· LOCAL RIGIDITY OF HOMOGENEOUS PARABOLIC ACTIONS: I. A MODEL CASE DANIJELA DAMJANOVIC1 AND...

LOCAL RIGIDITY OF HOMOGENEOUS PARABOLIC ACTIONS: I.

A MODEL CASE

DANIJELA DAMJANOVIC1 AND ANATOLE KATOK 2

ABSTRACT. We show a weak form of local differentiable rigidity for therank 2 abelian action of upper unipotents on SL(2, R) × SL(2, R)/Γ .Namely for a 2-parameter family of sufficiently small perturbations ofthe action, satisfying certain transversality conditions, there exists a pa-rameter for which the perturbation is smoothly conjugate to the actionup to an automorphism of the acting group. This weak form of rigidityfor the parabolic action in question is optimal since the action lives in afamily of dynamically different actions. The method of proof is based ona KAM-type iteration and we discuss in the paper several other potentialapplications of our approach.

1. INTRODUCTION

1.1. KAM/Harmonic analysis method in differentiable rigidity. In thispaper we develop an approach for proving local differentiable rigidity of

higher rank abelian groups, i.e. Zk × Rl, k + l ≥ 2, based on KAM-typeiteration scheme, that was first introduced in [5, 6]. In those papers local

differentiable rigidity was proved for Zk, k ≥ 2 actions by partially hyper-bolic automorphisms of a torus. Here we deal with certain non-hyperbolicactions of R2 and R3, namely actions of certain unipotent subgroups ofsemisimple Lie groups on homogeneous spaces by translations. From thepoint of view of general classification of dynamical systems those actionsare parabolic [10]. Before proceeding to specifics we will describe in generalterms the version of the method adapted to the use for parabolic actions.

First, an important feature of the parabolic case, that sets it apart fromall hyperbolic and many partially hyperbolic actions, comes from the factthat among homogeneous actions on semi-simple groups unipotent ones arenot stable. Unipotent abelian subalgebras of a given dimension k of asemisimple Lie algebra g have positive codimension d in the variety of k-dimensional abelian subalgebras of g. Thus among the non-homogeneous

perturbations of a unipotent Rk action α those conjugate to a (maybe dif-ferent) unipotent action are expected to have positive codimension, greateror equal than d. A natural way to find such perturbations is to consider

d-parametric families α(λ) of perturbed Rk actions and look for an action

1 Based on research supported by NSF grant DMS-0758555 .2 Based on research supported by NSF grants DMS-0505539 and DMS-0803880 .

1

2 DANIJELA DAMJANOVIC AND ANATOLE KATOK

conjugate to α or its homogeneous perturbation inside such a family. Thus,unlike the absolute rigidity established in [5, 6] (as well as earlier for higherrank hyperbolic actions in [11]), we will look for conditional rigidity. Letas point out that in the most traditional KAM results dealing with pertur-bations of translations or linear flows on the torus (elliptic systems in theterminology of [10]) one also has to consider parametric families with theexception of the low-dimensional cases (dimension one for maps and twofor flows) when rotation number serves as a modulus of conjugacy.

What is described below is a somewhat idealized scheme that highlightsessential features of the method. While dealing with concrete problems,including those considered in this paper, certain steps may be combinedand order of other steps may be reversed.

Step 1. Conjugacy equation is formally linearized at the target. Solutionof the linearized equation is attempted by finding an inverse on a propersubspace of data.

Step 2. Obstructions for solving the linearized conjugacy equation for a par-ticular element of the action are found.

There are infinitely many obstructions for solving the linearized (coho-mological) equations for an individual generator of the action. This is acrucial difference with standard KAM for translations where obstruction isunique. In some cases (including those considered in this paper) those ob-structions are invariant distributions for the action element. This has to dowith parabolicity of our examples. If some hyperbolicity is present, as in[5, 6], the linearized equations are twisted in some way and obstructionshave a somewhat different form.

While obstructions can be described by different means for our purposesdual description using harmonic analysis is preferred.

Step 3. All but finitely many obstructions vanish due to the commutationrelations. This is sometimes called the Higher-rank trick;At this step harmonic analysis is used crucially:

Fourier analysis for actions by torus automorphisms.

Unitary group representations for homogeneous actions.

The “philosophy” of the Higher rank trick is universal, its execution ishighly case-sensitive.

Step 4. The remaining finitely many parameters are absorbed by allowing(i) automorphism of the group, or(ii) a standard perturbation, or(iii) adjusting parameters.

Accordingly rigidity comes in different flavors.

RIGIDITY OF PARABOLIC ACTIONS 3

Step 5. a) Linearized equation is solved. This involves glueing of solutionsconstructed in certain invariant subspaces of functional spaces such as:

Cyclic spaces of characters for the torus.

Irreducible representation spaces for homogeneous actions.

b) Tame estimates are obtained for the solution. This means finite loss ofregularity in the chosen collection of norms in the Frechet spaces, such asCr or Sobolev norms.

Step 6. The perturbation can be split into two terms due to the commutationrelations: one for which the linearized equations are satisfied, and the other“quadratically small” with tame estimates. This step also uses harmonicanalysis.

Step 7. Conjugacy provided by the solution of the linearized equation trans-forms the (modified in the conditional case) perturbed action into an actionquadratically close to the target with a fixed loss of regularity in the esti-mates.

Step 8. The process is iterated and the product of conjugacies convergesto a conjugacy between the (modified) original and perturbed action (withadjusted parameters if necessary).

The last two steps are completely independent of the specific ways pre-vious steps are performed and depend only on the conclusions reached atthose steps.

Since we deal with the higher rank actions the basic iteration scheme issomewhat similar to that applied by Moser [14] to commuting rotations ofthe circle (which is the basis of the scheme used later in [6]).

1.2. Conjugacy problem and linearization. Now we describe Step 1 in

complete generality. Let G = G1 × G2 × · · · × Gm, where Gj (j = 1, . . . , m)are simple Lie groups. Let X = G/Γ where Γ is an irreducible cocompactlattice in G. Let g denote the Lie algebra of G and gj Lie algebras of Gj.Let U1, . . . , Ud be some commuting elements of g. We consider an action

α : Rd × X → X by left translations on X whose generating vector fields

are U1, . . . , Ud, namely Uk = ∂α(t1,t2,x)∂tk

|(0,0).

A smooth perturbation α of the action α is generated by commuting vec-tor fields Uk = Uk + Wk (k = 1, . . . , d) on X which are small perturbationsof Uk’s.

Given a vector field Y and a diffeomorphism f we define h∗Y(x) :=(Dh)h−1(x)Y ◦ h−1(x). Define operators L and M in the following way:

(1.1) Vect∞(X)L−→ Vect∞(X)d M−→ Vect∞(X)d×d


(1.2) HL−→ ((h1)∗U1, . . . , (h1)∗Ud) , (Y1, . . . , Yd)

M−→ ([Yi, Yj])d×d

where h1 = exp H and Vect∞(X) denotes the space of C∞ vector fields onX.

Obviously M ◦ L = 0. Denote by L → M the non-linear sequence ofoperators defined by (1.1) and(1.2). The ”conjugacy problem” or the ”per-turbation problem” is the question of whether the sequence L → M is exacton some small open sets.

Linearizing the sequence L → M at H0 = 0 and at (U1, . . . , Ud) ∈Vect∞(X)d we obtain the linearized sequence L → M as follows:

(1.3) Vect∞(X)L−→ Vect∞(X)d M−→ Vect∞(X)d×d

(1.4) HL−→ (LU1

H, . . . ,LUdH) , (W1, . . . , Wd)

M−→ ([LUjWi −LUi

Wj])d×d

Again, obviously M ◦ L = 0.The following Proposition is immediate from the fact that M is simply

a linearization of the commutation condition and that we are interested inperturbations in the space of commutative actions. It will be used later.

