular points and glueing in copies of the projective space of codimension one, one constructs a smooth action on a manifold that is diffeomorphic to the affine action on the infratorus outside of the singular points, see [13]. This can be considered as the "standard smooth model" of the infratorus action. Examples of infratori that are smooth manifolds can be found in [14,Section 2.1.4].
We formulate and present our results in two parts. The reason is that the first part is likely to hold with proper modifications (in particular, allowing more general infratori) in greater generality, namely, in the setting similar to that of [16] where no connection is assumed between the rank k ≥ 2 of the action and the dimension of the ambient manifold. Most steps in the proof work in that generality and remaining difficulties, while substantial, are of technical nature. The second part heavily relies on existence of codimension-one stable manifolds and hence is specific for the maximal rank setting.
The first part (Theorem 1) states in particular that modulo a finite permutation any such action is "arithmetic", i.e. there is a measurable isomorphism between the restriction of the action to each ergodic component of a certain finite index subgroup Γ ⊂ Z m−1 and a Cartan action by affine automorphisms of the torus T m or its factor T m ± . This isomorphism has nice topological and geometric properties that are described in detail below. This provides solutions of most conjectures and problems from [10,Sections 8.1]. The second part asserts that the restriction of the above mentioned isomorphism to each ergodic component of the group Γ extends to a continuous map between an open set in M and the complement to a finite set on T m or T m ± that is a topological semi-conjugacy (a factormap) between α and α 0 (Theorem 2). Furthermore, this map can be modified on a set of arbitrary small measure and then extended to a homeomorphism between an open set in M and the complement to a finite set on T m or T m ± . This has implications for the topology of M, in particular disproving Conjecture 4 from [10].
Technically the present paper builds upon the results of [10,16]. We use background information from those papers without special references.

Formulation of results
1.1. The arithmeticity theorem. Theorem 1. Let α be a C r , 1 + θ ≤ r ≤ ∞ maximal rank positive entropy action on a smooth manifold M of dimension m ≥ 3.
Then there exist: • disjoint measurable sets of equal measure R 1 , . . . , R n ⊂ M such that R = n i=1 R i has full measure and the action α cyclically interchanges those sets. Let Γ ⊂ Z m−1 be the stationary subgroup of any of the sets R i (Γ is of course isomorphic to Z m−1 ); • a Cartan action α 0 of Γ by affine transformations of either the torus T m or the infratorus T m ± that we will call the algebraic model; • measurable maps h i : R i → T m or h i : R i → T m ± , i = 1, . . . , n; such that (1) h i is bijective almost everywhere and (h i ) * µ = λ, the Lebesgue (Haar) measure on T m (correspondingly T m ± ); (3) for almost every x ∈ M and every n ∈ Z m−1 the restriction of h i to the stable manifold W s x of x with respect to α(n) is a C r−ǫ diffeomorphism for any ǫ > 0. (4) h i is C r−ǫ in the sense of Whitney on a set whose complement to R i has arbitrary small measure; those sets will be described in the course of proof; in particular, they are saturated by local stable manifolds.  [10]. In fact, description of Cartan (maximal rank) actions on the torus via units in the algebraic number fields given in [12] provides more precise information. We consider the weakly mixing case first.
Corollary 1. Let α be a C 1+θ , θ > 0 weakly mixing maximal rank positive entropy Z m−1 action. There exists a totally real algebraic number field K of degree m, that is a simple extension of Q uniquely determined by α, and, for any system of generators of α, an (m − 1)-tuple of multiplicatively independent units λ 1 , . . . , λ m−1 in K such that the Lyapunov characteristics exponents for those generators of α are where φ 1 , . . . , φ m are different embeddings of K into R.
In the general case one applies Corollary 1 to the restriction of the action to the stationary subgroup γ for each of the sets R i . Those restrictions for i = 1, . . . , n are isomorphic and hence have the same entropy that is also equal to the entropy of α(γ) for any γ ∈ Γ with respect to the non-ergodic measure µ. Let k be the index of Γ. Since k-th power of any element of Z m−1 lies in Γ and every element is a power of a generator one immediately obtains description of exponents in the general case.
Corollary 2. Lyapunov exponents of any element of a maximal rank action α have the form where λ is a unit in a totally real algebraic number field of degree m, and k is a positive integer that depends only on α but not on the element. Here as before φ 1 , . . . , φ m are different embeddings of the field into R.
Since entropy of an action element is equal to the Mahler measure of the corresponding unit we can use exponential below estimate for the Mahler measure for totally real fields [28,6] to obtain a lower bound on entropy.
Corollary 3. The entropy of any element of a weakly mixing maximal entropy action on an m-dimensional manifold is bounded from below by cm, where c is a universal constant.

1.2.2.
Entropy and isomorphism rigidity. Eigenvalues of an integer matrix A, when simple and real, determine its conjugacy class over R and hence over Q. Assume that det A = ±1. That in turn determines a conjugacy class of the automorphism of the torus F A up to a common finite factor or finite extension. By [12,Theorem 5.2] for a broad class of Z k , k ≥ 2 actions by automorphisms of a torus, that includes all Cartan actions, measure theoretic isomorphism (with respect to Lebesgue measure) implies algebraic isomorphism.
Notice that passing to a finite factor or finite extension does not change entropy. Likewise the entropy for affine actions with the same linear parts are the same. By symmetry the entropy does not change if all generators of an action are replaced by their inverses. Theorem 1 allows to show that entropy function determines a maximal rank action action on a finite index subgroup up to a measurable isomorphism with above mentioned trivial modifications. We call the next statement a corollary, despite the length of the argument needed to deduce it from Theorem 1. The point is that the argument is purely algebraic and deals only with linear actions, modulo choosing appropriate finite index subgroups.
Corollary 4. Let α and α ′ be two maximal rank actions. Assume that they are both weakly mixing and their entropy functions coincide. Then restrictions of α and α ′ to a certain subgroup Γ ⊂ Z m−1 of finite index are finite factors of measurably isomorphic actions, possibly with replacing all generators of one action by their inverses.
Proof. Let α 0 and α ′ 0 be the algebraic models for α and α ′ . Weak mixing implies that for both of them n = 1. Now take finite coversα 0 andα ′ 0 (if necessary) that are actions by affine transformations of T m . Take finite index subgroups of Z m−1 for whichα 0 andα ′ 0 act by automorphisms. Taking the intersection of those subgroups obtain a finite index subgroup Γ 1 for which bothα 0 andα ′ 0 act by automorphisms. Those restrictions still have identical entropy functions. Now there is a subgroup Γ 2 of finite index such that eigenvalues for all generators of both actions are positive. Let Γ = Γ 1 ∩ Γ 2 . Since eigenvalues are simple they are thus determined for the Γ action by Lyapunov exponents. But by [12,Proposition 3.8] irreducible (in particular, Cartan) actions by automorphisms with the same eigenvalues of their generators are algebraically conjugate to finite factors of the same action.
