Introduction - Pennsylvania State UniversityLet Gbe a simple split connected Lie group of rank n>1 and let Γ ⊂ Gbe a lattice. The Weyl chamber ﬂow on X= G/Γ is the left action

1

LOCAL RIGIDITY OF PARTIALLY HYPERBOLIC ACTIONS OF

Zk AND Rk, k ≥ 2. I. KAM METHOD AND ACTIONS ON THE

TORUS

DANIJELA DAMJANOVIC AND ANATOLE KATOK1

Abstract. We show C∞ local rigidity for Zk (k ≥ 2) partially hyperbolic ac-

tions by toral automorphisms using a generalization of the KAM (Kolmogorov-Arnold-Moser) iterative scheme. We also prove the existence of irreduciblegenuinely partially hyperbolic higher rank actions on any torus T

N for anyeven N ≥ 6.

1. Introduction

1.1. Algebraic actions of higher rank abelian groups and various notionsof rigidity. Differentiable (Cr, r ≥ 1) rigidity of the orbit structure and its vari-ations described below sets the higher rank abelian group actions apart from theclassical cases of diffeomorphisms and flows (actions Z and R) where only C0 orbitstructure may be stable under small perturbations.

In this paper we begin the study of differentiable rigidity for algebraic (homo-geneous of affine) partially hyperbolic actions of Zk and Rk, k ≥ 2. There are twoclasses of such actions where (generalized) rigidity is expected.

(1) Actions of Zk, k ≥ 2 by automorphisms or affine maps of tori or, moregenerally, (infra)nilmanifolds without nontrivial rank one factors for any finite cover,and suspensions of such actions.

(2) Left actions of a higher rank abelian group A on the double coset spaceC \ G/Γ where G is a connected Lie group, Γ ⊂ G a cocompact lattice and C acompact subgroup of the centralizer of A which intersects with A trivially. Fur-thermore, the Lie algebras of A, C and contracting subgroups of all elements of Agenerate the full Lie algebra g of G.

For an action of a finitely generated group Cr topology is defined as Cr topologyfor any finite set of generators. For a smooth action of a connected Lie group itis defined similarly with the vector fields generating the action of the Lie algebraplaying the role of the generators.

For actions of discrete groups the notions of rigidity which we consider are sum-marized as follows.

An action α of a finitely generated discrete group A on a manifold M is Ck,r,l

locally rigid if any sufficiently Cr-small Ck perturbation α is Cl conjugate to α, i.ethere exists a Cl close to identity diffeomorphism H of M which conjugates α to α:

(1.1) H α(g) = α(g) H

1Partially supported by NSF grant DMS 0071339.1991 Mathematics Subject Classification. Primary 37C85, 37C15, 58C15.

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2 DANIJELA DAMJANOVIC AND ANATOLE KATOK

for all g ∈ A. C∞,1,∞ local rigidity is often referred to as C∞-local rigidity. Thecase of C1,1,0 is known as C1-structural stability.

The definitions for continuous groups such as Rk one has to allow a “time change”i.e an automorphism ρ of the acting group close to id such that instead of (1.1) onehas

(1.2) H α(ρ(g)) = α(g) H.

A weaker notion in the case of actions of continuous groups is foliation rigidity:one requires the diffeomorphism H to map an invariant foliation F (e.g. the orbit

foliation O of the action α) to the perturbed foliation F ′ (e.g. orbit foliation O ofthe perturbed action α). The foliation rigidity in the case of the orbit foliation isusually called orbit rigidity. C1,1,0 orbit rigidity is called structural stability

For a broad class of hyperbolic algebraic actions of Zk and Rk, k ≥ 2 C∞-localrigidity has been established in [21]; see Section 1.2 for a discussion. In this paperwe prove for the first time rigidity for a class of partially hyperbolic actions of Zk,namely for type (1) actions on tori.

The orbits of partially hyperbolic algebraic actions are parts of homogeneousisometric foliations. For actions of type (2) differentiable rigidity should be un-derstood in a modified sense allowing not only a linear time change in A but alsoan algebraic perturbation of A inside the isometric foliation. Let us consider somecharacteristic examples.

Definition 1. Let G be a simple split connected Lie group of rank n > 1 and letΓ ⊂ G be a lattice. The Weyl chamber flow on X = G/Γ is the left action of amaximal Cartan subgroup on X .

The Weyl chamber flows are prime examples of normally hyperbolic (Anosov)actions for which rigidity has been proved in [21].

The following two special cases can serve both as illustrations and as key modelsfor the general problem for type (2) partially hyperbolic actions.

(2A) Restriction of the Weyl chamber flow on a homogeneous space of a splitsimple group of R-rank greater than one to a lattice L in the maximal Cartansubgroup A.

(2B) Restriction of the Weyl chamber flow on a homogeneous space of a splitsimple group of R-rank greater than one to a connected codimension one subgroupH ⊂ A in a general position.

In both of those cases the embedding of the acting group (L or H) into A can beperturbed thus creating non-trivial perturbations of the action. Call these actionsthe standard perturbations. The proper modification of the notions of rigidity inthis case is the statement that any small perturbation of the action is conjugate toa standard one.

1.2. Old and new approaches to rigidity. In this paper we introduce a newmethod of proving rigidity for smooth actions of higher rank abelian groups (i.e. Zk

and Rk for k ≥ 2 or, more generally, Zk ×Rl with k+ l ≥ 2) on compact manifolds.

1.2.1. The a priori regularity method. This older method introduced in [21] canvery briefly described as follows. One considers an action with sufficiently stronghyperbolic properties. The C0 orbit structure (or an essential part of it) of suchan action is rigid (structural stability, Hirsch-Pugh-Shub theory [9]). Moreover

LOCAL RIGIDITY OF ACTIONS ON THE TORUS 3

the C0 (orbit) conjugacy is unique transversally to the orbits. The method thenconsists of establishing a priori regularity of the conjugacy. The central idea isthat locally in the phase space there are invariant geometric structures on invariantfoliations for both the original and perturbed action. The simplest example of sucha structure is flat affine connection (local linearization); in general the structure ismore complicated, but it still has a finite-dimensional Lie group of automorphisms.Such a structure appears due to the fact that some elements of the action contractthe foliations in question and this is true both in the rank one and higher rankhyperbolic situations. Higher rank is crucial in showing that the conjugacy, whichis a priori only continuous preserves the invariant structure and hence is smooth inthe direction of each foliation. This in turn relies on the fact that in certain criticaldirections the unperturbed action acts by isometries on each leaf of the foliation.This in particular excludes the case of actions by automorphisms of the torus andof a nil-manifold in the presence on non-trivial Jordan blocks.

This method of establishing regularity of a conjugacy (whose existence is ob-tained from a different kind of reasoning) appears as an ingredient in the proofs oflocal rigidity for actions of certain groups other than abelian [26, 6]. Notice that theearliest application of a version of this method can be found in [15] in the contextof proving rigidity of some standard lattice actions on tori.

Both for the Rk actions and for case of general lattice actions considered in[6] there are certain residual directions left where regularity has to be establishedseparately. In the former case this follows from the cocycle rigidity which allowsto straighten out the time change [19]. Notice that cocycle rigidity also holds forsome partially hyperbolic actions [20].

If one tries to follow this approach for a partially hyperbolic action with semisim-ple linear part one starts with structural stability of the neutral foliation for theunperturbed action. Then two difficulties arise. First, this foliation for perturbedaction is not necessarily smooth and hence smoothness of the foliation conjugacyalong the stable and unstable directions of various elements of the action does notguarantee the global regularity, Second, even if one assumes smoothness of theneutral foliation, or equivalently only considers perturbations along this foliationcocycle rigidity is not sufficient.

Notice that there is no local (in the space) rigidity in this case. Not only nogeometric structure on the neutral foliation is preserved by individual elements ofthe perturbed action, but locally higher rank condition does not help either. Inother words, one can perturb the higher rank locally (say, in a neighborhood of aperiodic orbit) and change its topological structure in a dramatic way.

1.2.2. The KAM/Harmonic analysis (group representations) method. In our newmethod we do not start from a conjugacy of low regularity. Instead we constructone of high regularity by an iterative process as a fixed point of a certain non-linearoperator. We use an iterative procedure, namely an adapted version of the KAMprocedure motivated by the procedure used by Moser in [27] for commuting circlediffeomorphisms. Moser first noticed that commutativity along with simultaneousDiophantine condition was enough to provide a smooth solution to certain over-determined system of equations. Perhaps surprisingly, Moser’s philosophy turnedout to be applicable to a very different class of Zk (k ≥ 2) actions, namely actionsby toral automorphisms.


The first key observation is that the linearized equation in this case is ”almost”a twisted cocycle equation over the unperturbed linear action. It is possible tosolve such equations due to the ”higher rank trick” which lies at the root of cocyclerigidity [19, 20] and which is here adapted to the twisted case (Section 3.4).

The second key observation is that it is possible to obtain an approximate so-lution to the linearized equation if the linearized equation can be approximatedby a twisted cohomology equation over the unperturbed action. This reduces toapproximating a certain cocycle over the perturbed action by a cocycle over the un-perturbed action. We construct this approximation cocycle explicitly by adaptingand developing a method from [13].

As a byproduct of our approach we remove the restriction of semi-simplicity forthe unperturbed action. Our method works when the unperturbed action, whetherhyperbolic or partially hyperbolic, has nontrivial Jordan blocks. A disadvantage ofour method is the requirement that the perturbation is small in Cl topology for acertain large l while the method of [21] when it is applicable can deal with smallperturbations in C1 topology.

In this paper we carry out the new approach in the case of actions on the torusby automorphisms, affine maps and suspensions of such actions.

In the last section we outline the work in progress for the actions on type (1) onnilmanifolds and actions of type (2). While we expect to handle the former caseby a variation of the KAM/group representation method, at present the difficultiesin pursuing a similar approach even for such simple type (2) actions as (2A) and(2B) look formidable. Instead we are pursuing a different geometric method basedon non-commutativity of stable and unstable foliations which is not applicable toactions of type (1). Thus we expect KAM and geometric approaches to be comple-mentary in solving the local differentiable rigidity problem for partially hyperbolicactions.

Results of this paper are announced in [3].in [3] we announced thecombined method

We would like to thank Elon Lindenstrauss who pointed out an inaccuracy in anearlier version of this paper. As a result Section 3.5 has been re-worked.

1.3. Statement of results.

1.3.1. Rigidity. Let GL(N,Z) be the group of integer N × N matrices with de-terminant ±1. Any matrix A ∈ GL(N,Z) defines an automorphism of the torusTN = RN/ZN which we also denote by A.

An action α : Zk × TN → TN by automorphisms is given by an embeddingρα : Zk → GL(N,Z) so that

α(i, x) = ρα(i)x

or any i ∈ Zk and any x ∈ TN . We will write simply α(i) for ρα(i).

An action α′ : Zk × TN ′

→ TN ′

is an algebraic factor of α if there exists anepimorphism h : TN → TN ′

such that h α = α′ h.An action α′ is a rank-one factor if it is an algebraic factor and if ρα′(Zk) contains

a cyclic subgroup of finite index.An ergodic action α has no nontrivial rank-one factors if and only if ρα(Zk)

contains a subgroup isomorphic to Z2 which consists of ergodic elements (see forexample [29]).

Theorem 1. Let α : Zk × TN → TN be a C∞ partially hyperbolic action of Zk

(k ≥ 2) by automorphisms of N -dimensional torus with no non-trivial rank-one


factors. Then there exists a constant l = l(α,N) ∈ N such that for any Cl-small C∞

perturbation α of α there exists a C∞ map H : TN → TN such that α H = H α,i.e α is C∞,l,∞-locally rigid.

The constant l in the above theorem depends on the dimension of the torus andthe given linear action, and it will be precisely defined in the proof.

We note that the result above extend to corresponding affine actions. While anyaffine map of the torus without eigenvalue one in its linear part has a fixed pointand hence is conjugate to its linear part, the case of commuting affine maps is moregeneral (for examples of affine actions without fixed points see [10]). However onereduces this case to the automorphism case by passing to a subgroup of finite rankwhich has a fixed point and observing that the centralizer of an ergodic partiallyhyperbolic action by toral automorphisms is discrete.

We also obtain rigidity for suspensions of actions considered in Theorem 1. (Forthe detailed description of the suspension construction see for example [19, Section2.2] or [2, Section 5.4])

Theorem 2. Suspension of a Zk action by toral automorphisms with no rank onefactors is C∞,l,∞-locally rigid up to an automorphism of the acting group Rk wherel is the same as in Theorem 1.

1.3.2. Existence. As for the existence of actions for which our method is essentialwe show:

Theorem 3. Genuinely partially hyperbolic (i.e partially hyperbolic, not Anosov)Z2 actions by toral automorphisms exist: on any torus of even dimension N ≥ 6there are irreducible examples while on any torus of odd dimension N ≥ 9 there areonly reducible examples. There are no examples on tori of dimension N ≤ 5 andN = 7.

Remark 1. In addition to the general construction for even dimensions in dimen-sion 6 we give an explicit example.

Remark 2. The existence of genuinely partially hyperbolic irreducible actions inany odd dimension is ruled out by the fact that any A ∈ SL(N,Z) with irreduciblecharacteristic polynomial and N odd, is hyperbolic [30]. We give also much simplerproof of this fact.

Remark 3. One can also give similar conditions for existence of genuinely partiallyhyperbolic actions of Zk for k > 2 by ergodic automorphisms. For example, anirreducible action of such kind exists in any even dimension starting from 2k + 2.

2. Setting of the problem and overview of the KAM scheme

2.1. Higher-rank actions by ergodic toral automorphisms. An automor-phism A of the torus TN induces a map given by the transpose matrix At on thedual group ZN . If en(x) = e2πin·x are the characters then At acts as:

eAtn(x) = e2πiAtn·x = e2πin·Ax = en(Ax)

which implies that(f A)n = f(At)−1n

for Fourier coefficients of any C∞ function f on TN . We call (At)−1 the dual mapon ZN . To simplify the notation in the rest of the paper, whenever there is no


confusion as to which map we refer to we will denote the dual map by the samesymbol A. When we wish to clearly distinguish the two maps, we with use for thedual map notation A∗. Note that the dual map A induces a decomposition of RN

into expanding, neutral and contracting subspaces, i.e.