PROPOSITION 1. If U = U + W is a d-tuple of commuting vector fields, then forr ≥ 0:

‖M(W)‖r ≤ C‖W‖0‖W‖r+1.

This shows that the image of M is quadratically small with respect toW, so having a tame splitting (see Section 2 for definition of ”tame”) forthe L → M sequence is the same as having an approximate solution to thelinearized version of the conjugacy problem.

Finally, one may try to reduce the sequence L → M to a simpler se-quence, for functions rather than vector fields:

(1.5) C∞(X)L−→ C∞(X)d M−→ C∞(X)d×d

where by L → M we denote the following sequence of operators:

(1.6) fL−→ (LU1

f , . . . ,LUdf ) , (g1, . . . , gd)

M−→ ([LUjgi −LUi

gj])d×d

Now the general strategy of approaching the local perturbation problem

for a general Rk action can be roughly summarized as:

(1.7) L → M splits =⇒ L → M splits =⇒ L → M is exact.

Note. All the splittings above as well as the exactness should be tame, seeSection 2 for exact definitions.

The paper is dedicated to proving the first statement that L → M splitsand the two implications in (1.7) for certain actions under some additional


conditions: certain small modifications and smaller splitting spaces. Thesesmall modifications will be simply coordinate changes in the acting groupand considering smaller spaces is necessary because actions we deal withhave non-trivial first cohomology, see Step 4 of the general scheme. Thegenerality in which these statements are proved increases from left to right.

• The splitting of L → M is proved for certain type of examples (theMain example for the next section) via the splitting construction inSection 3.

• The first implication in (1.7) is discussed in the Section 4 and reliesvery much on the fact that the actions we consider here are upperunipotent, but applies to all the examples we describe in this paper.

• The second implication in (1.7) is proved through the KAM-typeiteration scheme which is very general, in Section 6 and Section 7.

1.3. Description of some unipotent actions. The choice of examples be-low is not accidental. They represent exactly those unipotent homogeneous

actions of Rk, k ≥ 2 for which Step 2 of the general scheme has been per-formed based on representation theory for SL(2, R) [8] and SL(2, C) [12].Steps 3, 4 and 5a work for all three cases. Steps 5b and 6 for the main ex-ample below are carried out in Sections 3 and 4. Only the first part (tameestimates and splitting for functions) is specific for that example. Reductionfrom vector fields to functions (Section 4) is applicable to all three cases andis valid in even more general setting. The last two steps are carried out inSections 6 and 7 and, as we already pointed out, are based only on theconclusions of the previous steps and are hence of general character.

We will discuss remaining steps (5b and 6) for examples 1.3.2 and 1.3.3as well as other applications of our method, both in progress and potential,in Section 9.

1.3.1. Main example: SL(2, R) × SL(2, R). This is the model referred to inthe title of the paper. Let G = SL(2, R) × SL(2, R), Γ be an irreduciblecocompact lattice in G, X = G/Γ, gj = sl(2, R) (j = 1, 2), and g = LieG =sl(2, R) × sl(2, R).

The Lie algebra gj has a basis Uj+, U

j0 and U

j− for j = 1, 2, where:

U1+ =

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

and U2

+ =

0 0 0 00 0 0 00 0 0 10 0 0 0

U10 =

1/2 0 0 00 −1/2 0 00 0 0 00 0 0 0

, U2

0 =

0 0 0 00 0 0 00 0 1/2 00 0 0 −1/2


U1− =

0 0 0 01 0 0 00 0 0 00 0 0 0

, U2

− =

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

.

Our model parabolic action

αRR : R2 × X → X

is the action by left translations on X generated by commuting unipotentelements

U1 = U1+, U2 = U2

+.

More precisely, if (t1, t2) denote coordinates in R2 then

(1.8) αRR(t1, t2, x) =

1 t1 0 00 1 0 00 0 1 t2

0 0 0 1

· x

We denote the pair of vector fields (U1+, U2

+) generating the action αRR by

U and the pair (U1−, U2

−) by V.

1.3.2. Example on SL(2, R) × SL(2, C). Let G = SL(2, R) × SL(2, C), Γ bean irreducible cocompact lattice in G, X = G/Γ. g1 = sl(2, R), and g2 =sl(2, C). The Lie algebra g1 is generated by U1

+, U1− and U1

0 as in the example

above, while g2 is generated by U2+, (U2

+)′ = iU2+, U2

−, (U2−)′ = iU2

−, U20 , (U2

0) =iU2

0 , where:

U2+ =

0 0 0 00 0 0 00 0 0 10 0 0 0

U2

− =

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

U2

0 =

0 0 0 00 0 0 0

0 0 12 0

0 0 0 − 12

.

Our example is the action αRC of R3 on X, by left translations, generatedby commuting unipotent elements

U1 = U1+, U2 = U2

+, U3 = (U2+).′

We denote the triple of vector fields (U1+, U2

+, (U2+)′) generating the action

αRC by U, and the triple (U1−, U2

−, (U2−)′) by V.

1.3.3. Example on SL(2, C). Let G = SL(2, C), Γ be an irreducible cocom-pact lattice in G, X = G/Γ. In this case m = 1, g = g1 = sl(2, C), andg1 is generated by U1

+, (U1+)′ = iU1

+, U1−, (U1

−)′ = iU1−, U1

0 , (U10)

′ = i(U10)

′,where

U1+ =

(

0 10 0

)

, U1− =

(

0 01 0

)

, U10 =

(

1/2 00 −1/2

)

.


Our third example is the action αC of R2 on X by left translations, generatedby commuting unipotent elements

U1 = U1+, U2 = (U1

+)′.

Remark 1. If the first Betti number of Γ is not zero there are additionalobstructions in the cohomology problem and they correspond to certainspecific classes of perturbations of action αC.

1.4. Cocycle rigidity of parabolic (unipotent) actions. David Mieszkowski[12] studied real valued cocycles over the actions of unipotent subgroupsdescribed above. He proved vanishing of the obstructions and constructedsmooth solutions in all three cases, obtained tame estimates for both prod-uct cases (actions αRR in the main example 1.3.1 and αRC in Example 1.3.2)and for the former also gave a partial proof for a splitting of an almost co-cycle into a cocycle and a small residue with tame estimates.

The starting point in [12] for the main example and Example 1.3.2 is thedescription of obstructions and solution of cohomological equations forhorocycle flows (upper-triangular unipotent action on SL(2, R)/Γ) by L.Flaminio and G. Forni [8]. In those cases presence of one SL(2, R) factoris essential. Solutions are constructed for the unipotent element from thatfactor and estimates are carried out. Additional generator(s) serve an aux-iliary purpose to guarantee the vanishing of obstructions for the solutionof the cohomological equations for the first factor. In fact, specific choicesof the second factor in these two cases has to do with the fact that those arethe only cases where irreducible lattices exist in the product. Example 1.3.3on SL(2, C)/Γ is completely different as we will remark later in Section 9.1.

The space L2(SL(2, R)/Γ) is decomposed into the direct sum of irre-ducible representation spaces for SL(2, R); representations of principal,complementary and discrete series are treated separately; each carries ob-structions for solutions of the cohomological equation

LUh = g

for the nilpotent U =(

0 10 0

)

∈ sl(2, R).

Thus, not only general perturbations but already time changes of the horo-cycle flow are highly unstable (infinitely many moduli of smooth conju-gacy) in contrast with the Diophantine elliptic case.

For an R2 action generated by unipotents exp U1 and exp U2 the basiccocycle condition has the form

LU2f = LU1

g.

(and similarly for more generators).Integrals with respect to the Haar measure are obvious obstructions and

solution, if it exists, is unique up to an arbitrary constant.


The higher rank trick is applied in each irreducible representation space(except for the trivial one and a single other exception for G = SL(2, C))and allows to find h such that

LU1h = f , LU2

h = g

with tame estimates for an appropriate family of Sobolev norms.