Let us show that Lyapunov exponents are in turn determined by the entropy function, possibly with replacing all generators by their inverses. To see that, notice first that entropy function is not differentiable exactly at the union of kernel of the Lyapunov exponents, the Lyapunov hyperplanes. Thus it determines every Lyapunov exponent up to a scalar multiple. In the Cartan case for each Lyapunov exponent χ there is exactly one Weyl chamber C χ where this exponent is positive and all other negative. Inside this Weyl chamber entropy is equal to χ. This Weyl chamber and its opposite −C χ are determined from the configuration of Lyapunov hyperplanes as the only two whose boundaries intersect all Lyapunov hyperplanes except for Ker χ. Thus for every Lyapunov exponents χ of α there is en exponent χ ′ of α ′ such that is equal to either α or −α. Let us show that for all exponents the sign is the same. For, suppose that for two exponents χ 1 and χ 2 of α there are exponents χ 1 and −χ 2 of α ′ . then in the Weyl chamber C χ 1 χ 2 is negative, hence in this Weyl chamber the entropy of α ′ is at least χ 1 − χ 2 , i.e greater than the entropy of α. Thus all exponents of α and α ′ are either equal or have opposite signs. In the latter case we can change generators of α ′ to their inverses and obtain actions with equal exponents.
In the case of maximal rank actions that are not weakly mixing one restricts the action to the stationary subgroup Γ for each of the sets R i and to apply Corollary 4 to each of those sets. An obvious additional invariant is the index of Γ. It can be determined, for example, from the discrete spectrum of the action. This spectrum determines how different elements of the action interchange the sets R i A conjugacy between restrictions of the action of γ to different sets R i can be effected by using the action of elements from corresponding cosets of Γ. Hence Corollary 4 can be simultaneously applied to those sets.
Corollary 5. Let α and α ′ be two maximal rank actions. Assume that they have the same discrete spectrum and their entropy functions coincide. Then restrictions of α and α ′ to a certain subgroup Γ ⊂ Z m−1 of finite index are finite factors of measurably isomorphic actions, possibly with replacing all generators of one action by their inverses. Any such cocycle over a C ∞ maximal rank action can be transferred to a cocycle over a finite extension of the linear model the same way as is described in the proof of [16,Theorem 2.8]. This proof works verbatim in our case and produces cocycle rigidity. Corollary 6. Any Lyapunov Holder (corr. Lyapunov smooth) real valued cocycle over a C r , 1 + θ ≤ r ≤ ∞ maximal rank action is cohomologous to a constant cocycle via a Lyapunov Holder (corr. Lyapunov smooth) transfer function (with the obvious proviso that the Lyapunov regularity of the transfer function is less that regularity of the action).
1.3. The topology theorem. Let L denotes either a torus T m or the infratorus T m ± ; Theorem 2. Let α be a C r , 1 + θ ≤ r ≤ ∞ maximal rank positive entropy action, then (1) the sets R 1 , . . . , R n in Theorem 1 can be chosen inside open Γinvariant subsets O 1 , . . . , O n , that are also interchanged by α; (2) each map h i extends to a continuous maph i : is a regular point in the infratorus then there exists an arbitrary small parallelepiped P x (in some linear coordinates) containing x such that on the boundary of P x the map h i is invertible and the inverse is a diffeomorphism on every face of P x ; (4) if x ∈ F is a singular point in the infratorus T m ± then there exists an arbitrary small projective parallelepiped P x (the factor of a centrally symmetric parallelepiped in some linear coordinates by the involution t → −t) containing x such that on the boundary of P x the map h i is invertible and the inverse is a diffeomorphism on every face of P x ; Remark 6. Notice that any singular point in L must be in the exceptional set F because topology of a small neighborhood of a singular point is different from that of points in a manifold.
1.4. Topological corollaries. Theorem 2 allows to make conclusions about topology of manifolds that admit maximal rank positive entropy actions. Right now we list only some of those properties that can be derived quickly. More detailed discussion of the consequences of Theorem 2 will appear in a separate paper.
Corollary 7. Let M be a connected manifold of odd dimension m ≥ 3 that admits a maximal rank positive entropy action of Z m−1 . Then if M is orientable it is homeomorphic to the connected sum of the torus T m with another manifold. If M is non-orientable, its orientable double cover is homeomorphic to the connected sum of the torus T m with another manifold.
In particular, in both cases the fundamental group π 1 (M) contains a subgroup isomorphic to Z m .
Proof. Consider the orientable case first. In odd dimension the infratorus T m ± is not orientable and the same is true to its complement to a finite set or, equivalently, to the complement to the union of finitely many small balls. Since an open subset of an orientable manifold is orientable the open subset S = Int h −1 i R ⊂ M is orientable and hence L is the torus. Now take a disc D ⊂ T m that contains the set F and consider closed set H −1 (T m \ D). Its boundary is a sphere and thus M is the connected sum of T m and a manifold that is obtained by glueing a disc to the boundary of M \ H −1 (T m \ D). Now assume that M is non-orientable and take the orientable double coverM of M. Let I :M →M be the deck transformation. Consider lifts of the elements of the maximal rank action α toM . Each element f has two lifts f 1 and f 2 = f 1 I. The involution I commutes with all lifts. The group Γ consisting of all lifts is either abelian or its commutator is the group of two elements generated by I Let us show that Γ has a finite index abelian subgroup isomorphic to Z m−1 . If Γ is already abelian then it is isomorphic to the direct product Z m−1 × Z/2Z. Otherwise consider generators of the action α and let f 1 , . . . , f m−1 be their lifts toM . Centralizer Z(f i ) of each of those elements in Γ is either the whole of Γ, or an index two subgroup. This follows from the fact I 2 = Id and that I is in the center of Γ since this implies that the product of any two elements not in Z(f i ) belongs is a finite index abelian subgroup of Γ that belongs to its center. Notice that the index of Z in Γ is at most 2 m−1 . Since the only finite order element of Γ is I, Z is isomorphic to Thus Z m−1 acts onM by lifts of elements of α. This is obviously a maximal rank positive entropy action so that from the argument for the orientable caseM is the connected sum of torus with another manifold. Since π 1 (M ) embeds into π 1 (M) the former a subgroup isomorphic to Z m .
A very similar argument allows to partially extend Corollary 3 to actions that are not weakly mixing. Proof. As before, let us consider the orientable case first. The obvious below estimate the entropy for the elements of the action of Γ on each ergodic component, divided by the number n of ergodic components that is equal to the index of Γ. Hence this number needs to be estimated from above. repeating the argument about connected sums from the proof of Corollary 7 we deduce that M is homeomorphic of the connected sum of n copies of the torus T m and another manifold. Hence by Mayer-Vietoris theorem β 1 (M) ≥ mn. Now Corollary 3 implies the needed estimate.