(2.1) RN = V1 ⊕ V2 ⊕ V3

such that all three subspaces are A-invariant and

(2.2)

‖Aiv‖ ≥ Cρi‖v‖, ρ > 1, i ≥ 0, v ∈ V1

‖Aiv‖ ≥ Cρ−i‖v‖, ρ > 1, i ≤ 0, v ∈ V3

‖Aiv‖ ≥ C(|i| + 1)−N‖v‖, i ∈ Z, v ∈ V2

The dual orbits of A are O(n) = Ain|i ∈ Z. It is not difficult to see that theergodicity of A is equivalent to all non-trivial dual orbits being infinite.

Indeed, suppose there exists a non-zero n ∈ ZN with a finite orbit, i.e. Amn = n

for some m. Then the function f =∑m−1

i=0 eAin is an A-invariant non-constantfunction, which contradicts ergodicity.

Conversely, if A is not ergodic and if f =∑

n fnen is an invariant function then

fAn = fn for all n ∈ ZN , therefore if the orbit of n is infinite then fn = 0. Thisimplies that f is constant. Thus A is ergodic.

It is an immediate consequence of the above characterization of ergodicity thatthe toral automorphism given by matrix A is ergodic if and only if A has no rootsof unity its spectrum. This follows from the fact that the dual map has a root ofunity in the spectrum if and only if there exists a non-trivial n ∈ ZN with finiteorbit. In particular, ergodic toral automorphisms are partially hyperbolic maps inthe sence that they have non-trivial expanding and contractiong directions.

Any action α of Zk by toral automorphisms, induces a dual action α∗ on ZN

given by:

ρα∗(i)n = (A∗1)

i1 (A∗2)

i2 ..(A∗k)ikn

for i = (i1, .., ik) ∈ Zk and n ∈ ZN . Then for every f =∑

n∈ZN fnen we have:

f(ρα(i)x) =∑

n∈ZN

fαinen

where, again to simplify the notation (whenever there is no confusion) we use thenotation αin for the dual action ρα∗(i)n on ZN . So we will use the same notationfor the elements of the action and the elements of the dual action whenever it isclear form the context which action we refer to.

Since we are interested in actions with no rank-one factors we will assume furtheron that the action α satisfies the following:

• There exist g1, g2 ∈ Zk with Adef= ρα(g1) and B

def= ρα(g2) such that any

AlBk for a non-zero (l, k) ∈ Z2 is ergodic. These will be referred to asergodic generators.

The following Lemma shows that producing a conjugacy for an ergodic generatorsuffices for the proof of Theorem 1.

Lemma 2.1. Let α be a Zk action by automorphisms of TN such that for someg ∈ Zk the automorphism α(g) is ergodic. Let α be a C1-small perturbation of α


such that there exists a C∞ map H : TN → TN which is C1 close to identity andsatisfies:

(2.3) α(g) H = H α(g).

Then H conjugates the corresponding maps for all the other elements of the actioni.e. for all h ∈ Zk we have:

α(h) H = H α(h)

Proof: Let h be any element in Zk, other than g. It follows from (2.3) andcommutativity that

α(g) H = H α(g)

whereH = α(h)−1 H α(h).

Thereforeα(g) H H−1 = H H−1 α(g)

Denote the lift of H H−1 to RN by id+ Ω where Ω is a C1-small N -periodic C∞

map, and use the same notation α(g) for the lift of α(g). Then:

(2.4) α(g) (id+ Ω) = (id+ Ω) α(g)

If there are no Jordan blocks the equation (2.4) reduces to several equations of thefollowing type:

λω = ω A

where A = α(g) and ω is C∞ function. Then passing to the dual action (while,as mentioned before, we use the same notation for the dual and for the originalaction) we have:

λωn = ωAn,

which impliesλiωn = ωAin

for all i ∈ Z. Therefore∞∑

i∈Z

λiωn =∑

i∈Z

ωAin

Since A is ergodic and ω is C∞ the right hand side converges absolutely. The lefthand side can converge only if ωn = 0 for all n 6= 0. Since λ 6= 1 we also haveω0 = 0 i.e. ω ≡ 0. This implies Ω ≡ 0 and H ≡ H .

If there are Jordan blocks for A, the argument above is still sufficient to deducethat Ω has to be 0. Namely, if there is, say, a 3-Jordan block, than the equation(2.4) reduces to:

λ 1 00 λ 10 0 λ

ω1

ω2

ω3

ω1 Aω2 Aω3 A

This implies:λω1 + ω2 − ω1 A = 0

λω2 + ω3 − ω2 A = 0

λω3 − ω3 A = 0

¿From the third equation above, using ergodicity of A as in the case of a simpleeigenvalue, we deduce that ω3 = 0. Substituting into the second equation we obtainω2 = 0 and finally, using this fact in the first equation above, we obtain ω1 = 0.


One can obviously by induction obtain Ω = 0 for Jordan blocks of any dimension.Therefore H = H and

H = α(h)−1 H α(h)

for an arbitrary h ∈ Zk.

In what follows we will make a frequent use of the following fact known asKatznelson lemma:

Lemma 2.2. (Katznelson, [22]) Let A be an N×N matrix with integer coefficients.Assume that RN splits as RN = V ⊕V ′ with V, V ′ invariant under A and such thatA|V and A|V ′ have no common eigenvalues. If V ∩ ZN = 0 then there exists aconstant γ such that d(n, V ) ≥ γ‖n‖−N for all n ∈ ZN where ‖ · ‖ is Euclideannorm and d is Euclidean distance.

Remark 4. This can be viewed as a version of the Liouville’s theorem aboutrational approximation of algebraic irrationals, i.e.|α−m

n | ≥ Cn−N for any non-zerointegers m and n, where α is an irrational first order root of an integer polynomialof degree N . The proof of this classical result inspires the proof of Lemma 2.2.

Proof: Any polynomial p sufficiently close to the minimal polynomial f of A onV satisfies the condition p(A)n 6= 0 for all n ∈ ZN , n 6= 0 because its null spaceis contained in V and V ∩ ZN = 0 by assumption. Then one can construct apolynomial fQ with rational coefficients of that kind, and since A is an integermatrix we have ‖fQ(A)n‖ > 1

q (the choice is made as |aj − rj/q| ≤1

qQ , where aj

are coefficients of f , rj/q coefficients of fQ and q ≤ Qk) for any non-zero n. Nowif nV is the projection of n to V then:

fQ(A)n = fQ(A)(n− nV ) + (fQ(A) − f(A))nV

This implies 1q ≤ C(d(n, V ) + ‖n‖

qQ ). Then by choosing Q = C‖n‖ where C is a

constant depending on A, the estimate follows:

d(n, V ) >1

Cq≥

1

CQk> C1‖n‖

−k > C1‖n‖−N

with C1 being a positive constant depending only on A.

Remark 5. In particular, if A is ergodic and V = V3⊕V2 from (2.1) then V ∩ZN =0. Then the above Lemma implies that

‖π1(n)‖ ≥ γ‖n‖−N

where π1(n) is the projection of n to V1, the expanding subspace for A.

2.2. Overview of the KAM scheme. Let α be a linear action as described inTheorem 1. Let α be its small perturbation (the topology in which the perturbationis made will become apparent from the proof). The goal is to prove the existenceof a C∞ map H : TN → TN such that α H = H α.

We can consider the problem of finding a conjugacy as a problem of solving anon-linear functional equation:

N (α,H)def= α H −H α = 0


Following the ideas of the elementary Newton method and assuming the existence ofa linear structure in the neighborhood of the id, we may view id as an approximatesolution and look at the linearization of the operator N at (α, id):

N (α,H) = N (α, id) +D1N (α, id)(α − α) +D2N (α, id)(Ω) + Res(α− α,Ω)

= α− α+ α(Ω) − Ω α+ Res(α− α,Ω)

where Ω = H − id and Res(α−α,Ω) is quadratically small with respect to α−αand Ω. If one can find H so that the linear part of the equation above is zero, i.e.:

α(H − id) − (H − id) α = −(α− α)

than such H is a better approximate solution of the equation N (α,H) = 0 thanthe id is. After obtaining a better solution, the linearization procedure and solvingthe linearized equation may be repeated for the new perturbation leading to aneven better approximation. The difficulty that arises in particular applications ofthis iterative scheme is to produce such sequence of approximate solutions whicheventually converges to an exact solution of the non-linear problem in some rea-sonable function space. We now adapt this general scheme to our specific problemconcerning toral automorphisms.

Any map of the torus TN into itself can be lifted to the universal cover RN .For every g ∈ Zk, the lift of α(g) is a linear map of RN i.e. a matrix with integerentries and with determinant ±1, which is also denoted by α(g) . The lift of α(g) isα(g)+R(g) where R(g) is an N−periodic function for every g, i.e. R(g)(x+m) =R(g)(x) for m ∈ ZN . The lift of H is id + Ω with an N−periodic Ω.

In terms of Ω the conjugacy equation is:

(2.5) αΩ − Ω α = −R (id + Ω)

If Ω is a solution for the corresponding linearized equation

(2.6) αΩ − Ω α = −R

or at least its approximate solution i.e. if it solves (2.6) with an error which isquadratically small with respect to R, then the new perturbation defined as:

α(1) def= H−1 α H

is expected to be much closer to α than α i.e. the new error

R(1) def= α(1) − α

should be quadratically small with respect to the old error R.The comparison between the two errors usually cannot be realized in the same

function space. The norm of the old error in some function space may only becompared to the norm of the new error in some larger space. This is the loss ofregularity. In the proof that we present all the maps involved will be C∞. Wewill however have loss in the sense described above and thus we will work with thefamily Cr (r ∈ N). In general, if the loss of regularity is fixed and if the familyof spaces used in the iterative scheme admits smoothing operators (Section 4.1),then the interpolation inequalities hold and it is often the case that the iterativeprocedure can be set to converge to a solution of the non-linear equation [32] in allthe spaces involved.

The new error is:

R(1) = α(1) − α =[Ω α(1) − Ω α+ R (id + Ω) −R

]+

[R− (αΩ − Ω α)

]


The error term in the first bracket comes from the linearization of the problem andis easy to estimate providing Ω is of the same order as R. The difficulty lies inestimating the part of the error in the second bracket, namely solving the linearizedequation (2.6) approximately, with an error quadratically small with respect to R.

The linearized equation (2.6) actually consists of infinitely many equations cor-responding to different elements gi of the action:

(2.7) α(gi)Ω − Ω α(gi) = −R(gi)

and we need a common approximate solution Ω to all the equations above. If Rwere an α-twisted cocycle over the action α than it would be possible to view thissystem of equations as a twisted cocycle equation over the linear action α. At thispoint two problems arise. First is that R is an α-twisted cocycle not over α butover α thus this is not a twisted cocycle equation. Second is that even if the righthand side of (2.6) is a (twisted) cocycle over the unperturbed action α, the equation(2.6) is a twisted coboundary equation and solving it (with tame estimates) requiresmore care than an untwisted one.

Lemma 2.1 shows that it is enough to produce a conjugacy for one ergodicgenerator. The same conjugacy will work, due to commutativity and ergodicityassumptions, for all other elements of the action.

It is clear however that, in general, it is not possible to produce a C∞ conjugacyfor a single map, since a single genuinely partially hyperbolic (i.e. ergodic, nothyperbolic) toral automorphism is not even structurally stable. Indeed, Lemma3.1 in the next section shows that there are infinitely many obstructions to solvinglinearized equation for one generator. Therefore, we consider two ergodic gener-ators, and reduce the problem of solving the linearized equation (2.6) to solvingsimultaneously the following system:

(2.8)AΩ − Ω A = −RA

BΩ − Ω B = −RB

where A and B are ergodic generators: A = α(g1), B = α(g2) and

(2.9) RA = R(g1), RB = R(g2).

The system (2.8) splits further into several simpler systems using appropriate basisand the fact that A and B commute i.e. (2.8) is reduced to several equations of thekind:

(2.10)JAΩ − Ω A = Θ

JBΩ − Ω B = Ψ

where JA is a matrix consisting of Jordan blocks of A corresponding to an eigenvalueof A, JB is the corresponding block of B, and Θ and Ψ are small vector valuedN -periodic maps given by the perturbation maps RA and RB . In particular, if λ, µare simple eigenvalues of A,B respectively, than we have:

(2.11)λω − ω A = θ

µω − ω B = ψ

where θ and ψ are small N -periodic functions. In Lemma 3.3 we show that it ispossible to solve the linearized equation (2.8) and to estimate the solution with the


fixed loss of regularity if the following cocycle condition is satisfied by R:

(2.12) L(RA, RB)def= (RA B −BRB) − (RB A−ARB) = 0,

i.e. if R gives a twisted cocycle over the Z2 action generated by A and B.As mentioned above, R does not satisfy this condition, however the fact that

it satisfies the twisted cocycle condition over the perturbed action α implies thatit satisfies the cocycle condition (2.12) for the unperturbed action up to an errorwhich is quadratically small with respect to R. More precisely, Lemma 3.4 showsthat if α = α + R is a commutative action then L(RA, RB) is quadratically smallin C0 norm with respect to R.

The key step is to show that being almost a cocycle over α implies being closeto a cocycle over α. In other words, we construct a projection R of R to the spaceof twisted cocycles over α so that their difference R = R−R is quadratically smallwith respect to R. More precisely, in Lemma 3.6 we show that even if RA andRB do not satisfy the solvability condition (2.12), it is still possible to approximateboth by maps which satisfy the condition (2.12) with an error bounded by the sizeof L(RA, RB) with a fixed loss of regularity.

Lemma 3.6, Lemma 3.3 and Lemma 3.4 combined give an approximate solutionto the linearized equation (2.8).

In Section 4.1 the smoothing operators are introduced to overcome the fixed lossof regularity at each iterative step and further in Section 4 the iteration processis set and is carried out producing a C∞ conjugacy which works for the Z2 actiongenerated by the two ergodic generators, thus according to Lemma 2.1 it works forall the other elements of the Zk action.