For G = SL(2, R) × SL(2, R) or SL(2, R) × SL(2, C) algebraic descriptionfor the irreducible representations is used, similarly to Flaminio–Forni [8].

But for G = SL(2, C) the analytic description of the irreducible represen-tations is used.

Mieczkowski obtained tame estimates with respect to standard Sobolevnorms for both product cases.

But for the SL(2, C) case Mieczkowski only obtained tame estimates withrespect to incomplete Sobolev norms based on two embeddings of SL(2, R)into SL(2, C).

Spectral gap is essential for obtaining tame estimates in any of the cases.It plays the role of Diophantine conditions.

1.5. The main result. As we already mentioned, in this paper we carry outthe program outlined in Section 1.1 for the action αRR in the main exampleof Section 1.3.1.

Let U = (U1+, U2

+) be generators of the action αRR in the main exam-

ple. Recall that V = (U1−, U2

−) denotes the pair of ”opposite” unipotentelements. Given a family of actions generated by vector fields U(λ) =(U1(λ), U2(λ)) we define a map Φ(U) : R2 → R2 such that Φ(U) takeseach λ to the pair of real numbers (Ave1

−(U1(λ)), Ave2−(U2(λ))) which are

averages of U1(λ) and U2(λ) in the directions U1− and U2

− with respect tothe Haar measure.

We use the following norms for a family U(λ) = (U1(λ), U2(λ)) of pairsof vector fields on X:

• For a fixed λ, ‖U(λ)‖r denotes the maximum of the Cr norms ofU1(λ) and U2(λ).

• ‖U(λ)‖r,(m) is the maximum of the Cr norms of all the partial deriva-

tives of U in the λ variable of order less or equal to m.• For the ”averages” map Φ(U) : R2 → R2, the notation ‖Φ(U)‖(r)

stands for the usual Cr norm.

THEOREM 1 (Main Theorem). Let U = (U1+, U2

+) be as in the main example

1.3.1 and αRR be the R2 action generated by U (see (1.8)).There exist l > 0, ε > 0, K > 0, C > 0 and such that for every family α(λ) of

perturbations of αRR where λ ∈ B = B(0, ε) ⊂ R2, generated by a pair of vectorfields U(λ) = (U1(λ), U2(λ)) such that for λ ∈ B:

‖U‖0,(2) ≤ K, ‖Φ(U) − Id‖(1) ≤1

2Cε, and ‖U(λ) − U‖l ≤ ε


there exist λ ∈ B, an automorphism π of the acting group R2 and h ∈ Diff∞Xsuch that for all x ∈ X, a ∈ R2,

h(α(λ)(a, x)) = αRR(π(a), h(x)).

The work on examples 1.3.2 and 1.3.3 is in progress and appropriate localrigidity results will appear in a subsequent paper joint with Livio Flaminio.We refer to Section 9.1 for more detailed discussion of those examples.

A substantial part of the proof of the above result is presented in theform which can be used in even greater generality than the remaining twoexamples.

2. SOME ANALYTIC TOOLS

2.1. Graded Frechet spaces and tame operators.

Definition 1. By a graded Frechet space we mean a Frechet space X withthe collection of norms ‖ · ‖r (r ∈ N0) such that ‖x‖r ≤ ‖x‖r+k for everyr, k ≥ 0, for every x ∈ X.

Definition 2. A map L : X → Y between two graded Frechet spaces is r0-tame if for every x ∈ X and for every r ≥ 0 we have that ‖Lx‖r ≤ Cr‖x‖r+r0

Definition 3. Tame maps L : X → Y and M : Y → Z between gradedFrechet spaces are said to form a sequence of maps if ML = 0. We denotesuch a sequence of maps by L → M. If ImL = KerM we call the sequenceexact. We call the sequence L → M r0-tamely exact if there exists an r0-tamemap L′ from KerM to X such that LL′ = Id on KerM.

Definition 4. It is said that a sequence of tame maps XL−→ Y

M−→ Z has anr0-tame splitting on a subspace Y of Y if there exists r0-tame maps L′ : Y → Xand M′ : Z → Y such that LL′ + M′M = IdY.

2.2. Frechet spaces and gradings to be used in the proof of Theorem 1.The space X is compact, thus spaces of smooth functions and smooth vectorfields are graded Frechet spaces. We will use the following notation:

(1) C∞(X) denotes the graded Frechet space of C∞ functions with grad-ing given either by the usual Cr norms or by the Sobolev norms. Wewill always specify which grading we use, and the chosen grading

will be denoted by ‖ · ‖r. Similarly C∞(X)d and C∞(X)d×d denoted-tuples and d × d matrices, respectively, of C∞-functions on X.

(2) Vect∞(X) denotes the graded Frechet space of vector fields on X ofclass C∞ with grading given by the component-wise norms (Cr orSobolev). These we also denote by ‖ · ‖r.

(3) Vect∞(X)d and Vect∞(X)d×d denote d-tuples, and d × d matrices ofvector fields on X, respectively. The grading on this space is givenby norms which we again denote by ‖ · ‖r and which are again justmaxima over the corresponding coordinate wise norms.


2.3. Smoothing operators and some norm inequalities. The space X iscompact, thus spaces of smooth functions and smooth vector fields aregraded Frechet spaces. There exists a collection of smoothing operatorsSt : C∞(X) → C∞(X), t > 0, such that the following holds:

(2.1)‖StF‖s+s′ ≤ Cs,s′t

s′‖F‖s

‖(I − St)F‖s−s′ ≤ Cs,s′t−s′‖F‖s

For the construction of smoothing operators see [9]: Example 1.1.2. (2),Definition 1.3.2, Theorem 1.3.6, Corollary 1.4.2 .

Smoothing operators on C∞(X) clearly induce smoothing operators on

Vect∞(X)d, Vect∞(X)d×d via smoothing operators applied to coordinatemaps.

It is easy to see that averages of F with respect to the Haar measure onX, in various directions in the tangent space do not affect the propertiesof smoothing operators listed above, so without loss of generality we mayassume that St are such that averages of StF in directions of U+

k , U−k , U0

k arethe same as those of F.

The following inequality implies the statement of Proposition 1. It willalso be used later in Section 6 . Let F, G ∈ Vect∞(X). Then for r ≥ 0

(2.2) ‖[F, G]‖r ≤ Cr(‖F‖0‖G‖r+1 + ‖F‖r+1‖G‖0)

Also, existence of smoothing operators on a graded space induces theusual interpolation inequalities for the norms which make up the grading.This has been demonstrated in several papers, for example [16] and in amore general set-up in [9].

3. FIRST COHOMOLOGY AND TAME SPLITTING FOR FUNCTIONS

In this section we show that the exact sequence of maps

(3.1) C∞(X)L−→ C∞(X)2 ∩ KerA M−→ C∞(X)

whereL( f ) = (LU1

f ,LU2f )

M( f , g) = LU1g −LU2

f

and A( f , g) = (∫

X f dν,∫

X gdν), has an r0-tame splitting. The constant r0

depends on spectral gap of Casimir operators �1 and �2 on SL(2, R) ×SL(2, R)/Γ.

In [12] Mieczkowski proved tame estimates for solutions of the scalar co-homological equation, i.e. showed that the sequence M → L is r0-tamelyexact where r0 depends only on the lattice Γ. In the process he uses explicitobstructions to the solutions of cohomology problem for a single actiongenerator Uk, k = 1, 2. The obstructions come from work of Flaminio andForni [8]. It is important for our application to remark that the space ofcohomological obstructions for each Uk is found to be finite dimensional in


each irreducible representation (this follows immediately from [8]). Moreprecisely, depending on the kind of irreducible representation the space ofobstructions is either one or two dimensional.