In the non-orientable case we consider the orientable double cover, lift the action as in the proof of Corollary 7 notice that entropy does not change and use the estimate in the orientable case. Since the Betti number of the manifold is the same as of the orientable double cover, the inequality follows.

Remark 7.
Notice that there is no estimate from below that depends on dimension only as in the weak mixing case. An appropriately modified version of the suspension construction over a weakly mixing action on the torus involving maxing holes around fixed points similarly to [13] and connecting them by cylinders, similarly to the filling of holes descried in [15], produces examples with arbitrary low entropy.
Even-dimensional case is more complicated. While the case L = T m of course works the same way, if L = T m ± the manifold M may not be a connected sum with the infratorus as one of the components. Since in this case the infratorus is orientable the double cover trick does not work. Indeed, there are some simply-connected manifolds, for example, some K3 surfaces, that admit maximal rank actions. Still some conclusions can be drawn. Here is a simple example. Furthermore, there is a unique measurable system of smooth affine parameters on the leaves of a Lyapunov foliation [16,Proposition 3.3]. In fact, these affine parameters and conditional measures are closely related: the affine parameter is obtained by integrating the conditional measure.
The next fact used in the proof is [16, Proposition 4.2] that asserts that conditional measures (and hence affine structures) are invariant (in the affine sense) with respect to holonomy along complimentary directions that includes all remaining Lyapunov foliations.
Remark 8. While absolute continuity of conditional measures of an absolutely continuous hyperbolic measure, as well as connection between conditional measures and affine structures, are fairly general facts, holonomy invariance is a specific higher rank phenomenon: it fails already for non-algebraic area-preserving Anosov diffeomorphisms of two-dimensional torus.
Naturally, both conditional measures and affine parameters are defined up to a scalar multiple. Fixing a measurable normalization smooth along the leaves of the Lyapunov foliation in question produces a cocycle; different normalizations produce cohomologous cocycles.
Remark 9. Notice that in the proof of [10, Theorem 4.1] we use a special time change for the suspension action and for this time change the expansion coefficient of the original suspension action in the Lyapunov direction is indeed cohomologous to a constant. This however does not imply that expansion coefficient for the original action or its suspension is cohomologous to a constant. For example this is not the case for hyperbolic flows where W. Parry constructed synchronization time change that inspired our construction. [23] 2. Affine structures and Holonomies 2.1. Weak-mixing reduction. Let us start with the weakly mixing reduction which gives the decomposition claimed in Theorem 1. Let α : Z m−1 → Dif f (M m ) be a C 1+θ action as in Theorem 1. Take n ∈ Z m−1 such that µ is a hyperbolic measure for α(n) absolutely continuous w.r.t. Lebesgue measure. By Pesin Ergodic decomposition Theorem [24] there is k > 0 and a α(kn)-invariant set R 1 ⊂ M of positive µ-measure such that α(kn)|R 1 is a Bernoulli automorphism, in particular weakly mixing. Set n 1 := kn. By ergodicity of α(n 1 )|R 1 we have that for any m Since µ is an ergodic invariant measure for the whole action there are This is the decomposition and the finite index subgroup claimed in Theorem 1.
To simplify notation, let us assume for the rest of the paper that Γ = Z m−1 and that R 1 has full measure. Under this assumption we can prove the following.
Lemma 2.1. For any hyperbolic element n ∈ Z m−1 , (i) α(n) is Bernoulli and (ii) there is a set of full measure R such that for any Weyl chamber is a set of full measure.
Proof. Let us show first that α(n) is weakly mixing (and hence Bernoulli) for any hyperbolic element n ∈ Z m−1 . Using Pesin Ergodic decomposition Theorem we have a k > 0 and a setR ⊂ M of positive µ measure invariant by α(kn) such that α(kn)|R is weakly mixing (and Bernoulli). This set is an ergodic component of α(kn). Now remember that there is an element of the action that we previously denoted by n 1 such that α(n 1 ) is weakly mixing. Since it commutes with α(kn), it interchanges ergodic components of the latter map. If there is more than one ergodic component for α(kn), α(n 1 ) has a non-constant eigenfunction. Thus weak mixing of α(n 1 ) implies thatR o = M and k = 1, so that α(n) is weakly mixing and hence Bernoulli.
Let now C be a Weyl chamber and let n ∈ C. Observe that if P is a Pesin set for a hyperbolic measure then for every point x ∈ P there is an open neighborhood P x of a fixed size (a Pesin box) such that for every y ∈ P ∩ P x the local stable manifold of x intersects the local unstable manifold of y transversally (at a single point). Since α(n) is hyperbolic and weak mixing, by the previous observation, for a.e. x, y there is a non-negative integer k ≥ 0 such that W u C (α(kn)(y)) intersects transversally W s C (x) (simply take an iterate so that α(kn)(y) ∈ P ∩P x ). Now let k(x, y) be the minimum of such integers k. It is clear that k(α(n)(x), α(n)(y)) = k(x, y) for µ × µ-a.e. (x, y) and by the remark about Pesin boxes k(x, y) = 0 on a set of positive µ × µ measure. Now, since α(n) is weak mixing, we have that α(n) × α(n) is ergodic and hence k(x, y) = 0 a.e.
Finally this statement is equivalent to the statement (ii) of the lemma.

Affine structures.
In this subsection we shall define affine structures along the leafs of the invariant foliations and prove they are coherent, in the next subsection we shall see that holonomy maps are affine with respect this affine parameters.
Let χ be a Lyapunov exponent of α and W = W χ be the corresponding Lyapunov foliation defined µ almost everywhere. There is a unique α-invariant family of smooth affine parameters defined on almost every leaf of W. Those affine structures change continuously within any Pesin set, see [10, Proposition 7.2] and hence they can be defined not only almost everywhere (at "typical" leaves with respect to recurrence/ergodic behavior of α) but at other specific important places such as leaves passing through periodic points that belong in a Pesin set. Those affine parameters are obtained by integrating telescoping products, see the proof of [9,Lemma 3.2]. But at the same time affine parameters define conditional measures of µ with respect to W.
By [16,Proposition 4.2] those affine structures are invariant with respect to the holonomy along leaves of the stable foliation of any generic singular element α(t), t ∈ R k for which χ(t) = 0.
We will present now the affine structures along stable manifolds of any Weyl chamber C and prove their coherence.