3. Approximate solution of the linearized equation

In this section we prove the existence of an approximate solution to the linearizedequation (2.8). We first introduce in Section 3.1 the notation that will be used. InSection 3.2 we introduce the obstructions for solving single linear equation in (2.11)and show that, providing the obstructions vanish, there exists a C∞ solution withtame estimates. In the hyperbolic case this is twisted version of arguments in [19]and in the partially hyperbolic case it uses a ”twisted” version of the arguments in[31].

In section 3.4 we show that those obstructions vanish providing the maps RA andRB in (2.8) satisfy the solvability criterion (2.12). Then in Lemma 3.4 we show thatthis criterion is almost satisfied, up to an error which is quadratically small withrespect to R. Finally in Section 3.5 we show that the operator L defined in (2.12)has a tame inverse which implies that RA and RB can be approximated by mapsthat satisfy the solvability criterion and this produces an approximate solution to(2.8).

3.1. Notation.

(1) Let f be a C∞ function f =∑

n fnen.

• ‖f‖adef= sup

n|fn||n|

a, a > 0.

• Djdef= ∂

∂xj, Dk def

= Dk11 Dk2

2 ...DkN

k , |k| =∑N

i=1 ki, k = (k1, .., kN )

‖f‖Crdef= max

0≤l≤rsup

x∈TN ,|k|=l

|Dkfx|, r ∈ N0.


• The following relations hold (see for example Section 3.1. of [24]):

‖f‖r ≤ C‖f‖Cr and ‖f‖Cr ≤ Cr‖f‖r+σ

where σ > N + 1, and r ∈ N0.

• For a map F with coordinate functions fi(i = 1, .., k) define: ‖F‖adef=

max1≤i≤k ‖fi‖a. For two maps F ,G define ‖F ,G‖adef= max ‖F‖a, ‖G‖a.

‖F‖Cr and ‖F ,G‖Cr are defined similarly.

(2) In definitions above, |n|def= max‖π1(n)‖, ‖π2(n)‖, ‖π3(n)‖, where ‖ · ‖ is

Euclidean norm and πi(n) are projections of n to subspaces Vi(i = 1, 2, 3)from (2.2), that is to the expanding, neutral, and contracting subspacesof RN for A (or B; we will use the norm which is more convenient in aparticular situation; those are equivalent norms, and the choice does notaffect any results).

(3) For n ∈ ZN we say n is mostly in i(A) for i = 1, 2, 3 and will write n→i(A)if the projection πi(n) of n to the subspace Vi corresponding to A is suffi-ciently large:

|n| = ‖πi(n)‖.

Notation n→1, 2(A) will be used for n which is mostly in 1(A) or mostlyin 2(A).

(4) Given a complex number λ and a function ϕ on the torus, define the twistedcoboundary operators:

∆λAϕ

def= λϕ− ϕ A

∆λAϕn

def= λϕn − ϕAn,

In what follows λ will usually be an eigenvalue of A, and µ will usuallydenote an eigenvalue of B, so we will usually use the following simplernotation:

∆λ def= ∆λ

A, ∆µ def= ∆µ

B

∆λϕndef= ∆λ

Aϕn, ∆µϕndef= ∆µ

Bϕn

(5) Define the following operators for a map F of the torus

∆AFdef= AF − F A.

(6) For functions θ, ψ and maps F , G define the following operators:

L(θ, ψ)def= ∆µθ − ∆λψ

L(F ,G)def= ∆BF − ∆AG


(7) We introduce the notation for the following sums:

∑+

A

ϕndef=

∞∑

i=0

λ−(i+1)ϕAin

∑–

A

ϕndef=

−1∑

i=−∞

λ−(i+1)ϕAin

∑A

ϕndef=

+∞∑

i=−∞

λ−(i+1)ϕAin

Here A is used to define the dual action and ϕn are Fourier coefficients fora C∞ function ϕ. The corresponding notation will be used for B insteadof A and µ instead of λ, both being corresponding eigenvalues of A and B.Also define:

∑

l–

def=

−1∑

l=−∞

,∑

l+

def=

∞∑

l=0

and∑

l

def=

l=+∞∑

l=−∞

(8) In what follows, C will denote any constant that depends only on the givenlinear action with chosen ergodic generators and on the dimension of thetorus. Cx,y,z,.. will denote any constant that in addition to the above de-pends also on parameters x, y, z, ...

3.2. Estimates for the solutions of twisted one-cohomology equations. Wefirst produce a solution to a twisted one-cohomology equation with estimates forthe norm of the solution. We follow closely the proof of Veech for the untwistedcase [31].

Lemma 3.1. Let θ be a C∞ function on the torus and λ ∈ C, λ 6= 1. Let A be aninteger matrix in GL(N,Z) defining an ergodic automorphism of TN such that forall non-zero n ∈ ZN the sums along the dual orbits are zero i.e.

(3.1)∑

A

θn =+∞∑

i=−∞

λ−(i+1)θAin = 0

Than the equation:

(3.2) ∆λω = λω − ω A = θ

has a C∞ solution ω and the following estimate:

(3.3) ‖ω‖a−δ ≤Ca

δν‖θ‖a

holds for δ > 0, ν = aN + 1 and a > | log |λ||log ρ . Here ρ > 1 is the expansion rate for

A from (2.2). Also:

(3.4) ‖ω‖Cr ≤ Cr‖θ‖Cr+σ

for any r > 0 and σ an integer number greater than maxN + 1, | lg |λ||lg ρ .

Proof: The equation λω − ω A = θ in the dual space has the form:

λωn − ωAn = θn, ∀n ∈ ZN


For n = 0, since λ 6= 1, we can immediately calculate ω0 = θ0

λ−1 . For n 6= 0 thedual equation has two solutions

ω+

(−)n = +

(−)

∑

i≥0(i≤−1)

λ−(i+1)θAin, n 6= 0.

Each sum converges absolutely since θ is C∞ and all non-zero integer vectors havenon-trivial projections to expanding and contracting subspaces for A due to the

ergodicity of A. By assumption∑ A

θn = 0 , ∀n 6= 0 i.e. ω+n = ω−

ndef= ωn. This

gives a formal solution ω =∑

ω+n en =

∑ω−

n en. We will estimate each ωn using

both of its forms in order to show that ω is C∞.For n→3(A), that is, n is mostly in the contracting subspace, we use the ω−

n

form:

|ωn| =

∣∣∣∣∣∣

∑

k≤0

λ−(k+1) θAkn

∣∣∣∣∣∣≤

∑

k≤0

|λ|−(k+1)|θAkn| ≤ ‖θ‖a

∑

k≤0

|λ|−(k+1)|Akn|−a

≤ ‖θ‖a

∑

k≤0

|λ|−(k+1)‖Akπ3(n)‖−a

≤ ‖θ‖aC−a

∑

k≤0

|λ|−(k+1)ρak‖π3(n)‖−a

≤ Ca‖θ‖a|n|−a

providing a > log |λ|log ρ .

Similarly, if n→1(A), using the form ωn = ω+n , the estimate |ωn| ≤ Ca‖θ‖a|n|

−a

holds, providing a > log |λ|−1

log ρ .

If n→2(A) and |λ| ≥ 1 using the form ω+n of the solution we have:

|ωn| ≤ ‖θ‖a

∑

k≥0

|λ|−(k+1)|Akn|−a

≤ ‖θ‖aC−a

∑

k≥0

|λ|−(k+1)(1 + k)Nα‖π2(n)‖−a

However, the above sum need not converge. This is where we use the fact that A isergodic. Namely, no integer vector can stay mostly in the neutral direction for ’toolong’. After the time which is approximately lg |n| the expanding direction takesover. The precise statement of this fact is the Katznelson lemma ( Lemma 2.2).

Namely, we have ‖π1(n)‖ ≥ γ|n|−Nfor some γ and all n. Therefore

|Akn| ≥ ‖Akπ1(n)‖ ≥ Cρk‖π1(n)‖ ≥ γCρk|n|−N ≥ γCρk−k0 |n|

for k ≥ k0 and ko = [ (1+N) log |n|log ρ ] + 1. This fact can be used to estimate most

terms of the series above, i.e. all but finitely many terms. For the rest we use thepolynomial estimate in (2.2) for vectors in V2:

|ωn| ≤ ‖θ‖a

k0−1∑

k=0

|λ|−(k+1)|k|Na‖π2(n)‖−a + C‖θ‖a

∞∑

k=k0

|λ|−(k+1)ρ−a(k−k0)|n|−a

Thus using that n→2(A) and |λ| > 1 we have:

|ωn| ≤ C‖θ‖a|k0|Na+1|n|−a + C‖θ‖a|n|

−a

Now by choice of k0: k0 ∼ log |n|, and we obtain the following estimate:

|ωn| ≤ Ca(log ‖n‖)Na+1|n|−a‖θ‖a


For |λ| < 1 the same estimate follows using the second form for ωn i.e. the negativesum and the fact that A−1 is also an ergodic toral automorphism thus going back-wards in time the contracting direction takes over, that is, we use the Katznelsonlemma for A−1. Therefore for all n ∈ ZN we have:

|ωn| |n|a−δ ≤

Ca

δν‖θ‖a

for a > | log |λ||log ρ and δ > 0. This implies the estimate (3.3) for ‖ω‖a−δ. The estimate

for Cr norms follows immediately using the norm comparison from Section 3.1. Inparticular if θ is C∞, ω is also C∞.

3.3. Orbit growth for the dual action. We now establish crucial estimates forthe exponential growth along individual orbits of the dual action and its implica-tions on estimating the Cr norm of specifically defined functions. The existence ofsuch estimates in case of Zd actions with d ≥ 2 relies fundamentally on the higherrank assumption i.e. on the ergodicity of all non-trivial elements of the action.

Lemma 3.2. Let α be a Zd action by ergodic automorphisms of TN . Then thereexist constants τ > 0 and C > 0 depending on the action only, such that:

a) For every integer vector n ∈ ZN and for all k ∈ Zd:

|αkn| ≥ C expτ‖k‖|n|−N

b) For any C∞ function ϕ on the torus, any non-zero n ∈ ZN and any vectory ∈ Rd the following sums

SK(ϕ, n) =∑

k∈K

ykϕαkn,

where yk def=

∏di=1 y

ki

i , converge absolutely for any K ⊂ Zd.

c) Assume in addition to the above that for an n ∈ ZN and for every k ∈ K =K(n) ⊂ Zd we have P (‖k‖)|αkn| ≥ |n| where P is a polynomial of degree N . Then:

(3.5) |SK(ϕ, n)| ≤ Ca,y‖ϕ‖a|n|−a+κy,α

for any a > κy,αdef= N+1

τ

∑di=1 | log |yi||.

d) If the assumptions of c) are satisfied for every n ∈ ZN then the function

S(ϕ)def=

∑

n∈ZN

SK(n)(ϕ, n)en

is a C∞ function if ϕ is. Moreover, the following norm comparison holds:

(3.6) ‖S(ϕ)‖Cr ≤ Cr,y‖ϕ‖Cr+σ

for r > 0 and any σ > N + 2 + [κy,α].

Proof: Proof of a). When d = 1 i.e. the action is given by a single ergodictoral automorphism, then the inequality a) is an immediate consequence of theKatznelson lemma (Lemma 2.2) and we have already used this fact in the previoussection.

When d ≥ 2, we first notice that it is enough to obtain the constant τ and toshow the exponential estimate in a) in the semisimple case i.e. when the action isgenerated by matrices A1, .., Ad which are simultaneously diagonalizable over C. Ifthe action is not semisimple, only polynomial growth may occur in addition, thus


the same estimate holds with slightly smaller τ and with possibly larger C (formore deatails see [12])

Here we give the proof in the case when the action is irreducible which showsthe main idea, but is technically simpler and we refer to [13] for the proof in thegeneral case.

In the case when the action is irreducible we may project a non-trivial n ∈ ZN

to the Lyapunov directions corresponding to non-zero Lyapunov exponents of theaction. Lyapunov exponents are defined as:

χi(k) =

d∑

j=1

kj ln |λij |

where k = (k1, ...kd) ∈ Zd, i = 1, .., r, and λ1j , ..λrj are the eigenvalues of Aj for j =1, .., d. Individual Lyapunov directions are irrational and due to the irreducibilityassumption each of the projections of the vector n to the Lyapunov directions isnon-trivial. Thus one may apply the Katznelson lemma to each of these projectionsand choose τ as the minimum of the function maxχi(t) for t on the unit sphere inRd. This minimum has to be positive.

Indeed, if the minimum was non-positive, it would have to be zero since∑

i χi(t) =0 for all t. This implies the existence of a line l in Rd such that for all points on l allLyapunov functionals take value zero. The line l then cannot contain any integervectors k ∈ Zd because of the ergodicity of all the non-trivial ellements αk of theaction. Thus there are lattice points arbitrary close to l, in other words, there isan integer ergodic matrix of determinant 1 whose eigenvalues in absolute value areall arbitrary close to 1. If all the eigenvalues of an integer matrix are of absolutevalue one, then they are roots of unity. Hence by taking a sufficiently high powerone can obtain an integer matrix arbitrary close to the identity matrix which thenhas to be the identity. This clearly contradicts ergodicity. Therefore, τ is strictlypositive.

Thus using the following norm:

‖αkn‖χ =

r∑

i=1

‖ni‖ expχi(k)

where ni are projections of n to the corresponding Lyapunov directions, and theKatznelson lemma, we obtain the needed estimate

|αkn| ≥ C‖αkn‖χC expτ‖k‖mini

‖ni‖ ≥ C expτ‖k‖‖n‖−N .