Below we give a general construction of a tame splitting for the productexamples, in the situation where for one action generator there is a finitedimensional space of obstructions for the cohomology problem in each ir-reducible representation.

Irreducible unitary representations of SL(2, R)× SL(2, R) are tensor prod-ucts:

Hµ ⊗Hθ

of the irreducible unitary representations Hµ andHθ for SL(2, R). For i = 1 or 2, µ (resp. θ) is the value which the Casimiroperator �1 in the first component (resp. the Casimir operator �2 in thesecond component) of the product SL(2, R) × SL(2, R) takes on Hµ (respHθ).

The space L2(SL(2, R) × SL(2, R)/Γ) decomposes as the direct integralof non-trivial irreducible unitary representationsHµ,θ of SL(2, R)× SL(2, R)with respect to Stiltjes measure ds(µ, θ). Similarly, spaces Ws(H) decom-pose for every s ≥ 0 which implies that every f ∈ Ws(H) has a decompo-sition

f =∫

⊕

m(µ1, µ2) fµ1,µ2ds(µ1, µ2)

where fµ1,µ2 ∈ Ws(Hµ1,µ2) and m(µ1, µ2) is multiplicity with which Hµ1,µ2

occurs in H.

Remark 2. Only in this section the notation ‖ · ‖r is used for Sobolev norms.In the rest of the paper ‖ · ‖r stands for Cr norms.

Sobolev norm of f ∈ Ws(H) is then given by:

‖ f‖2s =

∫

m(µ, θ)‖ fµ,θ‖2s ds(µ, θ)

This implies that for f , g ∈ Ws(H) uniform Sobolev estimates:

‖ fµ,θ‖t ≤ Cs,t‖gµ,θ‖s for every (µ, θ)

imply global Sobolev estimate:

‖ f‖t ≤ Cs,t‖g‖s .

This reduces the proof of the existence of tame splitting in Corollary 1below to proving the existence of a tame splitting in each irreducible Hµ,θ

in such a way that the constants involved in tame estimates can be chosenso that they do not depend on (µ, θ).

When SL(2, R)× SL(2, R)/Γ is compact the above direct integral decom-positions into irreducible representations reduces to infinite discrete sums.

We expect that the Theorem below will be used for other actions, in par-ticular for αRC from Example 1.3.2, so we formulate it and prove it in somegenerality.


THEOREM 2. Let G1 and G2 be Lie groups and let Hµ and Hθ be irreducible uni-tary representations of G1 and G2, respectively. Let H = Hµ ⊗Hθ and assumeU1 ∈ g1 and U2 ∈ g2 commute. Assume that the space of U1 invariant distri-butions I s

1(Hµ) of Sobolev order s is finite dimensional, spanned by distributions

D1, . . . , Dm. Define Di( fµ ⊗ gθ) := Di( fµ)gθ for i = 1, . . . , m. Further, assumethat there exists s0 ≥ 0, σ ≥ 0 independent of the pair µ, θ, such that the followinghold for k = 1, 2 and any s sufficiently large:

(1) If f = U1h ∈ Ws(H) then Di( f ) = 0 for all i = 1, . . . , m.(2) If Di( f ) = 0 for all i = 1, . . . , m, then the equation U1h = f has a

solution h ∈ Wt(H), t < s − σ, such that ‖h‖t ≤ Ct,s‖ f‖s .(3) There exists s0 such that if Ukh = 0 with h ∈ Ws0(H), then h = 0.(4) If U2h = f ∈ Ws(H) and if h ∈ Ws0(H), then h ∈ Wt(H), t < s − σ

and ‖h‖t ≤ Ct,s‖ f‖s ,

where Ct,s are independent of the pair µ, θ.

Then if U2 f − U1g = φ for f , g, φ ∈ Ws(H), Ave( f ) = Ave(g) = 0, thereexists h ∈ Wt(H) and there exists a constant C such that

‖ f − U1h‖t ≤ Ct,s‖φ‖s

‖g − U2h‖t ≤ Ct,s‖φ‖s

‖h‖t ≤ Ct,s‖ f‖s

The constant Ct,s depends only on the constants which appear in the estimatesof the assumptions (2) and (4) and can be chosen independently of the pair (µ, θ).

Remark 3. Our original proof of Theorem 2 for the case of the action αRR

from the main example was based on a specific and explicit construction ofprojections in each irreducible representation which used detailed descrip-tions of the representations. This proof completed and corrected attemptedproof of Theorem 18 in [12] that in fact contains an error because it uses onlyone of the two obstructions that appear in any irreducible representation ofSL(2, R) of the principal series.

The shorter and more general proof presented below was inspired byour discussions with Livio Flaminio.

Proof: For each i = 1, . . . , m there exists γi ∈ Ws(Hµ) such that: Di(γi) =1 and ‖γi‖s ≤ 2. By applying a linear map on the m-tuple (γ1, . . . , γm) wecan obtain an m-tuple of maps (γ1, . . . , γm) satisfying Dj(γi) = δij. Thenorm of the linear map is bounded above by the maximum of the norms ofDi, i = 1, . . . , m.

Thus without loss of generality we may assume that we have γ1, . . . , γm ∈Ws(Hµ) such that for any i = 1, . . . , m, ‖γi‖s ≤ C (where C is a con-stant multiple of the maximum of the norms of Di’s on Ws(H)), such thatDi(γi) = 1, and Dj(γi) = 0 for i 6= j.

Let Vs1 be the intersection of kernels of Di’s in Ws(H).

The following is the central ingredient in the proof of the Theorem.


LEMMA 1. There exists a linear map R : Ws(H) → Wt(H) such that:

(1) R is trivial on Vs1 .

(2) ‖R f‖s ≤ C‖ f‖s

(3) Dj(R f ) = Dj( f ) for all j = 1, . . . , m.(4) R commutes with U2.

Proof: For f = fµ ⊗ gθ , let

R f =m

∑i=1

γi ⊗ Di( f ).

Then since Di( f ) is bounded, and each γi has the Ws(H) norm bounded bya constant, we get ‖R f‖s ≤ C‖ f‖s. It is obvious by the definition that R istrivial on Vs

1 .

Further from Dj(γi) = δij we have Dj(R f ) = ∑i Dj(γi)Di( f ) = Dj( f ).

To check commutativity we use the fact that U2 commutes with Di:

R(U2 f ) =m

∑i=1

γi ⊗ Di(U2 f ) =m

∑i=1

γi ⊗ U2Di( f ) = U2(R f ).

This concludes the proof of the claim. �

Now for f ∈ Ws(H), we may define P f := f − R f . Then from theproperties of R obtained in the claim we have that P : Ws(H) → Vs

1 and‖P f‖s ≤ C‖ f‖s, and P commutes with U2 as well.

Given U2 f − U1g = φ, by assumption (1) and the properties of R weget U2R f = Rφ. Then by assumptions (4) and (3) we have the estimate‖R f‖t ≤ C‖Rφ‖s and by the bound for R we get ‖R f‖t ≤ C‖φ‖s.

Now by assumption (2) we have that P f = U1h for some h ∈ Wt(H)with ‖h‖t ≤ C‖P f‖s ≤ C‖ f‖ f . On the other hand we have from U2 f −U1g = φ that U2(P f ) = U1g + Pφ since P is identity on Vs

1 . Substitut-ing P f = U1h we get U1(U2h − g) = Pφ. Since for sufficiently large s,U2h − g ∈ Ws0 , by assumptions (1), (2) and (3) we have that ‖U2h − g‖t ≤C‖Pφ‖s ≤ C‖φ‖s. �

The assumptions of Theorem 2 are checked using the results of [8] sim-ilarly to the way the same results were used by Mieczkowski in his proofof vanishing of the first cohomology over the action αRR modulo averages[12]. Below we summarize conclusions of [8] adapted to the product casein the form which is sufficient for the purpose of the current paper.