Since Lyapunov hyperplanes are in general position any combination of signs of Lyapunov exponents, except for all positive or all negative, is possible for elements of the action α, and hence there are no resonances. In particular for every Lyapunov exponent χ there is a Weyl chamber C χ such that inside C χ , χ is the only positive Lyapunov exponent. Hence the stable manifolds W s Cχ (x) have codimension one. For any Weyl chamber C the manifolds W s C (x) are the intersections of W s Cχ (x) over those χ that are negative in C. In particular, leaves of any Lyapunov foliation are intersection of n − 1 codimension one stable manifolds.
Moreover, we can take elements of the action with pinched Lyapunov spectrum.
Let C be a Weyl chamber and let s = s(C) be the dimension of W s C . Given s ≥ 1, let D s be the group of invertible diagonal matrices on R s and let Emb 1+ǫ (R s , M) be the space of C 1+ǫ embeddings of R s into M with the topology of C 1+ǫ convergence on compact subsets, observe that in this way Emb 1+ǫ (R s , M) is a polish space.
From [15,Proposition 2.7] we have that there is a unique family of smooth affine structures on the leaves of foliation W s C . Proposition 2.2. Let α be a C 1+θ action as in Theorem 1. Then there are ǫ > 0, a set of full measure R ⊂ M and a measurable map sends the standard basis into the frame of invariant spaces E χ i where E s C (x) = E χ 1 ⊕ · · · ⊕ E χs for some numeration of the Lyapunov exponents. (4) There is a cocycle of diagonal maps of R s , A : Z m−1 × R → D s such that H α(n)(x) • A(n, x) = α(n) • H x for every n ∈ Z m−1 and x ∈ R. Such a family is unique modulo composition with a diagonal map D : R → D s , i.e. ifĤ is another affine structure then for a.e. x, In general, if α is C r , the affine structures can be taken C r−δ for any δ > 0.
We want to prove coherence of the affine structures built in Proposition 2.2 along stable manifolds.
There is a set of full measure R ⊂ M such that if x, y ∈ R and y ∈ W s C (x) then H −1 y • H x is an affine map with diagonal linear part.
Proof. Take some n in the Weyl chamber C such that W s C (x) is the stable manifold for α(n). Let us number the Lyapunov exponents so that χ 1 (n) < 0, . . . χ s (n) < 0 for n ∈ C. Take L a Luzin set of continuity of z → H z of µ-measure close to 1. Then there is a set of full measure R n ⊂ M such that whenever x, y ∈ R n then there are iterates l i → +∞ such that α(l i n)(x), α(l i n)(y) ∈ L for every i. This can be found using Birkhoff ergodic theorem as long as L has µ-measure larger that 1/2. The set of full measure R we claim in the proposition is the intersections of R n 's for finitely many choices of n ∈ C according to some pinching of the Lyapunov spectrum to be determined later.
Take x, y ∈ R n with y ∈ W s C (x). Put A (l) (x) = A(ln, x) and similarly with y. We have that ) −1 and by continuity on Luzin sets and since d(α(l i n)(x), α(l i n)(y)) → 0 (1)) stands for the sup C 1 norm on the unit ball).
We have also that A (l) (x) =: diag(λ (l) k (x)) and we know that lim Let P = (P 1 , . . . , P s ), take P k and let us show that ∂ j P k ≡ 0 for j = k. For given l, we have that Applying (2.2) and that l i A (l i ) (x)(B(1)) = R s , we get that if λ k /λ i > 1 then ∂ j P k ≡ 0 on R s . Hence take n k,j ∈ C such that χ k (n k,j ) > χ i (n k,j ) and we get that ∂ j P k ≡ 0 on R s for j = k.
So, we have that P k (v) = P k (v k ) only depends on the kth variable. Denote with P ′ k the derivative of P k . Then again using formula (2.2) and arguing as before we get that for any R > 0, In particular this implies that for any t ∈ R which gives the claim since the left hand side does not depend on t and then P ′ k is constant and hence P k is affine. So we get that P is an affine map with diagonal linear part.
Using uniqueness of affine structures and absolute continuity of the invariant measure we get that: The main point in the Lemma is to prove that for a typical x, W s C 1 (x) is a coordinate plane in the affine structure of W s C 2 (x). Ones we settle this then the Lemma follows from uniqueness of affine structures.
For a.e. point x we may assume that (Lebesgue) almost every point y in W s C 2 (x) is a regular point, moreover we may assume also that for Lebesgue a.e. point y ∈ W s C 2 (x), (Lebesgue) a.e. point z ∈ W s C 1 (y) is a regular point. This is just absolute continuity of invariant foliations plus the fact that conditional measures are equivalent to Lebesgue.
Let us denote with H z : R s → W s C 2 (z) the affine structures on W s C 2 (z) = W s C 2 (x) based at z. Let x be a point like in the previous paragraph and y a regular point in W s C 2 (x). By Proposition 2.2, item (3) we have that there is a coordinate plane V such that D 0 H x (V ) = T x W s C 1 (x) Let us consider the manifold W = H −1 x (W s C 1 (y)) ⊂ R s and let us show that this manifold is a plane parallel to V . Let us assume that (Lebesgue) a.e. z ∈ W s C 1 (y) is regular point. We shall show that for Lebesgue a.e. point a ∈ W , T a W = V . By Proposition 2.2, item (3) we have that D 0 H z (V ) = T z W s C 1 (z), and since W s and hence T a W = V for Lebesgue a.e. a ∈ W . Since W is a C 1 manifolds then we have that T a W = V for every a ∈ W and hence W is a plane parallel to V as wanted. uniquely extends to a smooth foliation, indeed an affine foliation in the affine coordinates of W s C 2 (x) A corollary of the above is the following, Corollary 2.5. Let C 1 and C 2 be two Weyl chambers and let C 3 be the Weyl chamber such that E s is a linear subspace in the affine structures along W s C 1 (x) and W s C 2 (y) tangent to the space corresponding to E s C 3 . Now Hopf argument can be applied and we obtain Corollary 2.6. Affine structures on the leaves of the Lyapunov foliation W are invariant with respect to the holonomy along the leaves of W C .

2.3.
Uniformity of the holonomies. Now we want to show that holonomy maps along unstable manifolds between two stable manifolds are almost everywhere defined w.r.t. Lebesgue measure on stables.