Proof of b). The claim in b) follows from the estimate in a) and the fast decayof Fourier coefficients:

|SK | ≤ ‖ϕ‖a

∑

k∈K

|y|k|αkn|−a ≤ Ca‖ϕ‖a|n|Na

∑

k∈K

(|y|sgn ke−aτ )‖k‖

The last sum clearly converges providing a > maxi=1,..,d| log |yi||

τ , and for a C∞

function ϕ we can choose a as large as needed.Proof of c). Here we use the norm where ‖k‖ = max1≤i≤d |ki| for k = (k1, .., kd) ∈

Zd. We choose an appropriate k0 ∈ N in order to split the sum SK(ϕ, n) into twosums: the finite one with ‖k‖ < k0 where the polynomial estimate for |αkn| can beused, and the infinite one for ‖k‖ > k0 where the exponential estimate obtained in


a) prevails. The estimate in a) allows us to locate k0 so that it is not very large.Namely, as in Lemma 3.1 we have:

|αkn| ≥ Cγ expτ‖k‖|n|−N

≥ Cγ expτ(‖k‖ − k0) expτk0|n|−N

≥ C expτ(‖k‖ − k0)|n|

providing ‖k‖ ≥ k0 >N+1

τ log |n|.

After choosing k0 = [N+1τ log |n|] + 1 split the sum SK as follows:

SK(ϕ, n) = S‖k‖<k0(ϕ, n) + S‖k‖≥k0

(ϕ, n)

To estimate the first sum we use the polynomial estimate for k ∈ K(n):

|S‖k‖<k0(ϕ, n)| ≤C‖ϕ‖a

∑

k∈K:‖k‖<k0

|y|kP (‖k‖)a|n|−a

≤C‖ϕ‖akd0

d∏

i=1

max|yi|, |yi|−1k0kdaN

0 |n|−a

≤Ca‖ϕ‖akd(1+aN)0 |n|−a+κ1

≤Ca‖ϕ‖a(log |n|)d(1+aN)|n|−a+κ1

≤Ca

δν‖ϕ‖a|n|

−a+κ1+δ

for any δ > 0 and any a > κ1 = N+1τ

∑di=1 | log |yi||. The constant ν = ν(N, d, a)

depends only on the dimension of the torus, rank of the action and a. For thesecond sum we use the exponential estimate obtained in a):

|S‖k‖≥k0(ϕ, n)| ≤ C‖ϕ‖a

∑

k∈K:‖k‖≥k0

|y|k|αkn|−a

≤ C‖ϕ‖a|n|−a

d∑

i=1

∑

k∈K:|ki|≥k0

|yi|kie−aτ(|kj|−k0)

j=d∏

i6=j=1

|yj |kje−aτ |kj|

≤ C‖ϕ‖a|n|−a

d∑

i=1

Ca,y max|yi|, |yi|−1k0

≤ Ca,y‖ϕ‖a|n|−a+κ2

for any a > κ2 = N+1τ max1≤i≤d | log |yi||.

Thus by combining the estimates obtained above we have |SK(ϕ, n)| ≤ Ca,y‖ϕ‖a|n|−a+κy,α

for any a > κy,αdef= N+1

τ

∑di=1 | log |yi||.

Proof of d)If for every n the set K(n) is chosen to satisfy the assumptions of c) then theestimate (3.5) clearly implies:

‖S(ϕ)‖a−κy,α≤ Ca,y‖ϕ‖a

which has as a consequence the Cr estimate (3.6) with loss of N +2+[κy,α] deriva-tives.


3.4. Higher rank trick. In this section we establish a criterion for the existenceof a common solution to equations (2.8). It is easy to check that if A and Bcommute and there exists a solution to (2.8) then L(RA, RB) = 0. We show thatif in addition every non-trivial AlBk is ergodic then the condition L(RA, RB) = 0is not only necessary but also sufficient for the existence of a solution to (2.8).This argument is essentially the argument of [19] adapted to the twisted and notnecessarily semisimple situation. After showing the vanishing of obstructions, thetame estimates for the common solution of (2.8) follow from Lemma 3.1.

Lemma 3.3. If L(RA, RB) = ∆BRA − ∆ARB ≡ 0, where RA, RB are C∞ mapsdescribed in Section 2.2 (see (2.7) and (2.9)) , then the equations (2.8)

∆AΩ = RA

∆BΩ = RB

have a common C∞ solution satisfying:

(3.7) ‖Ω‖Cr ≤ Cr‖RA, RB‖Cr+σ ,

for any r > 0 and σ > M0 = maxN+2, [m0(A,B)], where m0(A,B) is a positiveconstant defined explicitely in (3.24) below, depending on the eigenvalues of A andB.

Proof:(i) The semisimple case. Assume that A and B are simultaneously diagonaliz-

able. Then the equations L(RA, RB) ≡ 0 and (2.8) split into finitely many equationsof the form

(3.8) L(θ, ψ) = ∆µθ − ∆λψ ≡ 0

and

(3.9)∆λω = θ

∆µω = ψ

where θ, ψ are C∞ functions and λ and µ are corresponding eigenvalues of A andB, respectively.

The assumption L(θ, ψ) ≡ 0 implies ∆µθ ≡ ∆λψ , which after passing to to thedual action implies:

∑B

(∆µθn) =∑

B

(∆λψn)

∑A

(∆λψn) =∑

A

(∆µθn)

Consider now the first equation above. Since all the sums involved converge abso-lutely we have

∆µ∑

B

θn = 0

which implies ∑B

(∆λψn) = 0

Using Lemma 3.2 b) i.e. absolute convergence of the sums, this implies that theobstruction for ψ is not only multiplied by µ under the action of B, but is also


multiplied by λ under the action of A, i.e. λ∑ B

ψn =∑ B

ψAn. By iterating thisequation we obtain:

λk∑

B

ψn =∑

B

ψAkn,

for every k ∈ Z. Therefore

(3.10)∑

k∈Z

λk∑

B

ψn =∑

k∈Z

∑B

ψAkn

The series in the left hand side of (3.10) does not converge unless∑ B

ψn = 0 whilethe right hand side of (3.10) converges absolutely by Lemma 3.2 b). Therefore∑ B

ψn = 0, ∀n 6= 0. Similarly∑ A

θn = 0, ∀n 6= 0.By Lemma 3.1, the formal solutions of each equation in (3.9) are C∞ functions.

Moreover, they coincide. Indeed, if ω solves the second equation i.e. ∆µω = ψ then

∆λ∆µω = ∆λψ = ∆µθ

Since operators ∆λ and ∆µ commute this implies

∆µ(∆λω − θ) = 0

As in Lemma 2.1 the ergodicity of A and B implies that ∆µ and ∆λ are injectiveoperators on C∞. Therefore ∆λω = θ i.e. ω solves the first equation as well.

(ii) The general case. If A and B are not simultaneously diagonalizable thenchoose a basis in which A has its Jordan normal form. Since A and B commuteany root space for A is B-invariant. Therefore, in this basis B has a block diagonalform. Let JA = (aij) be anm×m matrix which consist of blocks of A correspondingto the eigenvalue λ, i.e. let aii = λ for all i = 1, ..,m and ai,i+1 = ∗i ∈ 0, 1 for alli = 1, ..,m− 1. Let JB = (bij) be the corresponding block of B where bii = µ forall i = 1, ..,m (µ is an eigenvalue of B) and bij = 0 for all m ≥ i > j ≥ 1. Thenbecause of the fact that A and B commute, by simply comparing coefficients, it iseasy to obtain the following relation which the coefficients of A and B must satisfy:

(3.11) ∗ibki = ∗kbk+1,i+1

for any fixed k between 1 and m− 1 and for all i = k + 1, ..,m− 1.For any such pair of blocks JA and JB the equations (2.8) split into equations

of the form:

(3.12)JAΩ − Ω A = Θ

JBΩ − Ω B = Ψ

and and the condition L(RA, RB) = 0 splits as

(3.13) JBΘ − Θ B = JAΨ − Ψ A

Let the coordinate functions of Θ and Ψ be θi and ψi (i = 1, ..,m), respectively.Then we look for functions ωi (i = 1, ..,m) which will solve the equations aboveand whose norm can be compared to the norm of Θ and Ψ.

Since JA and JB are upper diagonal, it is easy to obtain ωm. Namely, from (3.13)we have that θm and ψm satisfy the condition L(θm, ψm) = ∆µθm − ∆λψm = 0.Therefore, using part (i) , we have that there exist ωm which solves simultaneouslythe last of m pairs of equations in (3.12), namely the equations ∆λωm = θm and∆µωm = ψm. Morover, the estimate:

(3.14) ‖ωm‖a−δ ≤Ca

δν‖θm, ψm‖a ≤

Ca

δν‖Θ,Ψ‖a


follows from Lemma 3.1 for a ≥ max | log |λ||log ρ , | log |µ||

log η where ρ and η are growth

rates in the hyperbolic direction corresponding to A and B, respectively. Now the(m− 1)-st pair of equations in (3.12) is:

(3.15)∆λωm−1 + ∗m−1ωm = θm−1

∆µωm−1 + bm−1,mωm = ψm−1

while the cocycle condition for θm−1 and ψm−1 from (3.13) is:

(3.16) ∆µθm−1 + bm−1,m = ∆λψm−1 + ∗m−1ψm−1

By substituting θm = ∆λωm and ψm = ∆µωm into (3.16) we obtain that

L(θm−1 − ∗m−1ωm, ψm−1 − bm−1,mωm) = 0

where the norm of the both functions on which the operator L acts, due to theestimate (3.14), can be bounded by the norm of Θ and Ψ with a small loss. Nowwe may use the part (i) again to conclude that there exists some ωm−1 solving thesystem (3.15) and such that the following estimate holds:

(3.17) ‖ωm−1‖a−2δ ≤Ca

δ2ν‖Θ,Ψ‖a

Now we proceed by induction. Fix k between 1 and m− 2 and assume that for alli greater than k, we have obtained a solution ωi with the appropriate estimate, i.e.for every i = k + 1, ..,m we have a C∞ function ωi which solves the i− th pair ofequations of (3.12):

(3.18)

∆λωi + ∗iωi+1 = θi

∆µωi +

m∑

l=i+1

bilωl = ψi

and that the following estimates hold:

(3.19) ‖ωi‖a−(m−i+1)δ ≤Ca

δ(m−i+1)ν‖Θ,Ψ‖a

We wish to find ωk that solves the k-th pair of equations in (3.12):

(3.20)

∆λωk + ∗kωk+1 = θk

∆µωk +

m∑

i=k+1

bkiωi = ψk

providing that the k-th equation in (3.13) is satisfied by θk and ψk i.e.

(3.21) ∆µθk +

m∑

i=k+1

bkiθi = ∆λψk + ∗kψk+1

Now we use the fact that all the subsequent pairs of equations are solved i.e. wesubstitute all θi for i = k + 1, ..,m and the ψk+1 into (3.21) using their expressionas in (3.18). This implies:

∆µθk +

m∑

i=k+1

(bki∆λωi + ∗ibkiωi+1) = ∆λψk + ∗k∆µωk+1 +

m∑

i=k+1

bk+1,i+1ωi+1


Since A and B commute, we can use the equations (3.11) for the coefficients andthe linearity of operators ∆λ and ∆µ, to simplyfy the above expression to:

∆µ(θk − ∗kωk+1) = ∆λ(ψk −m∑

i=k+1

bkiωi)

Thus the functions θk − ∗kωk+1 and ψk −∑m

i=k+1 bkiωi satisfy the solvability con-dition, they are C∞ and therefore we may use the part (i) again to conclude thatthe pair of equations (3.20) has a common C∞ solution ωk which, as a consequanceof assumptions (3.19), satisfies the estimate

(3.22) ‖ωk‖a−(m−k+1)δ ≤Ca

δ(m−k+1)ν‖Θ,Ψ‖a

Since k is arbitrary number between 1 and m − 1 it follows that there exists asolution Ω to (3.12) providing the condition (3.13) is satisfied. This can be repeatedfor all corresponding blocks of A and B. Since the maximal size of a Jordan blockis bounded by N we obtain the following estimate for the norm of the solution Ωof the system ∆AΩ = RA, ∆BΩ = RB :

(3.23) ‖Ω‖a−Nδ ≤Ca

δNν‖RA, RB‖a

providing L(RA, RB) = 0, and for any a satisfying: a > m0 = m0(A,B) where

(3.24) m0(A,B)def= max

imax

| log |λi||

log ρ,| log |µi||

log η

where the first maximum is taken over all pairs of eigenvalues λi, µi of A andB, respectively. Of course, the constant Ca has been changing throughout theprocedure. It only depends on the matrices A and B and the dimesion N of thetorus, besides a.

As before, by fixing δ and using the norm comparison this implies the estimate(3.7) for Cr norm as well, with the loss of maxN + 2, [m0(A,B)] derivatives.

The solvability criterion is not necessarily satisfied for given RA and RB , i.e..L(RA, RB) = Φ,Φ 6= 0. However we see easily in the following Lemma that Φcannot be large if α + R is a commutative action. It is quadratically small withrespect to R. If L has a tame right inverse (this is the content of Section 3.5) thenR can be approximated by a cocycle over the linear action generated by A and Bwith an error quadratically small with respect to R.

Lemma 3.4. If α = α+ R is a commutative action, with α linear, then

(3.25) ‖L(RA, RB)‖C0 ≤ C‖R‖C0‖R‖C1

for C∞ maps RA = R(g1), RB = R(g2).

Proof:αA αB = αB αA

(A+RA) (B +RB) = (B +RB) (A+RA)

RA (B +RB) −BRARB (A+RA) −ARB

Therefore

L(RA, RB) = −RA B +BRA +RB A−ARB

= − (RB (A+RA) −RB A) + (RA(B +RB) −RA B)


Then the estimate (3.25) follows from this easily (see for example Appendix II [23]).

3.5. Estimates for the right inverse of the operator L. In this section weconstruct a tame right inverse for the operator L. Namely, given smooth functionsθ, ψ and ϕ satisfying the equation L(θ, ψ) = ϕ, in this section we construct tame

solution for the same equation, namely we construct C∞ functions θ and ψ suchthat L(θ, ψ) = ϕ and such that their norms are bounded by certain norms of ϕ.The estimates that we obtain are tame in the sense that the loss of derrivatives isa constant independent of the functions involved. The tame estimates are obtainedby using the growth estimates for the orbits of the dual action from Section 3.3.