THEOREM 3. Let H = Hµ ×Hθ be an irreducible representation of SL(2, R)×SL(2, R). Let U1 = U1

+, U2 = U2+ and X be as in the main example. Then the

space of U1-invariant distributions I s1(Hµ) is either one or two dimensional and

(1)-(4) of the Theorem 2 are satisfied in such a way that constants Cs,t > 0, σ ands0 can be chosen independently of µ and θ.


Remark 4. All three constants obtained in the Theorem above depend onlyon Γ, that is on spectral gap of Casimir operators �1 and �2 in the first andsecond component.

The tame splitting of the L → M sequence is now an easy corollary ofthe previous two theorems:

COROLLARY 1. Let U1 = U1+, U2 = U2

+ and X be as in the main example.For h, f , g ∈ C∞(X) let L(h) = (LU1

h,LU2h) and let M( f , g) = LU2

f −LU1

g. Then there exist r0 > 0 and there exist r0-tame operators (with respect toCr norms) L′ and M′ such that LL′ + M′M = I on C∞(X)2 ∩ KerA whereA( f , g) = (

∫

X f dν,∫

X gdν) and ν is the Haar measure on X.

Proof: For H = Hµ ⊗Hθ and f , g ∈ Ws(H) Theorem 2 and Theorem 3

imply that there exists h ∈ Wt(H) such that ‖h‖t ≤ Ct,s‖ f‖s , ‖ f − U1h‖t ≤Ct,s‖φ‖s and ‖g − U2h‖t ≤ Ct,s‖φ‖s. Thus we define L′( f , g) = h andM′(M( f , g)) = ( f − U1h, g − U2h). Since by Theorem 3 the constants inthe estimates do not depend on the representation Hµ ×Hθ, it is clear thatafter defining L′ and M′ on all of

⊕Hµ ⊗Hθ by gluing L′ and M′ in eachirreducible representation, the tame estimates in Sobolev norms for L′ andM′ remain true. From these the tame estimates in the Cr norms follow in astandard way with the loss of r0 derivatives where r0 depends only on thedimension of X and the lattice Γ. �

4. TAME SPLITTING FOR VECTOR FIELDS

As mentioned in the introduction, due to Proposition 1 the existence oftame splitting of the L → M-sequence in (1.4) implies the existence of an ap-proximate right inverse for the linearization L of our initial conjugacy prob-lem and this is the necessary ingredient in any attempt to solve a conjugacyproblem by linearization.

In this section we show the existence of a tame splitting LL′ + M′M = I

on Vect∞(X)d ∩KerA for the sequence L → M for the action αRR from main

example, where A is a specifically defined functional on Vect∞(X)d. To dothis we choose a basis in the Lie algebra g and by looking at the equation(1.4) in this basis we reduce the splitting problem to several simpler prob-lems. Those simpler problems inductively reduce to existence of the tamesplitting for functions i.e. for the sequence L → M to which the results ofthe previous section apply.

We give a unified proof for all three examples described in Section 1.3that splitting of functions implies splitting for vector fields, in anticipationof forthcoming tame splitting results for functions for actions αRC and αC

from Examples 1.3.2 and 1.3.3. Furthermore, the scheme of this proof canbe adapted to even more general situations of unipotent homogeneous ac-tions.


Reader should be warned though that uniformity comes at a price ofcertain clarity since in fact in the formulas written in a generic way m equalseither one or two, and d equals two or three. Rewriting the calculations foreach specific case will make them more compact and easier to visualize.

Let Gj (j = 1, 2) be either SL(2, R) or SL(2, C) and consider one of the

actions described in Section 1.3. By Uj0, U

j−, U

j+, (U

j−)′, (U

j+)′ we denote the

vector fields that are equal correspondingly to U0, U−, U+, U′−, U′

+ in the

j − th component of the product G1 × G2, j = 1, 2, and zero elsewhere.We will use generic notation α for the actions αRR, αRC and αC. Genera-

tors of α are as follows:

(1) Main example 1.3.1: G = SL(2, R)× SL(2, R); U1 = U1+, U2 = U2

+.

(2) Example 1.3.2: G = SL(2, R)× SL(2, C); U1 = U1+, U2 = U2

+, U3 =(U2

+)′.(3) Example 1.3.3: SL(2, C); U1 = U1

+, U2 = (U1+)′.

For H ∈ Vect∞(X) let H = ∑mj=1 h

j0U

j0 + h

j+U

j+ + h

j−U

j− and ∗ ∈ {0, +,−}.

Then:in case (1) m = 2 and h

j∗ are real-valued functions;

in case (2) m = 2 and h1∗ are real-valued and h2

∗ are complex-valued;in case (3) m = 1 and and h1

∗ are complex-valued.Now we use both notations for the Lie derivative: LXY = [X, Y] simul-

taneously. We have(4.1)

LUkH =

m

∑j=1

LUkh

j0U

j0 + h

j0[Uk, U

j0]+LUk

hj+U

j+ + h

j+[Uk, U

j+]+LUk

hj−U

j− + h

j−[Uk, U

j−]

From the definition of Uk it follows that [Uk, Uj+] = 0 for any j, k, and

[Uk, Uj0] =

−iUj+, Uk = (U

j+)′

−Uj+, k = j, Uk = U

j+

0, otherwise,

[Uk, Uj−] =

2iUj0, Uk = (U

j+)′

2Uj0, Uk = U

j+

0, otherwise.

Let

(4.2) ψkj =

i, Uk = (Uj+)′

1, Uk = Uj+

0, otherwise.

Then from (4.1) we have:

(4.3) LUkH =

m

∑j=1

(LUkh

j0 + 2ψkjh

j−)U

j0 + (LUk

hj+ − ψkjh

j0)U

j+ +LUk

hj−U

j−.


This completely describes the linear operator L : H 7→ (LU1H, . . . ,LUd

H)

with respect to the basis given by Uj+, U

j0, U

j−, j = 1, m. It is clear from this

description that j-th block of Lk is given by:

(4.4) Lkj =

LUk−ψkj 0

0 LUk2ψkj

0 0 LUk

For a C∞ function h define Cj(h) by:

(4.5) Cj(h) = (ψ1jh, ψ2jh, . . . , ψdjh),

and let L be the following map: h ∈ C∞(X) 7→ (LU1h, . . . ,LUd

h) ∈ C∞(X)d.Then the j-th block of L is given by:

(4.6) Lj =

L −Cj 00 L 2Cj

0 0 L

.

By denoting

(4.7) Cj =

I −Cj 00 I 2Cj

0 0 I

,

and using the notation L for the operator

L 0 00 L 00 0 L

, it follows that

(4.8) Lj = L+ Cj.

For operators Mlk, that are coordinates of the operator M defined in (1.3)

and (1.4), it follows from (4.1) that for Wl = ∑mj=1 wl

j0U

j0 + wl

j+U

j+ + wl

j−U

j−

and k 6= l expression LUkWl −LUl

Wk, has the form:

(4.9)

LUkWl −LUl

Wk =m

∑j=1

(LUkwl

j+ − ψkjwl

j0)U

j+ + (LUk

wlj0 + 2ψkjwl

j−)U

j0 + LUk

wlj−U

j−

−m

∑j=1

(LUlwk

j+ − ψljwk

j0)U

j+ + (LUl

wkj0 + 2ψljwk

j−)U

j0 + LUl

wkj−U

j−

For a d-tuple of C∞ functions (h1, . . . , hd) define Dj(h1, . . . , hd) by:

(4.10) Dj(h1, . . . , hd) = (ψkjhl − ψljhk)k,l=1,...,d

Let us denote by M the map which takes a d-tuple of C∞ functions (h1, . . . , hd)to (LUl

hk − LUkhl)k,l=1,...,d ∈ C∞(X)d×d. Then it is easy to see just directly

from the definitions of M,Dj, Cj and L that

(4.11) MCj + DjL = 0.