Proposition 2.7. Let C be a Weyl chamber. There is a set of full measure R := R C ⊂ M such that if x, y ∈ R and y ∈ W u C (x) then the holonomy along unstables Hol u,C x,y : W s C (x) → W s C (y) is defined for Lebesgue a.e. z ∈ W s C (x) and is affine, i.e. there is a diagonal linear map B preserving the frame such that for a.e. z ∈ W s C (x), there is a point Hol u,C x,y (z) ∈ W u C (z) ∩ W s C (y) and moreover Proof. Let as assume first that W s C is 1-dimensional. Let C 1 and C 2 be Weyl chambers such that E u C 1 ⊕ E u C 2 = E u C . By Lemma 2.4 we have that W u C 1 and W u C 2 are a pair of transverse linear sub-foliaitons of W u C . Hence there is a set of full measure R 0 such if x ∈ R 0 and y ∈ R 0 then there are regular points a, b ∈ W u C (x) such that a ∈ W u C 1 (x), b ∈ W u C 1 (y) and a ∈ W u C 2 (b). We have that Hol u,C x,y (z) = Hol u,C b,y • Hol u,C a,b • Hol u,C x,a (z), see Figure 1.
for some Weyl chamber C 4 we have by Lemma 2.4 that W u C and W s C 1 are transverse linear subfoliations of W u C 3 (x) and hence Hol u,C x,a : W s C (x) → W s C (a) is affine holonomy inside W u C 3 (x) which is everywhere defined. Similarly with Hol u,C b,y and Hol u,C a,b which settles the claim when W s C is 1-dimensional. Let us now assume that the dimension of W s C , s(C), is larger that one and assume by induction that we have proven the Proposition for dimension s < s(C). Then we have that there are Weyl chambers C 1 and C 2 such that E u We have that the dimensions s(C i ), i = 1, 2 of W s C i are strictly smaller than s(C). By the induction hypothesis, we have that for a.e. points x and y ∈ W u C (x) ⊂ W u C 1 (x), Hol u,C 1 x,y : W s C 1 (x) → W s C 1 (y) is everywhere defined and is affine. Taking x, y typical points, we have that Lebesgue a.e. point a ∈ W s C 1 (x) is regular, and Hol u,C 1 x,y (a) ∈ W u C 1 (a) ∩W s C 1 (y) is also regular, see Figure 2. Moreover, Hol u,C 1 x,y (a) = Hol u,C x,y (a). Indeed, for some Weyl chamber C 3 which gives by Lemma 2.4 that W u C and W s C 1 are a pair of transverse affine foliations in W u C 3 (x) = W u C 3 (y) and hence Hol u,C 1 x,y (a) = Hol u,C x,y (a) ∈ W u C (a) ∩ W s C 1 (y) ⊂ W u C 1 (a) ∩ W s C 1 (y), see Figure 2.
in the conditions of the previous paragraph. Hence Hol u,C x,y (a) is well defined and again by induction we have that Hol u,C 2 a,Hol u,C x,y (a) : is Lebesgue a.e. defined and is affine and Hol u,C 2 a,Hol u,C x,y (a) (z) ∈ W u C 2 (z) ∩ W s C 2 (Hol u,C x,y (a)).
Moreover, since E u C and E s C 2 are jointly integrable, arguing as before, we have that Hol u,C 2 a,Hol u,C x,y (a) (z) = Hol u,C x,y (z).
Which gives that Hol u,C x,y is Lebesgue a.e. defined and we get the Proposition.
When C is understood we shall note directly Hol u x,y = Hol u,C x,y . Proposition 2.8. For x, a, b ∈ R C , a ∈ W u C (x) and b ∈ W s C (x) we have that Hol s x,b (a) = Hol u x,a (b). Proof. The proof of this Proposition is very similar to the previous one. Assume without loss of generality that dimension of W s C is larger that 1 (if not consider W u C instead). Even though we will use lot of Weyl chambers to settle the Proposition, all holonomies here will be w.r.t. invariant manifolds of the Weyl chamber C. Figure 3. Proposition 2.8 Let C 1 and C 2 be Weyl chambers such that E s C 1 ⊕ E s C 2 = E s C . Then we have that there are regular points z 1 , z 2 ∈ W s C (x) such thatz 1 := Hol u x,a (z 1 ) andz 2 := Hol u x,a (z 2 ) are well defined and regular points and z 1 ∈ W s C 1 (x), z 2 ∈ W s C 2 (z 1 ) and b ∈ W s C 1 (z 2 ), see Figure 3. We know on one hand that Hol s x,b = Hol s z 2 ,b • Hol s z 1 ,z 2 • Hol s x,z 1 . We have also that Hol u x,a = Hol u z 1 ,z 1 = Hol u z 2 ,z 2 . Let C 3 and C 4 be Weyl chambers such that E u Then we have, using that x, a, z 1 ,z 1 ∈ W u C 3 (x) that z 1 = Hol u x,a (z 1 ) = Hol s x,z 1 (a) since all these holonomies take place inside W u C 3 (x). Following this argument we get also that z 1 , z 2 ,z 1 ,z 2 ∈ W u C 4 (z 1 ) and hencē z 2 = Hol u x,a (z 2 ) = Hol u z 1 ,z 1 (z 2 ) = Hol s z 1 ,z 2 (z 1 ). Finally we get that, since z 2 ,z 2 , b, Hol u . Putting all together we get the Proposition.
3. The arithmetic structure 3.1. The quotient. Given l ≥ 1, let D l be the group of invertible diagonal matrices on R l and let A l be the group of affine maps on R l whose linear term are in D l . In the sequel, when we say almost everywhere (a.e.) we mean w.r.t. Lebesgue measure unless another measure is clearly specified.
Let us summarize in the following Proposition what we have proven in the previous sections. The first 3 items are Proposition 2.2.
Proposition 3.1. Given any Weyl chamber C there is a set of full measure R C such that if x ∈ R C then there are C r affine parameters There is a cocycleα C : Z m−1 × R C → D s(C) such that for µ-a.e. x,

) [Lemma 2.4] For any intersection of stable spaces of different
Weyl Chambers, is a foliation by parallel planes in the affine parameters for every i. (6) [Proposition 2.7] If x, y ∈ R C and y ∈ W u C (x) then the holonomy along unstables map Hol u x,y : Hol s x,b (a) = Hol u x,a (b). Observe that in the affine coordinates the conditional measure is standard Haar measure with some normalization.
Let us fix by now a Weyl chamber C that we will not be moved. Considering −C we have affine parameters along the unstable foliation for points in R −C , take R = R C ∩ R −C , we shall assume also that R intersected with a.e. Lyapunov manifold has full Lebesgue measure, also we may need to reduce R finitely many times to sets of still full measure satisfying adequate properties. Let us omit any reference to C in the definitions above and denote with H s x the affine parameters along stable manifolds W s and H u x for the affine parameters along unstable manifolds W u .
We shall define a kind of covering or developing map for M, defined almost everywhere. Roughly, the idea is as follow, for a given x we defineĥ x : W s (x) × W u (x) → M,ĥ x (a, b) = Hol u x,b (a) = Hol s x,a (b) which is a point in W u (a) ∩ W s (b). Then we use affine parameters on W s (x) and W u (x) to define a map h x : R s ×R u → M. Since holonomies are only defined a.e. we need to take some care and that is what we do in the following paragraphs.