Applying Lemma 3.5 inductively, in Lemma 3.6 we obtain tame solution for theeqation L(Θ, Ψ) = Φ given that L(Θ,Ψ) = Φ for some C∞ maps Θ,Ψ,Φ on thetorus (see Section 3.1 for definitions of operator L ).

As a consequence of the existence of a tame right inverse for L, there exists anapproximate solution to the equation (2.8). Namely, the operator L is linear thus in

the set-up of Section 2.2, the difference maps RA = RA−RA and RB = RB−RB areC∞ maps of the same order as R and they satisfy the equation L(RA, RB) = 0. Theappropriate estimates are proved in subsequent Lemmas 3.5 and 3.6. Therefore, byLemma 3.3 there exists a common smooth solution to the equations ∆AΩ = RA and∆BΩ = RB which is of the same order as R. Then Ω is an approximate solutionof (2.8) because the error is of the order of RA and RB, due to the tame estimates

obtained in this section RA and RB are of the order of L(RA, RB) and this isquadratically small with r espect to R by Lemma 3.4. The quadratic estimate isthen used in Section 4 as the base for KAM iteration.

Even though the core of the two Lemmas in this section is the construction ofa tame right inverse, we state both Lemmas in the form which is most convenientfor application in Section 4.

Lemma 3.5. Let θ, ψ, ϕ be C∞ functions such that L(θ, ψ) = ∆µθ − ∆λψ = ϕ,then it is possible to split θ and ψ as

θ = θ − θ

ψ = ψ + ψ

so that L(θ, ψ) = 0, L(θ, ψ) = ϕ and the following estimates hold:

(3.26) ‖θ, ψ‖Cr ≤ C‖ϕ‖Cr+σ

for any r > 0 and any σ > Mλ,µ and

(3.27) ‖θ, ψ‖Cr ≤ C‖θ, ψ‖Cr+σ

for any r > 0 and any σ > Mλ,µ. As λ and µ are eigenvalues of A and B,

constants Mλ,µ and Mλ,µ depend only on A, B and the dimension of the torus andare precisely defined below (see (3.35) and (3.42)).

Proof:(i) Construction of θ, ψ, θ and ψ.


Let ωdef=

∑ωnen where

ωndef=

∑+

B

ψn, n→1, 2(B)

−∑

–

B

ψn, n→3(B)

for n ∈ ZN \ 0 and ω0 = (µ− 1)−1ψ0.Let

(3.28) ψdef= ∆µω = µω − ω B

Call n minimal and denote it by nmin if n is the lowest point on its B-orbit inthe sense that n→3(B) and Bn→1, 2(B) (for the definition of ”→” see Section3.1). There is one such minimal point on each non-trivial dual B-orbit. Now let

ψdef=

∑ ψnen where

(3.29) ψndef=

µ

∑B

ψn, n = nmin

0, otherwise

for n 6= 0 andψ0

def= 0. Then it is easy to check that

ψ = ψ + ψ

In part (ii) and (iii) bellow we will show that both ψ and ψ are smooth functions

such that ψ is of the order of ψ and ψ is the order of ϕ.Let us define θ as:

(3.30) θdef= ∆λω

Then it is easy to see that L(θ, ψ) = 0 since operators ∆λ and ∆µ commute due

to the commutativity of the generators A and B. Therefore by defining θ as:

(3.31) θdef= θ − θ

we obtain L(θ, ψ) = ϕ i.e.

(3.32) ∆µθ = ∆λψ + ϕ

Since operators ∆µ and ∆λ are bounded, if ψ is proved to be smooth with normcomparable to some norm of ϕ, then by Lemma 3.1 the same holds true for θ asa solution of the equation (3.32) (The operator ∆µ is injective on C∞ wheneverµ 6= 1. This fact is contained in the proof of the Lemma 2.1 and is a consequenceof the ergodicity of B).

(ii) Estimates for θ and ψ.The following estimate follows from the definition of ω in (i) and by using the

same method as in the proof of Lemma 3.1:

(3.33) ‖ω‖a−δ−κµ,B≤Ca

δν‖ψ‖a

for any a > κµ,B and any δ > 0, where

(3.34) κµ,Bdef=

N + 1

log η| log |µ||

The extra loss of κµ,B appears here because the obstructions∑B

ψ do not vanishand |µ| may be different than 1, just as in the proof of Lemma 3.2, parts b) and c).


Here, η > 1 is the constant coming from the exponential growth in the hyperbolicdirection for B (as ρ is for A in (2.2)). Since operators ∆µ and ∆λ are bounded in

any ‖ · ‖a norm, this also implies the following estimate for ψ and θ:

‖ψ, θ‖a−δ−κµ,B≤Ca

δν‖ψ‖a

which in particular implies the corresponding estimate (3.27) for Cr norms for ψ

and θ with the loss of

(3.35) Mλ,µdef= N + 2 + [κµ,B]

derivatives.(iii) Estimates for ψ and θ.

To estimate ψ we need to bound∑

B

ψn in case n→3(B) and Bn→1, 2(B) with

respect to ϕ. Since ∆µθ = ∆λψ + ϕ, the obstructions for ∆λψ + ϕ with respect toB vanish, therefore:

∆λ∑

B

ψn = −∑

B

ϕn

Iterating this equation with respect to A we obtain:

∑B

ψn + λ−l liml→∞

∑B

λ−lψAln = −l∑

i=0

∑B

ϕAin

¿From Lemma 3.2 b) the limit above is 0. By iterating backwards and applying thesame reasoning, we obtain:

(3.36)∑

B

ψn =∑

–

A∑

B

ϕn = −∑

+

A∑

B

ϕn

In the notation of Lemma 3.2 c), (3.36) implies that for n ∈ ZN which is minimalon its B orbit, we have:

ψn = SH+(ϕ, n) = −SH−(ϕ, n)

where H+ is the set of lattice points (l, k) in Z2 with positive l, and H− the setof points with negative l. Then according to Lemma 3.2 d), the needed estimate

for ψ with respect to ϕ follows if in at least one of the half-spaces H− and H+ thedual action satisfies some polynomial lower bound for every n = nmin.

In case Bn→2(B) for all l and all k we obviously have:

(3.37) |AlBkn| ≥ C|l|−N |k|−N |n|

thus the polynomial estimate needed for the application of part c) of Lemma 3.2 issatisfied both in H+ and H− for such n.

However in the other case i.e. when Bn→1(B) the same estimate holds eitherin H+ or in H−. This follows from the fact that in this case n is substantially largeboth in the expanding and in the contracting direction for B.

Let ni1 and ni3 be (large) projections of n to some expanding and contract-ing Lyapunov subspaces Vi1 and Vi2 for B with Lyapunov exponents χi1 and χi2 ,respectively, i.e. let

‖ni1‖ ≥ C|n| and ‖ni3‖ ≥ C|n|


where C is some fixed positive number. Then (assuming for the moment that α issemisimple) this implies:

(3.38)|AlBkn| ≥ C

r∑

i=1

expχi(l, k)‖ni‖

≥C(expχi1(l, k) + expχi3(l, k))|n|

where Now we notice that the union H of half-spaces (l, k) : χi1(l, k) ≥ 0 and(l, k) : χi3(l, k) ≥ 0 covers either H+ or H−. Namely, for any k ∈ Z, (l, k) is in

H if l(log |λi1 |

log |µi1 |−

log |λi3 |

log |µi3 |) ≥ 0 and this is true for l ≥ 0 or for l ≤ 0 depending on

the sgn(log |λi1 |

log |µi1 |−

log |λi3 |

log |µi3 |). Here λi3 , λi1 and µi3 , µi1 are corresponding eigenvalues

of A and B, respectively. Therefore, from (3.38) we obtain

|AlBkn| ≥ C|n|

in H+ or in H− if α is semisimple. If α is not semisimple than it decomposes aproduct of its semisimple and its unipotent part. For the semisimple part we usethe estimate above and in the unipotent part only a polynomial growth may occur.This implies that (3.37) holds in H+ or in H− for a general (non-semisimple) α.

Now choose the half-space in which the estimate (3.37) holds, that is choose oneof the sums SH+(ϕ, n) or SH−(ϕ, n). Than the assumptions of d) in Lemma 3.2are satisfied for one of the sums above SH+ or SH− and therefore the estimate forψ follows:

(3.39) ‖ψ‖a−δ−κ(λ,µ),α≤Ca

δν‖ϕ‖a

for any a > κ(λ,µ),α and any δ > 0, where

(3.40) κ(λ,µ),αdef=

N + 1

τ(| log |µ|| + | log |λ||)

Here, τ = τ(A,B) > 0 is the constant chosen as in the Lemma 3.2 a)

As we mentioned in part (i), by construction we have ∆µθ = ∆λψ + ϕ. This by

using Lemma 3.1 implies the following estimate for θ with respect to ϕ:

(3.41) ‖θ‖a−δ−κ(λ,µ),α≤Ca

δν‖ϕ‖a

for any a > κ(λ,µ),α and any δ > 0. lThis implies the Cr estimate (3.26) for ψ and

θ with the loss σ > Mλ,µ, where

(3.42) Mλ,µdef= N + 2 + [κ(λ,µ),α]

where κ(λ,µ),α is defined in (3.40).

Lemma 3.6. For two C∞ maps RA, RB with L(RA, RB) = Φ the splitting: RA =

RA + RA and RB = RB + RB exists and satisfies:

L(RA, RB) = 0, L(RA, RB) = Φ

(3.43) ‖RA, RB‖Cr ≤ Cr‖R‖Cr+σ

(3.44) ‖RA, RB‖Cr ≤ Cr‖Φ‖Cr+σ


for any r > 0 and σ > M = M(A,B,N) , where constant M depends only on thedimension of the torus and the linear action and is defined bellow (see (3.58)).

Proof: If A and B are semisimple than the statement follows directly fromLemma 3.5 as the condition L(RA, RB) = Φ splits into finitely many equations ofthe type

∆µθ − ∆λψ = ϕ

where θ, ψ, ϕ are C∞ functions.Now assume that A and B are not simultaneously diagonalizable and choose a

basis in which A is in its Jordan normal form with some non-trivial Jordan blocks.Then in the same basis B has block diagonal form and as in Lemma 3.3 we cantake m ×m blocks JA and JB corresponding to eigenvalues λ and µ of A and B,respectively, and split L(RA, RB) = Φ into equations:

(3.45) JBΘ − Θ B − JAΨ + Ψ A = Φ

where as in Lemma 3.3 we take JA = (aij) to be an m ×m matrix which consistof blocks of A corresponding to the eigenvalue λ, i.e. aii = λ for all i = 1, ..,m andai,i+1 = ∗i ∈ 0, 1 for all i = 1, ..,m − 1 and JB = (bij) to be the correspondingblock of B where bii = µ for all i = 1, ..,m (µ is an eigenvalue of B) and bij = 0for all m ≥ i > j ≥ 1. The equation (3.45) splits into m equations. For everyk = 1, ..,m we have the following equation which we call (EQ)k:

(3.46) (∆µθk +

m∑

i=k+1

bkiθi) − (∆λψk + ∗kψk+1) = ϕk

where θi, ψi and ϕi are coorinate functions of Θ, Ψ and Φ, respectively, in the basisin which A is in its Jordan normal form. In the special case when ϕk = 0 than wedenote the equation (3.46) by (EQ)0k. Clearly, for k = m the equation (EQ)m issimply

∆µθm − ∆λψm = ϕm

which by Lemma 3.5 implies the existence of the splitting

(3.47)θm = θm + θm = ∆λωm + θm

ψm = ψm + ψm = ∆µωm + ψm

where ωm, θm, ψm, θm = ∆λωm and ψm = ∆µωm are C∞ functions satisfying theestimates:

(3.48)

‖θm, ψm‖a−(δ+κ) ≤Ca

δν‖Φ‖a

‖ωm‖a−(δ+κ) ≤Ca

δν‖Θ,Ψ‖a

‖θm, ψm‖a−(δ+κ) ≤Ca

δν‖Θ,Ψ‖a

where we let κdef= maxκ(λ,µ),α, κµ,B (see the proof of Lemma 3.5, (3.40),(3.34)).

Now we proceed by induction. Fix k between 1 and m− 1 and assume that forall i = k + 1, ..,m we already have the splitting:

(3.49)θi = θi + θi = ∆λωi + θi

ψi = ψi + ψi = ∆µωi + ψi


where ωi, θi = ∆λωi, ψi = ∆µωi, θi and ψi are C∞ functions satisfying the followingestimates:

(3.50)

‖θi, ψi‖a−(m−i+1)(δ+κ) ≤Ca

δ(m−i+1)ν‖Φ‖a

‖ωi‖a−(m−i+1)(δ+κ) ≤Ca


‖θi, ψi‖a−(m−i+1)(δ+κ) ≤Ca


and such that θi and ψi satisfy the equation (EQ)0i (3.46).By substituting from (3.49) the expressions for θi and ψi for all i = k + 1, ..,m

into (3.46), we obtain:

(3.51) ∆µ(θk − ∗kωk+1) − ∆λ(ψk −m∑

i=k+1

bkiωi) = ϕk − ∗kψk+1 −m∑

i=k+1

bkiθi

Then using Lemma 3.5 again we can obtain ωk, θk and ψk such that

(3.52)

θk − ∗kωk+1 = ∆λωk + θk

ψk −m∑

i=k+1

bkiωi = ∆µωk + ψk

with estimates for θk and ψk following from (3.50) and (3.51):

(3.53)

‖θk, ψk‖a−(m−k+1)(δ+κ) ≤Ca

δ(m−k+1)ν‖Φ‖a

‖ωk‖a−(m−k+1)(δ+κ) ≤Ca


‖θk, ψk‖a−(m−k+1)(δ+κ) ≤Ca


where we define θk and ψk as

(3.54)

θk = ∆λωk + ∗kωk+1

θk = ∆λωk +

m∑

i=k+1

bkiωi

Now checking that θk and ψk are ”good”, i.e. that they satisfy the equation (EQ)0kis easily done just by substitution.

At this point however one has to use the coefficients relations (3.11) which werederived in Lemma 3.3 from the fact that A and B commute.