From definition of ψkj’s it follows that

(4.12) DjCj = 0

Also it follows from (4.9) that the operator M acts in the following way on

(Wj1, . . . , W

jd)

(4.13) Mj =

M −Dj 00 M 2Dj

0 0 M

where Wjl = wl

j0U

j0 + wl

j+U

j+ + wl

j−U

j− for l = 1, . . . , d.

Operator M is defined to be the operator which acts by Mj in each direc-

tion gj.Let

(4.14) Dj =

I −Dj 00 I 2Dj

0 0 I

,

and M =

M 0 00 M 00 0 M

. Then

(4.15) Mj = M + Dj

Let C ′j : (h1, . . . , hd) 7→ ∑

dk=1 ψkjhk and D′

j : (aij)i,j=1,...,d 7→ ∑dk=1 ψkj(ak1, . . . , akd

).

Also let

(4.16) D′j =

0 0 0−D′

j 0 0

0 12D′

j 0

, C′j =

0 0 0−C ′

j 0 0

0 12C ′

j 0

and define C′ and D′ to act by C′j and D′

j on each gj.

Since M and L are 0-tame it is immediate that both M and L are 0-tame.Now we are ready to state and prove the main splitting result.

PROPOSITION 2. Assume that for an action α generated by Uk, k = 1, . . . , d asdescribed in examples in Section 1.3, i.e. one of the actions αRR, αRC, αC , thereexists an r-tame splitting for the sequence of s-tame linear operators L → M i.e.there exist r-tame linear operators L′ and M′ such that

LL′ + M′M = I

on C∞(X)d ∩ KerA, where A is the average map.Then the exact sequence of linear operators M → L is 3s-tame and it admits

3r-splitting. i.e. there exist 3r-tame linear operators L′ and M′ such that

LL′ + M′M = I

on Vect∞(X)d ∩ KerA, where

(4.17) A = A − CC′A − D′DA


and A is the coordinate-wise average map.

Proof: First notice that it follows from (4.8) and (4.15) that the exact se-quence M → L is 3s-tame. By assumption there exist operators L′ and M′

such that

(4.18) LL′ + M′M = Id

on C∞(X)d ∩ KerA.Recall that for a C∞ function h, Cj(h) is defined by:

(4.19) Cj(h) = (ψ1jh, ψ2jh, . . . , ψdjh)

and for a d-tuple of C∞ functions (h1, . . . , hd), Dj(h1, . . . , hd) is defined by:

(4.20) Dj(h1, . . . , hd) = (ψkjhl − ψljhk)k,l=1,...,d

Below we use the notation L,M, L′ and M′ to denote operators whichact on gj-valued maps by L,M,L′ and M′, respectively, on each compo-

nent in Uj+, U

j0 and U

j− direction.

Recall that from (4.8), Lj = L+ Cj and from (4.15) Mj = M + Dj.Now define operators L′

j by:

(4.21) L′j =

L′ L′CjL′ −2L′CjL′CjL′

0 L′ −2L′CjL′

0 0 L′

Also, define operators M′j for j = 2, . . . , m

(4.22) M′j =

M′ M′DjM′ −2M′DjM′DjM′

0 M′ −2M′DjM′

0 0 M′

Operator M′j makes sense on the image of M, which consists of gj-valued

d × d skew-symmetric matrices. The following is obtained immediatelyfrom definitions above:

(4.23) L′j = L′(I − CjL

′j) and M′

j = (I − M′jDj)M′

Since L′ and M′ are assumed to be r-tame, and Cj and Dj are obviously0-tame, it follows immediately that L′ and M′ are 3r-tame.

Using the assumption (4.18), and (4.11), it follows that

(4.24) LjL′j + M′

jMj = I

on the kernel of the map −Aj + CjC′jAj + D′

jDjAj, where Aj denotes the

map which assigns to a given vector field the triple of averages in the three

directions Uj+, U

j0 and U

j−. Maps C′

j and D′j are defined in (4.16). Thus by

denoting

(4.25) Aj = Aj − CjC′jAj − D′

jDjAj


we have the required splitting on KerA:

(4.26) LjL′j + M′

jMj = I

where A is defined to be equal to Aj in each direction gj, j = 1, . . . , m, just

as L′ and M′ are defined to be L′j and M′

j in each gj. �

Corollary 1 asserts that assumptions of Proposition 2 hold for the actionαRR of the main example. This gives the principal ingredient for carryingout the iteration scheme.

COROLLARY 2. The sequence L → M in (1.4) of 1-tame operators for the actionαRR of the main example has an r0-tame splitting on the space Vect∞(X)d ∩KerA,where A = A − CC′A − D′DA and r0 depends only on the lattice Γ.

Remark 5. While it is obvious from the definition that KerA has finite codi-mension, on the surface it looks that this codimension is greater than d.In the next section we will show how to reduce the required number ofparameters.

5. DETAILED STUDY OF THE AVERAGE MAP A: CONSTRUCTION OF

COORDINATE CHANGES

Now we will carry out Step 4(i) and (ii) of the general scheme; notice thatpart (ii) of this step in our setting involves conjugating the unipotent actioninside the group G since unipotent homogeneous actions in our cases areall conjugate to the standard ones. Thus standard perturbations produceactions isomorphic to the original ones.

Part (iii) corresponds to taking parametric families in the Main Theoremwith the number of parameters exactly equal to the codimension of unipo-

tent homogeneous actions among all homogeneous actions of Rd.Recall that

C ′j : (h1, . . . , hd) 7→

d

∑k=1

ψkjhk

and

D′j : (aij)i,j=1,...,d 7→

d

∑k=1

ψkj(ak1, . . . , akd).

It is immediate from definitions that CjC ′j + D′

jDj = I. This implies

CjC′j + D′

jDj =

CjC ′j 0 0

0 Id 00 0 D′

jDj

Thus KerA in Vect∞(X)d coincides with

{W = (W1, . . . , Wd)|(A− CC ′A)(W+) = 0, (A−D′DA)(W−) = 0} =

{W = (W1, . . . , Wd)|D′DA(W+) = 0, CC ′A(W−) = 0}


Here W+ denotes the d-tuple of functions which represent components of

the d-tuple of vector fields W in the direction Uj+ and W− denotes the d-

tuple of components of W in direction Uj−.

In fact, because of the definition of ψkl (4.2) it is not difficult to see thatD′DA(W+) involves the averages of W in all the directions Ul for whichψkl = 0 and that CC ′A(W−) involves all the averages of W in the directionof Vl whenever ψkl 6= 0. In particular, in the notation of Theorem 1 we have:

CC ′A(W−) = Φ(W)(λ) · V

On the other hand, the following proposition shows that D′DA(W+) canbe assumed to be quadratically small by making a good choice of generat-ing vector fields for the action.

PROPOSITION 3. Let U = (U1, . . . , Ud) ∈ Vect∞(X)d, U = U + W, generatean abelian action α. Then there exists a linear transformation T acting on U =(U1, . . . , Ud) such that U := TU = U + W also generate the action α,

W = WA + C ′CA(W−) + We

where WA ∈ KerA, We is a constant, and the following estimates hold:

(1) ‖T − I‖ ≤ ‖W‖0;

(2) ‖W‖r ≤ C(1 + ‖W‖0)‖W‖r , r ≥ 0;

(3) ‖WA‖r ≤ C(1 + ‖W‖0)‖W‖r, r ≥ 0;

(4) ‖We‖ ≤ C‖W‖20.

Proof: Let k 6= l and akl denote the average of the component of Wk inthe direction of Ul.

Let

(5.1) B =

(

1 a12

a21 1

)

or

1 a12 a13

a21 1 a23

a31 a32 1

accordingly for d = 2 and d = 3.Then U = U + W = BU + W ′, where W ′ = W + (I − B)U. By definition

of B, D′DA(W ′+) = 0, C ′CA(W ′

−) = C ′CA(W−) and ‖W ′‖r ≤ C‖W‖r forr ≥ 0.