Let us fix x ∈ R and assume that W s (x) ∩ R and W u (x) ∩ R have full Lebesgue measure. Call . Observe that by item (6) for a.e. (z s , z u ), We have as an immediate corollary of the previous Lemma: Let Γ x be the group of L ∈ A m such that for Lebesgue a.e. (z s , z u ) ∈ R s × R u = R m . Γ x should be thought as the group of deck transformations of the "covering" h x . We call Γ x the Homoclinic Group since there is a correspondence between the points in W u (x) ∩ W s (x) and Γ x . It is a nice experience for the reader to understand the previous construction in the case of a hyperbolic automorphism of T 2 ± We consider R m with its natural additive group structure and let λ be Haar (=Lebesgue) measure on R s × R u = R m . It is the product of Haar measure on R s and Haar on R u . Lemma 3.5. For Lebesgue a.e.z = (z s , z u ) there is c x (z) > 0 and for any ǫ > 0 there is δ > 0 and a set K ǫ (z) ⊂ B δ (z), such that Proof. By Lemma 3.3 and Corollary 3.4, it is enough to prove the Lemma whenz = (0, 0). Given ǫ > 0, since x is a regular point it belongs to some Pesin set, and we may assume it is a density point of a Pesin set, hence we can take δ small so that for K ǫ equal the pre-image by h x of the Pesin set intersected with the ball of radius δ around 0 items (1) and (2) hold. For item (3) notice that the conditional measure of µ along stables and unstables is Haar measure (with some normalization). Hence, by holonomy invariance of the conditional measures, we have that, locally on Pesin sets, the measure µ is the product of Haar on stable and Haar on unstable which gives Haar in affine coordinates h x .
Applying item (3) of Proposition 3.1 both for s and u, the definition of h x and Oseledcs Theorem we get the following: |α x (kn)| 1/k → D(n) = diag(exp χ 1 (n), . . . , exp χ m (n)), as k → ±∞, where for a diagonal matrix D, |D| the matrix with entries its absolute value and |D| 1/k is its real positive k-th root.
(3) For any n ∈ Z m−1 , The last assertion follows from the definition of Γ x and Corollary 3.4. Proof. Given x let T r x ⊂ Γ x be the normal subgroup of translations in Γ x . We always regard R m with the standard inner product. Let E(x) ⊂ R m be the vector space generated by the translations in T r x . By Lemma 3.5 we have that T r x is discrete and hence the quotient E(x)/T r x is a torus. Let v(x) be the volume of E(x)/T r x . Notice that y → v(y) is a measurable map. Indeed, by Lemma 3.3 we have that for a.e. y there is a L x,y such that h x •L x,y = h y , and by the construction we get that we can choose L x,y in such a way that x → L x,y is measurable.
Let D x,y be the linear part of L x,y . Some linear algebra gives that D x,y T r y = T r x which gives the measurability of y → v(y).
On the other hand, by Lemma 3.6 and arguing as before we have that for any n ∈ Z m−1 and a.e. x ∈ M,α x (n)T r x = T r α(n)(x) . Let D(n) = diag(exp χ 1 (n), . . . , exp χ m (n)).
Let y be a typical point. Let d = dim E(y). It is clear from the previous analysis and ergodicity of α that d does not depend on y. Let us assume by contradiction that 0 < d < m then, since the Lyapunov exponents of α are in general position we can chose an element n ∈ Z m−1 such that for l → +∞ the action of the iterates D(ln) in the dth exterior product Λ d (R m ) expands the volume element of E(y) exponentially. Since the cocycleα y (ln) is asymptotically D(ln) we have thatα(ln) y also expands the volume element of E(y) exponentially. Hence we get that for l → +∞, v(α(ln)(y)) tends to infinity contradicting that by recurrence α(ln)(y) has to return to a region where v(y) is finite.
Hence we get that either d = 0 or d = m.
If d = 0 then Γ x has no translation part and hence is abelian (consider the homomorphism D 0 : Γ x → D m , D 0 L = derivative of L at 0). So, Γ x is conjugated by a translation to the action of a diagonal subgroup on R m .
Let F (x) = F ix(Γ x ) be the set of points fixed by all the elements in Γ x . Observe that we have that F (x) is an affine subspace parallel to the coordinate axes. We shall show that 0 ∈ F (x) for µ-a.e. x (i.e. F (x) is a linear subspace) and then reach a contradiction.
From Lemma 3.6 we get that F (α(n)(x)) =α x (n)F (x). Sinceα x (n) is diagonal and by ergodicity of α(n) we get that there is a linear subspace F paralell to the axes, independent of x and a unique vector p(x) perpendicular to F such that F (x) = p(x) + F for µ-a.e. x. Moreover p(x) is measurable,α x (n)F = F andα x (n)p(x) = p(α(n)(x)) for every n ∈ Z m−1 . Since by Lemma 3.6 the cocycleα x (n) is diagonal and asymptotically D(n) = diag(exp χ 1 (n), . . . , exp χ m (n)) we get, working with each coordinate of p(x) at a time that p(x) = 0.
So we have that F (x) = F is a linear subspace independent of x and 0 ∈ F = F (x). In particular Γ x ⊂ D m . Since h x |R s × {0} coincides with the affine parameter along the stable manifold of x we get that R s × {0} ⊂ F , similarly we get that {0} × R u ⊂ F and hence F = R m , i.e. Γ x = {id} is trivial.
In particular, by Corollary 3.4 we get that h x : R m → M is 1-1 Lebesgue a.e. Let ν = (h x ) * µ. By Lemma 3.5, ν is a a probability measure equivalent to Lebesgue measure and invariant by the action α 0 (n) := h −1 x • α(n) • h x . By Lemmas 3.3 and 3.6 we have that α 0 (n) is affine for every n. But this is a contradiction since affine maps on R m do not admit positive entropy invariant probability measures but (α 0 (νn), ν) is measurably isomorphic through h x to (α(n), µ). So we get that d = m and hence E(x) = R m . Recall that from Lemma 3.5 we know that T r x is discrete. Let us take a linear map and conjugate T r x to Z m and Γ x to Γ. Since T r x is normal in Γ x then we have that Z m is normal in Γ. Hence we have thatΓ = Γ/Z m is identified with a subgroup of affine maps on the torus T m = R m /Z m , and R m /Γ x ∼ R m /Γ = T m /Γ. Again, using Lemma 3.5 we have thatΓ cannot have any recurrence and hence it has to be finite, finishing the proof.
So we get that R m /Γ x is a well defined orbifold. By Corollary 3.4 we get that for any n ∈ Z m−1 . Let ν = (h x ) * µ be the pullback measure.