Since the maximal size of a Jordan block of A is less than N , we can estimateΘ and Ψ in the ‖ · ‖a−N(δ+κ) norm:

(3.55) ‖Θ, Ψ‖a−N(δ+κ) ≤Ca

δNν‖Φ‖a

and similarily for maps θ and ψ:

(3.56) ‖Θ, Ψ‖a−N(δ+κ) ≤Ca

δNν‖Θ,Ψ‖a

If repeated for all Jordan blocks, this produces the required splitting:

(3.57) RA = RA + RA RB = RB + RB


which satisfies the conditions and the estimates in the statement. Since we userepeatedly Lemma 3.5 the Cr estimates (3.43) and (3.44) hold for σ > M =M(A,B,N), where

(3.58) Mdef= N + 2 +N [maxmax

iκµi,B,max

iκ(λi,µi),α]

where the first maximum in i is taken over all eigenvalues µi of B (where κµi,B

are numbers defined in (3.34)) and the second maximum in i is taken over all pairsof corresponding eigenvalues λi and µi of A and B ( where κ(λi,µi),α are numbersdefined in (3.40)).

4. Proof of Theorem 1

Assuming α is a C∞ perturbation of α, and that the difference R = α − α issmall in some Cl norm (where l is fixed and will be determined in the proof, (4.12))we show that α is smoothly conjugate to α. The conjugacy is produced for the twoergodic generators and by Lemma 2.1 it works for all elements of the action.

4.1. Smoothing. Following the scheme described in Section 2.2 at each step of theiterative procedure we solve the linearized equation (2.8):

AΩ − Ω A = −RA

BΩ − Ω A = −RB

approximately. By results of the Section 3.5 we have that the linearized equation(2.8) has an approximate solution Ω which is C∞ although we can only compareits norm in Cr to the norm of R in Cr+σ where σ is large but fixed. The error isR and is comparable to L(RA, RB) thus it is quadratically small with respect to Rbut comparison again comes with fixed loss σ. The loss of derrivatives might be alarge number (depending of the hyperbolicity properties of the linear action), butit only depends on the dimension of the torus and the unperturbed linear action.To overcome this fixed loss of derrivatives at each step of the iteration process, itis standard (see for example [32]) to introduce a family of smoothing operators:SJ , J ∈ N. Then instead of solving approximately (2.8) we will solve approxi-mately the following system:

(4.1)AΩ − Ω A = −SJRA

BΩ − Ω B = −SJRB

For a C∞ function f =∑

n fnχn we define SJf as:

SJfdef=

∑

|n|<J

fnχn

Then the smoothing operators satisfy:

(4.2)‖SJf‖a+b ≤ Jb‖f‖a

‖SJf‖Ca+b ≤ Jb+σ‖f‖Ca

where a > b > 0 and σ > N + 1. Also:

(4.3)‖(I − SJ )f‖a−b ≤ J−b‖f‖a

‖(I − SJ )f‖Ca−b ≤ CJ−b+σ‖f‖Ca


for a > b > σ > N + 1. These simple smoothing operators are convenient in oursetting since they are well behaved with respect to the operator L. Namely, wehave:

L(SJf, SJg) = S Jξ(L(f, g)) + F> J

ξ

where ξ is a constant depending on A and B and the last term in the expressionabove consists of pieces of Fourier series for f and g involving only terms with|n| > J

ξ . This implies the following estimates:

(4.4)

‖L(SJf, SJg)‖a ≤ ‖S Jξ(L(f, g))‖a + CJ−b‖f, g‖a+b

‖L(SJf, SJg)‖Cr ≤ ‖S Jξ(L(f, g))‖Cr+σ + CJ−b+σ‖f, g‖Cr+b

≤ J2σ‖L(f, g)‖Cr + CJ−b+σ‖f, g‖Cr+b

for any a > 0 and b > σ > N + 1.

4.2. Iterative step and the error estimate. At each step of the iterative schemewe first choose an appropriate smoothing operator SJ . In order to solve approxi-mately (4.1) we use Lemma 3.6 to obtain the splitting:

SJRA = SJRA + SJRA

SJRB = SJRB + SJRB

so that L(SJRA, SJRB) = 0. Here, for better visualization, the notation f is used

instead the notation f of the Section 3.5. Now from Lemma 3.3 the system

AΩ − Ω A = −SJRA

BΩ − Ω B = −SJRB

has an approximate solution Ω such that:

(4.5)‖Ω‖Cr ≤ Cr‖SJRA, SJRB‖Cr+σ ≤ Cr‖SJRA, SJRB‖Cr+2σ

≤ CrJ3σ‖RA, RB‖Cr ≤ CrJ

3σ‖R‖Cr .

Here we used the estimates from Lemma 3.6 and the properties of smoothing (4.2).In what follows R is assumed to denote the pair of mappings R(g1) = RA andR(g2) = RB and ‖R‖Cr denotes the maximum of the corresponding norms of thetwo maps.

The estimate (4.5) holds for any σ large enough: σ > maxM,M0 so that the

estimates in Lemma 3.3 and Lemma 3.6 hold. Then we form Hdef= id+ Ω (since Ω

is made small throughout the iteration, H will is invertible) and

α(1) def= H−1 α H

The new error is

R(1) def= α(1) − α

and it has two parts:

• The error coming from solving the linearized equation only approximately:

E1 = SJR + (I − SJ)R

• The standard error coming from the linearization:

E2 = Ω α(1) − Ω α+ R (id+ Ω) −R


Estimate of E1. Using Lemma 3.6 and the properties of smoothing (4.4), forevery b > σ we have:

‖SJR‖C0 ≤ C‖L(SJRA, SJRB)‖Cσ ≤ C‖S JξL(RA, RB)‖

C2σ+ CJ−b+σ‖R‖Cσ+b

≤ C[J3σ‖L(RA, RB)‖C0 + J−b+σ‖R‖Cσ+b

]

Also, using (4.3):

‖(I − SJ )R‖C0 ≤ ClJ−l+σ‖R‖Cl

for any l ≥ σ. Let b = l − σ. Thus, using Lemma 3.4 we have:

(4.6) ‖E1‖C0 ≤ CJ3σ‖R‖C0‖R‖C1 + CJ−l+2σ‖R‖Cl

for l > 2σ.Estimate of E2. The first part of E2 is estimated as follows:

‖Ω α(1) − Ω α‖C0 ≤ C‖Ω‖C1‖α(1) − α‖C0 = C‖Ω‖C1‖R

(1)‖C0 ≤1

4‖R(1)‖C0 ,

thus this part is absorbed by ‖R(1)‖C0 providing ‖Ω‖C1 is bounded throughout theprocedure. The second part of E2 we estimate by using (4.5):

(4.7) ‖R(id+ Ω) −R‖C0 ≤ C‖R‖C1‖Ω‖C0 ≤ CJ3σ‖R‖C1‖R‖C0

Estimate of the new error R(1). By combining (4.6) and (4.7) we obtain anestimate for the new error:

(4.8) ‖R(1)‖C0 ≤ CJ3σ‖R‖C1‖R‖C0 + CJ−l+2σ‖R‖Cl

for any l > 2σ. From α(1) = H−1 α H using the fact that Ω is N-periodic andsatisfies the estimate (4.5) we have:

(4.9) ‖R(1)‖Cl ≤ ClJ3σ (1 + ‖R‖Cl)

Therefore we have obtained what usually constitutes the basis for the iteration (see[32] for example) i.e., the ”quadratic” estimate (4.8) for the ”low” norm of the newerror and the ”linear” estimate (4.9) for some ”high” norm of the new error withrespect to the initial error R.

4.3. Setting up the iterative process. To set up the iterative process we firstlet:

R(0) = R; α(0) = α; H(0) = id

Now construct R(n) inductively for every n: for R(n) we choose an appropriatenumber Jn to obtain SJn

R(n) which produces, after solving approximately thelinearized equation, new Ω(n). Then we define:

(4.10)

H(n) = id+ Ω(n)

α(n+1) =(H(n)

)−1

α(n) H(n)

R(n+1) = α(n+1) − α

Consequently:

α(n+1) =(H(n)

)−1

(H(n−1)

)−1

· · · (H(0)

)−1

α H(0) · · · H(n)

= H−1n α Hn


where Hndef= H(0) · · · H(n). To ensure the convergence of the process we set:

(4.11)‖R(n)‖C0 < εn = ε(k

n), ‖R(n)‖Cl < ε−1n

‖Ω(n)‖C1 < ε1/2n , Jn = ε

− 13(3σ+2)

n

where k =4

3. At this point we fix l:

(4.12) l = 23σ + 15

where σ = σ(A,B,N) = maxM,M0 is the constant for which all the estimatesobtained before hold ((3.58),(3.24)). The constant l is chosen so that the processconverges. We check the convergence in the subsequent section.

4.4. Convergence. By induction it is proved that all the above bounds (4.11) holdfor every n ∈ N:

‖R(n+1)‖Cl ≤ ClJ3σn

(1 + ‖R(n)‖Cl

)≤ ClJ

3σn

(1 + ε−1

n

)

≤ 2ClJ3σn ε−1

n ≤ 2Clε− 3σ

3(3σ+2)n ε−1

n < ε− 1

3−1n = ε

− 43

n

= (εn+1)−1

Using interpolation inequalities we have:

‖R(n)‖C1 ≤ Cl‖R(n)‖

1− 1l

C0 ‖R(n)‖1l

Cl

Along with (4.8) this implies:

‖R(n+1)‖C0 ≤ C

[ε− 3σ

3(3σ+2)n ε

1− 1l

n ε− 1

ln εn + ε

l−2σ3(3σ+2)n ε−1

n

]

= C

[ε− σ

3σ+2 +2(1− 1l )

n + εl−2σ

3(3σ+2)−1

n

]= C [εx

n + εyn]

≤ ε43n = εn+1

providing x >4

3, y >

4

3, i.e.

x = −σ

3σ + 2+ 2

(1 −

1

l

)>

4

3

y =l − 2σ

3(3σ + 2)− 1 >

4

3providing

Both inequalities above are satisfied for l ≥ 23σ + 15. We note here that withmore precision (by changing the rate of convergence) the constant l can be madesomewhat smaller (see Section 6).

Using (4.5) we check the C1 bound for Ω:

‖Ω(n+1)‖C1 ≤ C · J3σ+1n ‖R(n)‖C0 ≤ C · J3σ+1

n εn = C · ε− 3σ+1

3(3σ+2)+1

n

< ε23n = ε

12n+1

Thus for sufficiently small ‖R‖C0 and ‖R‖Cl the process converges to a solution

Ω ∈ C1 with ‖Ω‖C1 <1

4.

Now using interpolation inequalities, we show that Ω is C∞:


For arbitrary m from (4.9) we have:

‖R(n+1)‖Cm ≤ CmJ3σn

(1 + ‖R(n)‖Cm

)≤ ε

− 13

n

(1 + ‖R(n)‖Cm

)

‖R(n)‖Cm ≤ Cm

n−1∏

ν=1

ε− 1

3ν (1 + ‖R‖Cm) ≤ ε−1

n Cm

where in the second line above the constant Cm is Cm := Cm (1 + ‖R‖Cm). Letm = 3k. Then:

‖R(n)‖Ck ≤ Ck‖R(n)‖

23

C0‖R(n)‖

13

C3k < Ckε23nε

− 13

n = Ckε13n

‖Ω(n)‖Ck ≤ CkJ3σn ‖R(n)‖Ck ≤ Ckε

− 3σ3(3σ+2)

n ε13n = Ckε

δn

with δ =2

3(3σ + 2)> 0 and the constant Ck changing throughout the above

procedure, but depending only on k and the C3k norm of the initial perturbationR.

This implies the convergence of the sequence Hn in Ck norm for every k ∈ N i.e.the limit H is a C∞ map.

5. Proof of Theorem 2

Definition 2. A C∞ foliation F of a manifold M is C∞,l,∞-locally rigid if for anysufficiently Cl close C∞ foliation F ′ there exists a C∞ diffeomorphism H of Mtaking leaves of F to leaves of F ′.

Definition 3. An action α of Rk on M is C∞,l,∞-orbit foliation rigid if its orbitfoliation is C∞,l,∞-locally rigid.

Lemma 5.1. C∞,l,∞-local rigidity for an action γ of Zk by diffeomorphisms on amanifold M implies C∞,l,∞-orbit foliation rigidity for the suspension action α(γ)on the suspension manifold N(γ).

Proof: Let Γ : Zk → Diff∞(M) be a representation of Zk defining a Zk actionγ by diffeomorphisms of M . Let α(γ) : Rk × N(γ) → N(γ) be the correspond-ing suspension action with the orbit foliation F(γ) and let α be a Cl small C∞

perturbation of α(γ) with the orbit foliation F(γ) which is Cl close to F(γ).If the perturbation is sufficiently small the orbit foliation of the perturbation is

still transversal to the fibers M of the suspension manifold N(γ). Let Γ : Zk →

Diff∞(M) be the holonomy of the perturbed foliation F(γ), defining an action γof Zk on M . Then there exists a C∞ diffeomorphism H1 : N(γ) → N(γ) taking

leaves of the foliation F(γ) of the suspension action over γ to the leaves of F(γ).(Theorem 3 in Section 5.4 [2])

On the other hand, since γ and γ are sufficiently Cl close C∞ Zk-actions thereis a C∞ conjugacy h : M →M such that h γ = γ h according to our assumptionon the local rigidity of γ. This implies the existence of a C∞ orbit equivalence forthe corresponding suspensions, i.e there exists H2 : N(γ) → N(γ) taking leaves ofF(γ) to leaves of F(γ) (Theorem 2 in Section 5.4 [2])

Thus the C∞ diffeomorphism H = H2 H1 on N(γ) is an orbit equivalence foractions α(γ) and α which implies C∞,l,∞-orbit foliation rigidity for α(γ).