Let T = B−1. Then it is obvious from the definition of B that the estimate(1) holds for the norm of T − I.

Let U = TU = U + TW ′, let W = TW ′ and W” = (T − I)W ′. ThenW = W ′ + W”. This implies:

From (1): ‖W”‖r = ‖(T − I)W ′‖r ≤ C‖W‖0‖W‖r for r ≥ 0. This implies:

‖W‖r ≤ ‖W ′‖r + ‖(T − I)W ′‖r ≤ C(1 + ‖W‖0)‖W‖r


As for the operators D′DA and C ′CA the following is immediate fromthe definition of W:

D′DA(W+) = D′DA(W ′+) +D′DA(W”+) = D′DA(W”+)

C ′CA(W−) = C ′CA(W ′−) + C ′CA(W”+) = C ′CA(W−) + C ′CA(W”+)

Let WA := W − D′DA(W+) − C ′CA(W+). Then clearly WA ∈ KerA.Moreover, since

D′DA(W+) + C ′CA(W+) = D′DA(W”+) + C ′CA(W−) + C ′CA(W”+)

by denoting We = D′DA(W”+) + C ′CA(W”+) we obtain

W = WA + C ′CA(W−) + We

where We is constant and satisfies ‖We‖ ≤ C‖W‖20. �

6. ITERATIVE STEP

The following is an immediate corollary of the classical implicit functiontheorem.

LEMMA 2. There exists an open ball O = O(Id, R) in C1(Rd, Rd), there exists aneighborhood U of 0 ∈ Rd and a C1 map Ψ : O → U such that for every G ∈ O,G(Ψ(G)) = 0.

To simplify notations in this section we will denote the action αRR of themain example simply by α. Recall that Φ denotes the mapping which takesa 2-parameter family of perturbations of α generated by U1 + W1(λ) andU2 + W2(λ) where Ui := Ui

+ (i = 1, 2), to the pair of real numbers (µ1, µ2)where µi is the average of Wi(λ) in the Ui

− direction with respect to theHaar measure, for i = 1 or 2.

Recall that ‖Φ‖(r) denotes the usual Cr norm of Φ as a map from R2 to

R2, and that ‖W‖0,(r) stands the supremum of the Cr norms of W in the λ

variable over X. As before, we reserve the notation ‖W(λ)‖r for the usualCr norm on X of the vector field W(λ) for a fixed parameter λ.

In the following proposition we use indices n and n + 1 pertaining to theiterative step of the construction of conjugacy. This is done for convenienceof the reader since these same notations are used in the convergence proofin the next section. What is in fact proved here is that, given a family ofperturbations of the action α satisfying a certain set of conditions, one con-structs a conjugacy such that the new family of actions satisfies another setof conditions. The letter C is used generically to denote various constantsthat depend only on the actionα; indices such as r, indicate additional de-pendence of the corresponding parameter.

PROPOSITION 4. There exist constants r0 and C such that the following holds:


Let Un(λ) = U + Wn(λ) (λ ∈ R2) be a family of perturbations of U such thatfor all λ in a fixed closed ball B, r a natural number and t a positive real numbersuch that:

(1) ‖Wn(λ)‖0 ≤ εn < 1;(2) Φn := Φ(Wn) ∈ O has a zero at λn;(3) Wn(λ) is C2 in λ and ‖Wn(λ)‖0,(2) ≤ Kn;

(4) ‖Wn(λ)‖r0+r ≤ δr,n;

(5) tr0 ε1− 1

r+r0n δ

1r+r0r,n < C.

Then there exist a linear map Tn : R2 → R2 and a C∞ vector-field Hn on X,such that for hn := exp Hn, Un+1(λ) := (hn)∗(TnUn(λ)), λ ∈ B, Wn+1(λ) :=Un+1(λ) − U:

(a) ‖Hn‖r ≤ Crt2r0‖Wn(λn)‖r

(b) ‖Wn+1(λ)‖0 ≤ Kn‖λ − λn‖ + Errn+1(t, r)

Errn+1(t, r) := Cε2n + Cδ

r0+1r0+r

r,n ε2− r0+1

r0+r

n + Crt−rδr,n

+ Cr0t2r0 ε2− 1

r0+r

n δ1

r0+r

r,n + Cr0 t2r0 ε3− 1

r0+r

n δ2

r0+r

r,n

(c) ‖Wn+1(λ)‖r0+r ≤ Crt2r0 δr,n =: δr,n+1

(d) Φn+1 := Φ(Wn+1) satisfies

‖Φn+1 − Φn‖(0) ≤ Errn+1(t, r)

‖Φn+1 − Φn‖(1) ≤ Kntr0 εn + Errn+1(t, r).

If Φn+1 is in O, then it has a zero at λn+1 ∈ B which satisfies

‖λn+1 − λn‖ ≤ CErrn+1(t, r) + CKn(Kntr0 εn + Errn+1(t, r))2

(e) Wn+1(λ) is C2 in λ and

‖Wn+1(λ)‖0,(2) ≤ (1 + Ctr0 ε1− 1

r+r0n δ

1r+r0r,n )Kn =: Kn+1(t, r)

(f) ‖Tn − Id‖ ≤ εn.

Remark 6. The proof of this Proposition relies only on Proposition 3 andCorollary 2 which hold for the main example, and is otherwise general.

Proof: Within this proof we use the notation Wn(λ) := Un(λ) − U (in-stead of Wn(λ)) for the perturbation in the n-th step, in order to be consis-tent with the notation of Proposition 3.

By Proposition 3 and the assumption (1): given Un(λ) = U + Wn(λ)there exists Tn(λ) such that Tn(λ)Un(λ) = U + Wn(λ) and

Wn(λ) = WnA(λ) + Φn(λ) · V + Wn

e (λ)

where we used the fact that C ′CA(W(λ)− = Φ(W)(λ) · V and we used thenotation Φn := Φ(Un).


From Proposition 3 it follows that for all λ and all s ≥ 0 we have:

(6.1) ‖Wn(λ)‖s ≤ C‖Wn(λ)‖s

(6.2) ‖WnA(λ)‖s ≤ C‖Wn(λ)‖s

Due to loss of regularity which appears in estimates of Corollary 2 (recallthat the operators which appear in Corollary 2 are r0-tame) it is customaryto use the smoothing operators described in Section 2.1. Then we have:

Wn(λ) = StWnA(λ) + (I − St)Wn

A(λ) + Φn(λ) · V + Wne (λ)

so that WnA ∈ KerA.

By the choice of smoothing operators if WnA(λ) ∈ KerA, then Sn

t WA(λ) ∈KerA. Thus we can apply Corollary 2 to StW

nA(λ) to obtain

Hn(λ) := L′StWnA(λ)

such that(6.3)Wn(λ) = L(Hn(λ))+ M′M(StW

nA(λ))+ (I − St)Wn

A(λ)+ Φn(λ) ·V +Wne (λ)

From Corollary 2, for all r ≥ 0:

(6.4) ‖Hn(λ)‖r ≤ Cr‖StWnA(λ)‖r0+r ≤ Crt

r0‖WnA(λ)‖r ≤ Crt

r0‖Wn(λ)‖r

And for all s ≥ 0 and r ≥ 0:

(6.5) ‖(I − St)WnA(λ)‖s ≤ Ct−r‖Wn

A(λ)‖s+r ≤ Ct−r‖Wn(λ)‖s+r

From Proposition 3, Wne (λ) is a constant (depending on λ) and

(6.6) ‖Wne (λ)‖ ≤ C‖Wn(λ)‖2

0

Now by the assumption (2), there exists λn such that Φn(λn) = 0. Thenat λn we have