Proof. The only thing that needs proof in this Corollary is the property on the group and on ν. We know already that Z m is a finite index normal subgroup of Γ x . Letα 0 be the lifting of the action α 0 to the finite covering T m and let us lift also the measure ν to T m . By the generic position of the Lyapunov exponents for α we get thatα 0 is a restriction of a maximal Cartan action to a finite index subgroup and hence we get that the lifted measure is absolutely continuous w.r.t. Lebesgue and invariant and hence is Haar measure. Hence we get the claim on the measure. Again using thatα 0 is a maximal Cartan action on T m we get that the only possibility for Γ x /Z m is to be {±id} and we get the Corollary.
Proof of Theorem 1 . By the weak-mixing reduction subsection 2.1 we have the a set R 1 with µ(R 1 ) > 0 and a finite index subgroup stabilizing R 1 . By restricting the action to this finite index subgroup and normalizing the measure we may assume that the measure is weakmixing and hence by Lemma 2.1 we get that there is a set of full measure R 2 such that for any x ∈ R 2 , is a set of full measure. As a consequence of the construction of h x we get that the image of h x contains R 3 and hence has full measure, hence h x : (R m /Γ x , ν) → (M, µ) is an isomorphism conjugating α with α 0 . By Corollary 3.8 we have that R m /Γ x is either a torus or the infratorus T m ± . Take some x and define h = h −1 x . This gives the first part and items (1) and (2) Observe that we can use h x and its restriction to planes parallels to the axes as new affine parameters. This are still smooth parameters and with this new affine parameters holonomies are isometries.
For future use, let us summarize some properties of the measurable conjugacy.
Lemma 3.9. There is an α 0 -invariant set of full Lebesgue measure R ⊂ R m /Γ x in the infratorus such that for every v ∈ R the measurable conjugacy h x restricted to any invariant linear subspace v+E ⊂ R m /Γ x trough v coincides with the affine structure through W E (h x (v)) ⊂ M (W E (y) ⊂ M is the invariant manifold associated to E through y). In particular, for a.e. y and every Weyl chamber C, h −1 x |W u C (y) is a diffeomorphism onto h −1 x (y) + E u C (the corresponding unstable plane) and holonomies are isometries in this affine parameters.

Anosov actions
An action α : Z k → Dif f (M) is an Anosov action if there is n 0 ∈ Z k such that α(n 0 ) is an Anosov diffeomorphisms.
Theorem 3. Let (α, µ) be an action as in Theorem 1, i.e. a maximal rank action, assume furthermore that α is an Anosov action. Then α is smoothly conjugated to α 0 and hence M is indeed diffeomorphic to a (standard) torus.
We shall prove that the measurable conjugacy in Theorem 1 is indeed a homeomorphism.
Proof. Let x ∈ M be a regular point and consider h x : R s × R u → M which is defined almost everywhere with respect to Lebesgue measure on R s × R u , we consider here the Anosov element and take the Weyl chamber containing this Anosov element for the definition of h x . First of all observe that by definition we get that there is ǫ > 0 and δ > 0 small such that if (z s , z u ) is δ close to (0, 0) then 0, z u )). This implies that h x restricted to the δ neighborhood B δ (0, 0) of (0, 0) is continuous. Now, using Proposition 3.3 we get that for Lebesgue a.e. (w s , w u ) there is an isometry L such that if y = h x (w s , w u ) then L(0, 0) = (w s , w u ) and h x • L = h y a.e. In particular h x restricted to the δ neighborhood of (w s , w u ), B δ (w s , w u ) is also continuous since h y is continuous when restricted to the δ neighborhood B δ (0, 0) of (0, 0) and h x = h y • L −1 and L is an isometry. Since δ is fixed we get that the union of the δ balls around Lebesgue a.e. point is R s × R u and hence h x is continuous everywhere.
Following the same reasoning as in the proof of Theorem 1 we get that h x is indeed a covering map and taking the quotient by the group of deck transformations we get that h x is a homeomorphisms and a conjugacy between the affine action α 0 on an infratorus and the action α.
Observe that here the infratorus is a manifold, hence, applying the results in [26] or [27] on global rigidity of maximal Anosov rank actions, we get the smooth classification.

Proof of Theorem 2
From Theorem 1 we have a decomposition into weak mixing components, a corresponding finite index subgroup of Z m−1 and a measurable conjugacy h : (M, ν) → (L, λ) between α and an affine action α 0 when restricted to this finite index subgroup. Here we shall show how h coincides with a continuous onto map from an α-invariant open set O and L \ F for some finite α 0 -invariant set F satisfying the conclusion of Theorem 2.
The first step is to identify the open set O and the finite set F . Given a Weyl chamber C and a regular point x let W σ C (x), σ = s, u be the stable and unstable manifolds through x corresponding to this Weyl chamber. With a subscript W σ C,loc (x) we shall denote the local invariant manifold once a Pesin set is understood. Let C 1 , . . . , C m denote the Weyl chambers with only one positive exponent. We say that a closed set B ⊂ M is a box or a cube if it is homeomorphic to the unit cube in R m and its boundary ∂B is in the union of stable and unstable manifolds for different Weyl chambers, i.e. there are regular points x i,± , i = 1, . . . , m, such that We shall call each piece a face of the cube B (or of its boundary ∂B). We are assuming that x + i and x − i do not belong to the same stable manifold, if not take connected components.
Given a Pesin set P , if we can take x i,± l ∈ P close enough so that then we say that B is a good box and we get as a consequence that For any given Pesin set P and for ν a.e. point x ∈ P there is a sequence of good boxes B l , l ≥ 1, such that: (1) x ∈ B l ⊂ intB l−1 and ∩ l≥1 B l = {x}, (2) B l is diffeomorphic to the closed unit cube, h is defined a.e. w.r.t. Lebesgue measure on ∂B l and coincides with a diffeomorphism with C r norm bounded by a constant depending only on P and h(∂B l ) is the boundary of a linear cubê B l , (5) For i = 1, . . . m, W s C i ,loc (x) disconnects B l into two connected components named B ± i,l which are also boxes and h(∂B ± i,l ) is the boundary of a corresponding linear cubeB ± i,l . Moreover, the points x i,± l ∈ P can be further required to belong to a given full measure set (e.g. has a dense orbit in the support of ν, etc).
Proof. Consider x a density point on the Pesin set P intersected with the set of full measure in Lemma 3.9. Since W s C i ,loc (x) locally separates a neighborhood of x in two connected components, we can take the points x i,± l from the same set as x and from both sides of W s C i ,loc (x), approaching x.