It follows from Lemma 5.1 and Theorem 1 that for a given suspension actionand its small perturbation there exists a C∞ orbit equivalence taking orbits of theoriginal to the orbits of the perturbed action. This reduces the question of localrigidity of suspensions to considering small perturbations along the leaves of orbitfoliation only. However such perturbations are given by Rk valued cocycles and dueto the fact that suspensions over Zk actions by ergodic toral automorphisms areC∞-cocycle rigid (see [19] Section 4.2 for explanation of how cocycle rigidity of a Zk

action on the torus induces cocycle rigidity for a suspension) it follows that pertur-bations in orbit direction are conjugate to original action up to an automorphismof Rk. This completes the proof of Theorem 2.

6. Comments on finitely differentiable and analytic case

6.1. Finitely differentiable case. It is clear that all our arguments only use afinite number of derivatives so that our rigidity results in a modified form holdwhen the perturbed action α has only finitely many derivatives. We discuss nowspecific modifications which appear this way.

Let R be Cm. Then the estimate (3.7) for the approximate solution of thelinearized equation obtained in Lemma 3.3 still holds if m > σ, where σ > M0

and M0 is defined in (3.24). Similarily, the estimates (3.43) and (3.44) obtainedin Lemma 3.6 which are used later in the proof of the Theorem 1 (Section 4) holdwhen R is Cm and m > M , where M is defined in (3.58). Now the iterative proofin Section 4 aplies in the following setting:

Let α be a Zk action by automorphisms of TN as in the statement of Theorem1. Let σ ∈ N be a fixed number greater than constants M (3.58) and M0 (3.24)defined in Lemmas 3.3 and 3.6.

The convergence set-up is the same as in the Section 4.3, the modification onlycomes in determining the speed of the convergence. Instead of (4.11) now we set:

(6.1)‖R(n)‖C0 < εn = ε(k

n), ‖R(n)‖Cl < ε−1n

‖Ω(n)‖C1 < ε2−k

kn , Jn = ε

− k−1(3σ+2)

n

where 1 < k < 2. While making sure that these bounds hold for every n, we willobtain a lower bound on l which depends on the speed of convergence k.

We first check the bound for the Cl norm using the estimate (4.9):

(6.2) ‖R(n+1)‖Cl ≤ ClJ3σn (1 + ‖R(n)‖Cl

) ≤ C′lε

−(k−1)n ε−1

n = ε−1n+1

Using (4.5) it is easy to check the bound for Ω:

(6.3) ‖Ω(n)‖C1 ≤ Cε− k−1

3σ+2 (3σ+1)n εn ≤ ε2−k

n ≤ ε2−k

k

n+1

Now for the C0 bound we have as in the section 4.4, using (4.8) and the inter-polation estimates:

‖R(n+1)‖Cl ≤ ClJ3σn ‖R(n)‖C1‖R(n)‖C0 + ClJ

−l+2σn ‖R(n)‖Cl

≤ ClJ3σn ‖R(n)‖

2− 1l

C0 ‖R(n)‖1l

Cl + ClJ−l+2σn ‖R(n)‖Cl

≤ Clε−

3σ(k1)3σ+2

n ε2− 1

ln ε

− 1l

n + ε− (k−1)(−l+2σ)

3σ+2n ε−1

n

≤ ε−k+3− 2

ln + ε

(k−1)(l−2σ)3σ+2 −1

n


Thus if −k + 3 − 2l > k and (k−1)(l−2σ)

3σ+2 − 1 > k then we have ‖R(n+1)‖Cl ≤ εn+1.

The first condition gives l > 23−2k and the second one l > 2σ + k+1

k−1 (3σ + 2). Thefirst condition is satisfied already for l > 2. The second one actually gives thedependence on k. Namely, as the speed of convergence k approaches 2, the lowerbound for l approaches l0 := 11σ+ 6. Thus if one chooses l = l0 + δ for δ > 0, thenit is possible to choose the rate of convergence k = k(δ) so that all the bounds (6.1)hold for every n.

This implies the convergence of the procedure in C1 norm. Therefore, the per-turbation needs to be close to the initial action in Cl with l (only) strictly largerthan l0, in order to obtain a C1 solution.

In order to get more derrivatives for the soluton, we need to assume that theperturbation is more regular than Cl0+δ. Again, at this point we use interpolationinequalities. Assume that the initial perturbation is Cm and let 1 < r < m. As inSection 4.4 it is easy to check that one has ‖R(n)‖Cm ≤ Cmε

−1n ‖R(0)‖Cm . Therefore

we have:

‖R(n)‖Cr ≤ C‖R(n)‖1− r

m

C0 ‖R(n)‖rm

Cm < Cmε1−2 r

mn

‖Ω(n)‖Cr ≤ CJ3σn ‖R(n)‖Cr ≤ Cmε

−(k−1) 3σ(3σ+2)

n ε1−2 r

mn

Therefore in order to obtain a solution of order r we need that rm < 1− k

2 . So ifthe perturbation is C∞ then for any l > l0 the solution is C∞ with k chosen closeto 2 if l is close to l0 as described above.

Now if the perturbation is only Cm, and close to the unperturbed action in Cl

for some l > l0 then by choosing k so that l > 2σ + k+1k−1 (3σ + 2), that is choose

k close but larger than k0(l) = l+5σ+2l−3σ−2 ∈ (1, 2). Then the solution is Cr for every

r < (1 − k0(l)2 )m.

6.2. Analytic case. Now suppose the perturbed action α is real-analytic. A nat-ural question is whether the unique C∞ conjugacy with the linear action is alsoanalytic.

Both estimates in Lemma 3.1 and in part d) in Lemma 3.3 can be obtainedin the category of analytic spaces Aσ allowing arbitrary small loss of domain δ inall directions and assuming that there is no growth for the action in the neutraldirection.

The algebraic part of the proof of Lemma 3.5 holds too, thus one can define mapsψ, θ, ψ, θ. The analytic estimates for the ψ, and ψ with small loss of domain hold(for estimating ψ we need to use that θ, ψ are small with respect to L(θ, ψ) andthus quadratically small with respect to ψ which follows by using Caushy estimatesfor L(θ, ψ)). The problem is that the estimates for other functions, θ and θ canbe obtained by this method only in a domain which is in some directions muchsmaller than the initial one (multiplied by a constant less than one comming fromthe contraction of some directions by the action of A). Such a loss at every iterativestep cannot lead to a convergence. Thus, lack of analytic result is due to specificconstructions in Section 3.5.

7. Existence of genuinely partially hyperbolic actions

To prove the statement of Theorem 3 we first show that there are no examplesof genuinely partially hyperbolic Zk actions (irreducible or reducible) for N ≤ 5.It is proven in [30] that there are no irreducible automorphisms in odd dimensions


with non-trivial neutral direction. We will give below another proof of this fact.Further we show that there are irreducible examples of such actions in any evendimension N ≥ 6. By combining these results we obtain reducible examples in anyodd dimension N ≥ 9.

7.1. There are no examples on TN for N < 6. We show that it is not possibleto have a genuinely partially hyperbolic Z2 action on TN if N < 6:

Case N = 2. Since we exclude purely hyperbolic case, the two commutingmatrices would have to be pure rotations. Moreover, the ergodicity assumptionimplies that each would have to have two complex conjugate eigenvalues of absolutevalue one, that are not roots of 1. It is clear however that there are no such matricesamong integer matrices at all.

We note that it is of course possible to have one hyperbolic integer matrix ofdeterminant one in dimension 2, but no two commuting ones (that are not powersof each other) exist due to the Dirichlet Units Theorem [14].

Case N = 3. In this case, same as above, no genuinely partially hyperbolicmatrices exist. Namely, if there are two eigenvalues of absolute value 1 than thethird eigenvalue would have to be real and exactly one. This is excluded by theergodicity assumption.

As for the purely hyperbolic case, already for N = 3 a 2-action exists. Namely, ifA is a hyperbolic matrix with all three eigenvalues real, the Dirichlet Units Theoremimplies that the dimension of the centralizer of A is 2, thus there exists an integermatrix B multiplicatively independent from A such that AB = BA.

Case N = 4. In dimension 4 it is possible to have one matrix with desired prop-erties, namely a matrix that is integer and has two complex conjugate eigenvaluesof absolute value one, and two real eigenvalues λ and λ−1. To produce such matrixit is enough to choose a quadratic irreducible polynomial with two real eigenvaluesone bigger than 2 and the other less than 2 in absolute value. Than the substitu-tion x → x + 1

x gives a quartic polynomial which generates a matrix with desiredproperties. However, it is not possible to have two commuting matrices with theproperties above and with common neutral subspace (this last one is a necessaryrequirement for otherwise the action generated by the two matrices would be hy-perbolic). Indeed, if B commutes with A and has real eigenvalues µ and µ−1 thenbecause of irreducibility requirement, both λ and µ are irrational, thus we canchoose an integer vector (l, k) in Z2 such that λlµk is close to 1. But then the sameholds true for the remaining real eigenvalues. Moreover rotations in the neutraldirection are assumed irrational therefore by choosing k and l large enough theycan be made close to one, too. This implies that the matrix AlBk is integer matrixclose to identity. This is not possible unless the matrix is identity itself, in whichcase the action would have a non-ergodic element.

Case N = 5. Here, it is not possible to have even one matrix with desiredproperties. Since we exclude purely hyperbolic case, there are only two possibilities:either there are two complex conjugate eigenvalues of absolute value 1 and threereal eigenvalues (signature (3,1)) or there are two pairs of complex eigenvalues,(exactly) one of these is of absolute value 1, and the one remaining eigenvalue isreal (signature (1,2)). By carefully examining characteristic polynomials and usingthe irreducibility assumption we are able to eliminate both possibilities:


Signature (3,1). Let f(x) be:

f(x) = (x− λ1)(x− λ2)(x− λ3)(x − µ)(x− µ)

where λ1, λ2, λ3 are real and |µ| = |µ| = 1 are complex eigenvalues. Then

f(x) = [x3 − ax2 + bx− 1][x2 + cx+ 1]

where a =∑

i=1,2,3 λi, b =∑

i=1,2,3 λ−1i and c = 2Reµ. Then the coefficients of

f(x) are: 1, α = c − a, β = b − cb + 1, γ = −a + cb − 1, δ = b − c,−1. Since allshould be rational, and since α + δ = b − a and β + γ = (b − a)(1 + c). If b 6= athis implies that c is rational and this contradicts irreducibility. But b = a alsocontradicts irreducibility assumption since f(x) would have factor (x − 1) in thatcase.Signature (1,2). Similarly as above we write f(x) as:

(x− λ)(x − ν)(x − ν)(x − µ)(x− µ) =

(x− λ)(x2 − Ix+ 1)(x2 −Rx+ ρ2)

where I = 2Reµ, R = 2Reν = 2ρ cos θ and ρ = |ν|. If we denote: a = λ + R andb = λR+ ρ2 then the coefficients of f(x) are:

1, α = −a− I, β = aI + 1 + b, γ = −a− 1 − Ib, δ = b+ I,−1

Since α+δ = b−a and β+γ = (b−a)(1−I) and using the fact that the coefficientsare rational, this implies that either I is rational or b = a. The first possibilityclearly contradicts irreducibility of f(x). Using that λρ2 = 1, b = a implies either|ρ| = 1 or 1 + ρ2 − 2ρ cos θ = 0. The first option implies λ = 1 thus contradictsirreducibility. The second equation is equivalent to (sin θ)2 + (cos θ − ρ)2 = 0 andimplies |ρ| = 1 which, as above, contradicts irreducibility.

We note that the preceding discussion also implies that there are no reduciblegenuinely partially hyperbolic Z2 actions in dimension N ≤ 5 as well.

7.2. There are no irreducible examples on TN for N-odd. As mentioned inthe introduction to this section, there are no irreducible genuinely partially hyper-bolic toral automorphisms in any odd dimension, thus there are no such actions inany odd dimension either. This was proved in [30]. This fact also follows from asimple number theoretic argument. Namely, if we let p(x) be irreducible character-istic polynomial of an integer matrix A of degree N , let ψ be root of p of absolute

value 1 and let θdef= ψ + ψ−1, than for the corresponding number fields L = Q(ψ)

and K = Q(θ) we have:

|L : K||K : Q| = N

On the other hand K is real since θ = ψ + ψ−1 = ψ + ψ ∈ R so |L : K| ≥ 2. Butwe also have ψ2 − ψθ + 1 = 0 therefore |L : K| ≤ 2. This implies |L : K| = 2 thusN has to be even.

7.3. An irreducible example in dimension 6. The following example of irre-ducible genuinely partially hyperbolic Z2 action on the torus T6 was produced by S.Katok with the use of PARI program. For the background on use of algebraic num-ber theory for producing examples of higher rank actions by toral automorphismswe refer to [14].


Starting with an irreducible cubic polynomial x3 − 2x2 − 8x + 1 with two realroots of absolute value larger than two and one less than two in absolute value, bysubstituting x→ x+ 1

x we obtain an irreducible recurrent polynomial

f(x) = x6 − 2x5 − 5x4 − 3x3 − 5x2 − 2x+ 1

which has 4 real roots and a pair of complex conjugate roots. The complex conjugateroots have to be of absolute value 1.

LetK = Q(λ) be the number field corresponding to f(x). The fundamental unitsare λ1 = λ, λ2 = λ5 − 3λ4 − 2λ3 − λ2 − 4λ+ 1, λ3 = λ4 − 2λ3 − 6λ2 − λ + 1, λ4 =2λ5 − 6λ4 − 3λ3 − 6λ2 − 6λ. It turns out that λ4 produces a matrix with sameneutral subspace as A.

With respect to the basis 1, λ, λ2, λ3, λ4, λ5 in Z[λ] multiplications by λ1 andλ4 are given by the matrices

A =

0 1 0 0 0 00 0 1 0 0 00 0 0 1 0 00 0 0 0 1 00 0 0 0 0 1

−1 2 5 3 5 2

B =

0 −6 −6 −3 −6 2−2 4 4 0 7 −2

2 −6 −6 −2 −10 3−3 8 9 3 13 −4

4 −11 −12 −3 −17 5−5 14 14 3 22 −7

Since B = 2A5−6A4−3A3−6A2−6A, AB = BA, and since the minimal polynomialof the matrix B is recurrent: x6 + 23x5 + 16x4 − 60x3 + 16x2 + 23x+ 1 and alsohas 4 real roots, the two complex conjugate roots are of absolute value 1.

7.4. Existence of irreducible examples in any even dimension greater than6. In this section we use the Dirichlet units theorem to prove the main part of theTheorem 3 stated in Section 1.3, that is to produce examples of Z2 genuinelypartially hyperbolic actions in any even dimension N ≥ 6.

Let q(x) be an irreducible integer polynomial of degree n with all real roots.Let r(x) = q(x + 1

x)xn be the recurrent polynomial of degree 2n given by q. LetL = Q(ψ) and K = Q(θ) be the number fields corresponding to q and r respectivelywith ψ + ψ−1 = θ. Let σi be embeddings of K into C. Since all roots of r come inpairs, let σi(ψ)σn+i(ψ) = 1 for all i = 1, .., n. Let α = p(ψ) ∈ L with deg(p) ≤ 2nbe any other element in L. Then σi(p(ψ

−1)) = p(σi(ψ−1)) = p((σi(ψ))−1) =

p(σn+i(ψ)) = σn+i(p(ψ)). This implies that p(ψ) and p(ψ−1) have same norms:N(p(ψ)) = N(p(ψ−1) = σi(ψ)...σ2n(ψ). Therefore, p(ψ) is a unit if and only ifp(ψ−1) is.

Let UL and UK be groups of units of L and K, respectively. Define a homomor-phism f : UL → UK by

f(p(ψ))def= p(ψ)p(ψ−1)

We first show that f(UL) is indeed in UK . Since for any integer polynomial p(x)we have p(x)p(x−1) = P (x+ x−1) where P is a rational polynomial, it follows thatf(p(ψ)) = P (θ) thus f(UL) ⊂ K. Also p(ψ) being a unit implies that p(ψ−1) is also


a unit and thus P (θ) is a unit in UL. Since ψ2 − θψ + 1 = 0 we have |L : K| ≤ 2.With |L : L∩R| = 2 and the fact that K is real, this implies K = L∩R. Therefore,a real unit in UL must lie in UK . This proves that the image of f is in UK . It isthen easy to check that f is a group homeomorphism.

In order to obtain matrices that commute with matrix A whose characteristicpolynomial is r(x) it is enough to show that the kernel of f contains at least twoindependent units. We show that the kernel contains s independent units wheres is the number of roots of q(x) of absolute value bigger than 2 (which implies 2sreal roots for r(x)). The rest t = n− s roots of q(x) are of absolute value less than2 and they induce t pairs of complex conjugate roots of r(X) of absolute value 1.By Dirichlet units theorem the structure of groups UL and UK is known. Namely,

every unit in UL can be expressed as ρuj11 u

j22 ..u

js+n−1

s+n−1 , where ui are independentunits. The images f(ui) of fundamental units of UL are units in UK . By Dirichletunits theorem UK can have at most n − 1 independent units. Therefore, in theset S = f(ui)|i = 1, .., n+ s − 1 there can be at most n − 1 independent units,without loss of generality assume those are f(ui), i = 1, .., n− 1. This implies thatthere are at least s relations of the kind:

1 = f(u1)j1,kf(u2)

j2,k ...f(un−1)jn−1,kf(uk)jk

Where k = n, ., n+ s. Since f is a homomorphism, this implies there are s units inthe kernel of f :

vk = uj1,k

1 uj2,k

2 ..ujn−1,k

n−1 ujk

k

for k = n, .., n+ s. Moreover, all vk are independent since ui are.This implies that if s ≥ 2 than there exists a unit v = p(ψ) such that B = p(A)

commutes with A, A and B are multiplicatively independent, and B has the sameneutral eigenspace as A. Namely, since v is in the kernel of f we have : σi(f(v)) = 1for all i, i.e. 1 = σi(p(ψ))σi(p(ψ

−1))p(σi(ψ))p(σi(ψ−1)) = p(ψi)p(ψ

−1i ).

In particular, if ψi is a root of r of absolute value 1 than ψ−1i = ψi therefore

1 = p(ψi)p(ψi) = p(ψi)p(ψi) = |p(ψi)|. Thus B has neutral eigenvalues in thesame direction as A and the action generated by A and B is genuinely partiallyhyperbolic.

7.5. Existence of reducible examples in any odd dimension greater than9. Reducible examples of Z2 actions on torus TN viewed as Tn1×Tn2 (N = n1+n2)are obtained from generator gi of Z2 acting by Ai,j on Tnj , where i, j ∈ 1, 2 andAi,j are matrices in GL(nj ,Z). If such an action is genuinely partially hyperbolicthan one of the reduced actions (on tori of dimension n1 or n2) would have tobe genuinely partially hyperbolic. This, along with the preceding discussion ondimensions less than 5 implies that there are no reducible examples on the torus ofdimension N = 7.

In any odd dimension N ≥ 9, as mentioned before, there are no irreducibleexamples. However reducible examples exist as products of purely hyperbolic ac-tions (which exist already in dimension 3) and the irreducible examples producedpreviously in this section. For example on the nine-dimensional torus, in the con-struction above take n1 = 3, n2 = 6, let A1,2 and A2,2 be commuting matrices fromthe Section 7.3 and for A1,1 and A2,1 choose any two 3 × 3 commuting hyperbolictotally real integer matrices (for various examples see [14]).


8. Other partially hyperbolic actions

8.1. Extensions of the KAM/harmonic analysis approach. The method pre-sented in this paper provides a blueprint for proving differentiable rigidity, both foractions on infranilmanifolds and for actions of type (2) from Section 1.1. In bothof those cases C∞ cocycle rigidity holds and hence one may expect to solve thelinearized equation for the conjugacy problem (at least approximately).

For a success of the iteration procedure tame estimates for the approximatesolution of the linearized equation are needed as well as tame estimates for theerror of approximation. For the first kind of estimates one needs to estimate normsof the solutions of certain cohomology equations through the norms of the right-hand parts with a fixed loss of derivatives, say Cr norms through Cr+σ norms,where σ is a fixed number.

Tame estimates for the torus case follow form the analysis of decay of Fouriercoefficients and of dynamics that individual elements of the action induce on thegroup of characters. Even more importantly, for the second kind of estimates,i.e. for the error of approximation, a fixed loss of derivatives in the torus case isobtained via an explicit procedure using Fourier series and specific properties of thedual action of the whole acting group on characters.

The work on the nilpotent case is in progress. In order to carry out our approachin the nilpotent case Fourier analysis is substituted by arguments using represen-tations of nilpotent groups. The new elements include solving the first cohomologyequation via the non-abelian harmonic analysis, detailed analysis of the dual action,and induction on the nilpotent length.

The prospects of applying this method to actions of type (2) are less certain.In [20] C∞ cocycle rigidity for actions of type (2) is established using Hormanderelliptic theory. This only uses the fact that the directions along which solutions aredifferentiable generate the whole Lie algebra. This unavoidably leads to the loss ofat least half of the derivatives and makes resulting estimates useless in the KAMsetting. There is a hope however to improve these estimates taking into accountspecific form of the solutions.

An even more important difficulty in pursing this approach in the semisimplecases such as (2A) and (2B) appears at the second step. Since the approxima-tion is made at the dual level finding an appropriate projection requires detailedinformation about irreducible unitary representations of the group G.

8.2. The new geometric approach. Now we describe another work in progressfor certain actions of type (2), including (2A) and (2B). We use another approachwhich radically bypasses the difficulties of the KAM/harmonic analysis method.First notice that due to the rigidity of the neutral foliation in the semisimple case,which is established by the a priori regularity method, it is sufficient to consideronly perturbations along this foliation. For these actions a conjugacy betweensuch a perturbation and a standard perturbation can be found as a solution of thelinear cocycle equation over the perturbed action. It is known that obstructionsto solvability for those equations are periodic cycle functionals [17]. The programthen is to analyze the structure of these functionals for the unperturbed systemfirst, and to show that a sufficient collection of those functionals can be extractedfrom certain basic commutation relations in the Lie algebra. The question is purelycombinatorial. In order to guarantee vanishing of the obstructions one needs to


show that it is sufficient to use relations involving one-parameter subgroups insidethe stable subgroups of various elements. For SL(n,R), n ≥ 3 this follows forexample from [7]. The next step is to show that the same procedure can be carriedout for the perturbed action. This is a highly refined quantitative version of thearguments used to prove stable accessibility in the theory of partially hyperbolicsystems.

Once vanishing of the periodic cycle functionals is established a Holder solutionfor the coboundary equation can be found. Then the usual regularity along stabledirections and Hormander theorems guarantee that the solution and hence theconjugacy is C∞.

A model case where this method is known to work is that of totally non-symplectic(TNS) actions by hyperbolic toral automorphisms. This condition means that notwo Lyapunov exponents are negatively proportional. In this case cocycle rigidityfor a perturbed action can be established directly by local considerations plus asimple global homological argument without any recourse to Fourier analysis (see[18]). Of course, this is not needed for the proof of differentiable rigidity since suchan action is hyperbolic and the a priori regularity method is sufficient.

In [4] this method is carried out to establish cocycle rigidity for Weyl chamberflows on SL(n,R), SL(n,C)n ≥ 3, and, more interestingly, to the restrictions ofWeyl chamber flows on SL(n,R), SL(n,C)n ≥ 4 on any linear subspace of Rn−1

which contains a two-dimensional plane in general position with respect to theLyapunov functionals.

Notice that for partially hyperbolic actions this approach requires sufficient non-commutativity for various stable foliations for their brackets to generate the neutraldirections and hence it cannot be applied in principle to partially hyperbolic actionson the torus. Let us repeat that thus we expect the the KAM and the new geometricapproach to be complementary in solving the local differentiable rigidity problemfor partially hyperbolic actions.

References

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[2] C. Camacho, A.L. Neto, Geometric theory of foliations, Bikhauser 1985[3] D. Damjanovic and A. Katok, Local rigidity of actions of higher rank abelian groups and

KAM method ERA–AMS, 10 (2004), 142–154.[4] D. Damjanovic and A. Katok, Periodic cycle functionals and cocycle rigidity for Weyl

chamber flows, Discrete and continuous dynamical systems, to appear.[5] M. Einseidler, A. Katok and E. Lindenstrauss Invariant measures and the set of

exceptions in Littlewood’s conjecture, Annals of Mathematics, to appear.[6] D. Fisher and G.A.Margulis, Local rigidity for affine actions of higher rank Lie groups

and their lattices, preprint, 2004.[7] S. M. Green, Generators and Relations for the Special Linear Group over a Division Ring,

Proc. AMS 62 (1977), 227-232.[8] M. Guysinsky and A. Katok Normal forms and invariant geometric structures for dynam-

ical systems with invariant contracting foliatios. Math.Res. Lett. 5 (1998), no. 1-2, 149–163.[9] M. Hirsch, C. Pugh and M. Shub, Invariant manifolds. Lecture notes in mathematics,

583, Springer Verlag, Berlin, 1977.[10] S. Hurder Affine Anosov actions Michigan Math. J. 40 (1993), no. 3, 561–575.

[11] A. Katok and B. Hasselblatt Introduction to the modern theory of dynamical systems.Cambridge University Press, 1995.

[12] A. Katok and S. Katok Higher cohomology for Abelian groups of toral endomorphisms,Erg. Theory and Dynam. Systems, 15 (1995), 569-592,


[13] A. Katok and S. Katok Higher cohomology for Abelian groups of toral automorphismsII: the partially hyperbolic case, and corrigendum, to appear in Erg. Theory and Dynam.Systems,

[14] A. Katok, S. Katok and K. Schmidt Rigidity of measurable structure for Zd actions by

automorphisms of a torus. Comm. Math. Helv., 77 (2002), 718-745.[15] A. Katok and J. Lewis, Local Rigidity of certain groups of toral automorphisms, Israel J.

Math., 75 (1991), 203-241.[16] A. Katok and J. Lewis Global rigidity results for lattice actions on tori and new examples

of volume-preserving action Israel J. Math. 93 (1996), 253-280.[17] A. Katok and A. Kononenko, Cocycle stability for partially hyperbolic systems, Math.

Res. Letters, 3, (1996), 191-210.[18] A. Katok, V.Nitica and A.Torok, Non-Abelian cohomology of abelian Anosov actions,

Erg. Theory and Dynam. Systems, 20, (2000), 259–288.[19] A. Katok and R. Spatzier First cohomology of Anosov actions of higher rank abelian

groups and applications to rigidity Publ. Math. I.H.E.S., 79 (1994), 131-156[20] A. Katok and R. Spatzier, Subelliptic estimates of polynomial differential operators and

applications to rigidity of abelian actions. Math. Res. Lett. 1. (1994), no. 2, 193–202.[21] A. Katok and R. Spatzier Differential rigidity of Anosov actions of higher rank abelian

groups and algebraic lattice actions.Proc. Steklov Inst. Math. 216 (1997), 287-314.

[22] Y. Katznelson Ergodic automorphisms on Tn are Bernoulli shifts. Israel J. Math. 10 (1971),

186-195.[23] V. F. Lazutkin KAM theory and semiclassical approximations to eigenfunctions Springer-

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of algebraic actions on a torus. Dynamical systems. 7. J. Math. Sci. (New York), 95 (1999),no. 5, 2576–2582.

[30] F. Rodrigues Hertz Stable ergodicity of certain linear automorphisms of the torus Annalsof Mathematics(2004), to appear.

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[32] E. Zehnder Generalized implicit function theorems with applications to some small divisorproblems. I. Comm. Pure Appl. Math. 28, (1975), 91–140.

Erwin Schroedinger Institute, Boltzmanngasse 9, A-1090 Vienna, Austria

E-mail address: [email protected]

URL: http://www.math.psu.edu/damjanov

Department of Mathematics, The Pennsylvania State University, University Park,

PA 16802, USA

E-mail address: katok [email protected]

URL: http://www.math.psu.edu/katok a

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Introduction - Pennsylvania State UniversityLet Gbe a simple split connected Lie group of rank n>1 and let Γ ⊂ Gbe a lattice. The Weyl chamber ﬂow on X= G/Γ is the left action