Given a good box B, for σ = s, u let be the core of the box B. Let W (u,±) C i (x) be the separatrix of W u C i (x)\{x} that intersects ∂ ± C i B and for a regular point y ∈ W s C i (x) we define W u,± C i (y) accordingly. Finally, for r > 0 let W u,± C i ,r (y) be the segment of length r with respect to the affine parameters given by h (see Lemma 3.9) inside W u,± C i (y). Let us fix x a point as in Lemma 5.1, and l ≥ 1, we shall omit the subscript l in B l in the sequel. Define We have that the corresponding set is an open nonempty α 0 -invariant set, then by Berend's Theorem [2] we get that it is the complement to a finite α 0 -invariant set F . Observe that singular points of the infratorus are contained in F since points in L \ F have a cube neighborhood. We may also assume that because the faces of the boundary of B (respectively ofB) is formed by stable manifolds of different elements of the action passing trough points which can be taken to have dense orbit on the support of the measure and hence each face of the boundary is mapped eventually completely inside intB (respectively intB).
For a point x as in Lemma 5.1, let r i,± be the length of the separatrix Lemma 5.2. For 1 ≤ i ≤ m and for ν a.e. x and for any full Lebesgue w.r.t. ν-measure. In particular, for ν a.e. point in y ∈ B, The assertion on the transverse intersection is an immediate consequence of the first assertion. The first assertion is an immediate consequence of Lemma 3.9, that h is a measurable conjugacy between ν and λ and that the same assertion for the linear case is trivial.
Let E σ C j , σ = s, u, j = 1, . . . m be the corresponding stable and unstable invariant spaces for the linear action. Let us use the same notation for their projection on the infratorus L. Observe that as long as z + E σ C j ⊂ R m does not contain a point corresponding to a singular point of the infratorus, the natural projection p from z + E σ C j into L is one to one and onto the corresponding affine space E σ C j (p(z)). Given a boxB as in Lemma 5.1 andŷ ∈B, let E σ C j ,B (ŷ) be the connected component of E σ C j (ŷ)∩B containingŷ. Given a regular point y ∈ B recall that W σ C i ,B (y) is the connected component of B ∩ W σ C i (y) containing y. Lemma 5.3. For ν a.e. point y ∈ B, W s C i ,B (y) is a k-dimensional box and h(W s C i ,B (y)) = E s C i ,B (h(y)). Moreover, for ν a.e. y, z ∈ B with z ∈ W u C i ,B (y), Hol s y,z : W s C i (y) → W s C i (z) is such that Hol s y,z (W s C i ,B (y)) = W s C i ,B (z). Finally, W s C i ,B (y) disconnects B in two connected components, homeomorphic to boxes.
Proof. The first assertion follows from Lemma 3.9 and the constructions of the boxes in Lemma 5.2. The second is a direct consequence of the first and the same property for the linear case.
Finally, let us prove the third assertion. From Lemma 5.1 and the first part we get that h(∂B l ∪ W s C i ,B (y)) = ∂B ∪ E s C i ,B (h(y)) and on this domain h is a diffeomorphism by Lemma 3.9.
Taking B small enough so that it is in a neighborhood chart and using Schönflies Theorem [1,3,21,22] we get that the pair (B, W s C i ,B (y)) is homeomorphic to the pair (I m , I m−1 × {1/2}). Indeed by the Hcobordism theorem, it follows that it is diffeomorphic if m − 1 = 3, (i.e. m = 4).
Lemma 5.4. Given a set R of full measure, for every y ∈ B there is a sequence of boxes y ∈ B n+1 (y) ⊂ intB n (y), ≥ 1, such that (1) ∂B n (y) is contained in the union of stable manifolds for different Weyl chambers through points from R, (2) For each n ≥ 1, h(∂B n (y)), is the boundary of a parallelepiped, moreover diam(h(∂B n (y))) → 0 as n → ∞.
Proof. This is an immediate consequence of Lemma 5.3 and the traditional subdivision of boxes like in Heine-Borel theorem.
Proof. of Theorem 2 From Lemma 5.4 it follows that h uniquely extends to a continuous map from B ontoB. Indeed for y ∈ B take the nested sequence from Lemma 5.4 and define h(y) to be the limit point of h(∂B n (y)). Continuity follows since the preimage of the box bounded by h(∂B n (y)) is B n (y) that is a neighborhood of y for every n ≥ 0. From the definition of O and L \ F we get that h extends uniquely to a continuous map h : O → L \ F that semi-conjugates. Moreover, from Lemma 5.4 it also follows that for any z ∈ L \ F , h −1 (z) is the nested intersection of boxes and for λ a.e. z ∈ L this nested intersection is a point by Lemma 5.1.
For the rest of the proof of Theorem 2 we shall prove that the restriction of h to a suitable k-dimensional skeleton is a diffeomorphism and that this restriction extends to a homeomorphism of O onto L \ F .
We have the following topological lemma for the infratorus.
Lemma 5.5. Given ǫ > 0, and a boxB ⊂ L \ F as in Lemma 5.1 there is a bounded subset K ⊂ Z m−1 , R > 0 and a partition by rectangles C i , i = 1, . . . r of the complement of some neighborhood of the singularities L ǫ := 1≤i≤r C i , such that: ∂C i ⊂ n∈K, a∈±, 1≤j≤m α 0 (n)((E s C j ,R + ∂ a C j (B))) intL ǫ is homeomorphic to L \ F .
Proof. Consider L \ z∈F B ǫ (z) and the covering of this compact set by the iterates ofB, α 0 (n)(B), n ∈ Z m−1 . Take a finite subcover, i.e. a finite subset K ⊂ Z m−1 so that α 0 (n)(B), n ∈ K also covers. Now, L ǫ = n∈K α 0 (n)(B) admits a partition by rectangles R i as desired.
Let Sk = i ∂C i be the m − 1-dimensional skeleton defined by the partition from Lemma 5.5.
Observe that for other dimensions m − 1 the possible existence of exotic spheres and hence of nonstandard smooth embeddings of S m−1 into R m , [19,18] may preclude the possibility of extending h −1 diffeomorphically to some cell of the partition.
Proof. That h −1 restricted to Sk is a diffeomorphism is a consequence of Lemma 3.9. Hence we have a well defined skeleton Since C i ⊂ α 0 (n)(B) for some n ∈ K we have that h −1 (∂C i ) ⊂ α(n)(B) for some n ∈ K. Hence h −1 : ∂C i → α(n)(B) is an embedding of the m − 1-dimensional sphere into the a m dimensional cube α(n)(B). Now, Schönflies Theorem and Alexander trick gives that h −1 extends to a homeomorphism. The differentiable statement follows from the smooth Schönflies theorem, valid for m − 1 = 3, plus the nonexistence of exotic embeddings for the given dimensions. Lemma 5.6 and that intL ǫ is diffeomorphic to L\F finishes the proof of Theorem 2.