M(WnA(λn)) = M(Wn(λn)) + M(Wn

e (λn))

From Corollary 2, the operator M′ is r0-tame and M is 0-tame. Therefore,using Lemma 1, for any r > 0, and estimates (6.1) and (6.2), the followingestimates hold:

‖M′M(StWnA(λn))‖0 ≤ ‖M(StW

nA(λn))‖r0

≤ ‖M(WnA(λn))‖r0 + ‖M((I − St)Wn

A(λn))‖r0

≤ ‖M(Wn(λn))‖r0 + ‖M(Wne (λn))‖0 + ‖M((I − St)Wn

A(λn))‖r0

≤ C‖Wn(λn)‖r0+1‖Wn(λn)‖0 + C‖Wne (λn)‖0 + ‖(I − St)Wn

A(λn)‖r0

≤ C‖Wn(λn)‖r0+1‖Wn(λn)‖0 + C‖Wn(λn)‖20 + Crt

−r‖WnA(λn)‖r0+r

≤ C‖Wn(λn)‖r0+1‖Wn(λn)‖0 + C‖Wn(λn)‖20 + Crt

−r‖Wn(λn)‖r0+r

Set

En1 (λn) = Wn(λn)− L(Hn(λn)) = M′M(StW

nA(λn))+ (I− St)Wn

A(λn)+Wne (λn)


Then

(6.7)

‖En1 (λn)‖0 ≤ Ct−r(1 + ‖Wn(λn)‖0)‖Wn(λn)‖r + 2C‖Wn(λn)‖2

0+

C(1 + ‖Wn‖0)2‖Wn(λn)‖r0+1‖Wn(λn)‖0+

Crt−r‖Wn(λn)‖r0+r

Define Hn := Hn(λn). Let hn = exp Hn. Due to (6.4), the map hn isclose to the identity to the order of the initial perturbation. So by choos-ing sufficiently small perturbations initially, one can make sure that hn is adiffeomorphism on X. For this purpose, by the estimate (6.4) and interpo-lation estimates we have

‖Hn‖1 ≤ Ctr0 ε1− 1

r+r0n δ

1r+r0r,n

so it suffices to assume tr0 ε1− 1

r+r0n δ

1r+r0r,n < C, where C is a fixed constant. This

is assumption (5).Thus we may define

Un+1(λ) := hn∗(Tn(λ)Un(λ)) = hn

∗(Tn(λ)(U + Wn(λ)) = hn∗(U + Wn(λ))

and define αn+1(λ) to be the family of actions generated by the commutingd-tuple Un+1(λ).

We show now that this family of actions, with parameter λ satisfies thestatements of the Theorem.

Part (a) is immediate form (6.4).To prove the rest, we first linearize hn

∗(U + Wn(λ)). After linearization:

(6.8) Un+1(λ) := hn∗(U + Wn(λ)) = U + Wn(λ)− L(Hn) + En

0 (λ) + EnL(λ)

where L(Hn) = (LU1Hn, . . . ,LUd

Hn) as before, and En0 (λ) = [H, Wn(λ)]

and EnL(λ) is the error of linearization.

Thus the new error is:

(6.9) Wn+1(λ) = Wn(λ) − Wn(λn) + En1 (λn) + En

0 (λ) + EnL(λ)

To estimate norms of the new error we will make use of the interpolation

estimates: ‖Wn(λ)‖p ≤ C‖Wn(λ)‖1− ps

0 ‖Wn(λ)‖pss for p, s > 0.

By (6.4), interpolation estimates and assumptions (1) and (4), we have:(6.10)

‖En0 (λ)‖0 = ‖[Hn, Wn(λ)]‖0 ≤ C‖Hn‖0‖Wn(λ)‖1 + C‖Hn‖1‖Wn(λ)‖0

≤ Cr0 tr0(‖Wn(λn)‖0‖Wn(λ)‖1 + ‖Wn(λn)‖1‖Wn(λ)‖0)

≤ Cr0 tr0(‖Wn(λn)‖0‖Wn(λ)‖1 + ‖Wn(λn)‖1‖Wn(λ)‖0)

≤ Cr0 tr0(‖Wn(λn)‖0‖Wn(λ)‖1− 1

r0+r

0 ‖Wn(λ)‖1

r0+r

r0+r+

‖Wn(λn)‖1

r0+r

r0+r‖Wn(λn)‖1− 1

r0+r

0 ‖Wn(λ)‖0)

≤ Cr0 tr0 ε2− 1

r0+r

n δ1

r0+r

r,n


Using the assumption that ‖Wn‖0 ≤ εn < 1, (6.1), (6.4), interpolation esti-mates and assumptions (1) and (4), we have:(6.11)

‖EnL(λ)‖0 = ‖hn

∗(U + Wn(λ)) − (U + Wn(λ)) − [Hn, U + Wn(λ)]‖0

≤ Cr‖Hn‖0‖Hn‖1‖Wn(λ)‖1

≤ Cr0 t2r0‖Wn(λn)‖0‖Wn(λn)‖1‖Wn(λ)‖1

≤ Cr0 t2r0‖Wn(λn)‖0‖Wn(λn)‖1‖Wn(λ)‖1

≤ Cr0 t2r0‖Wn(λ)‖3− 1

r0+r

0 ‖Wn(λ)‖2

r0+r

r0+r

≤ Cr0 t2r0 ε3− 1

r0+r

n δ2

r0+r

r,n

By applying interpolation estimates to the estimate (6.7) for E1(λn), as-sumptions (1) and (4) and putting all the estimates together in (6.9), weobtain:

(6.12)‖Wn+1(λ)‖0 ≤ Kn‖λ − λn‖+ Cε2

n + Cδr0+1r0+r

r,n ε2− r0+1

r0+r

n

+ Crt−rδr,n + Cr0 t2r0 ε

2− 1r0+r

n δ1

r0+r

r,n + Cr0t2r0 ε3− 1

r0+r

n δ2

r0+r

r,n

where we used the fact that ‖Wn(λ) − Wn(λn)‖0 ≤ Kn‖λ − λn‖ by thebound in assumption (3).

For the Cr0+r norm of the new error we only need to have a ”linear”bound with respect to the corresponding norm of the old error. Thereforewe may use (6.8) for this purpose, and by (6.4), (6.11), (6.10), and (6.1), forany s ≥ 0 it follows that:

‖Wn+1(λ)‖s = ‖Wn(λ) − L(Hn) + En0 (λ) + En

L(λ)‖s ≤ Cst2r0‖Wn(λ)‖s

Therefore:

‖Wn+1(λ)‖r0+r ≤ Crt2r0 δr,n

The statement (e) follows from (6.4) and:

‖∂iλj

hn∗Un‖0 ≤ ‖hn

∗∂iλj

Un‖0 ≤ (1 + C‖Hn‖1)‖∂iλj

Un‖0

≤ (1 + Ctr0 ε1− 1

r+r0n δ

1r+r0r,n )Kn

for any i, j ∈ {1, 2}.Proof of (d): by (6.9)

Φn+1(λ)− Φn(λ) = Φ(En0 (λ) + En

L(λ))

This implies by (6.10) and (6.11) that ‖Φn+1 − Φn‖0 ≤ Errn+1(t, r). Also by(6.4) and assumption (3) we have:

‖∂λjΦn+1 − ∂λj

Φn‖(0) ≤ ‖∂λjΦ((Dhn − Id)Wn(λ))‖(0) ≤

‖∂λjWn(λ)‖0‖Hn‖0 ≤ Kntr0 εn


for any j ∈ {1, 2}. This implies:

‖Φn+1 − Φn‖(1) ≤ Kntr0 εn + Errn+1(t, r)

By Lemma 2 if Φn+1 is in O then it has a zero λn+1 and

‖λn+1 − λn‖ ≤ ‖Φn+1 − Φn‖(1)

Thus we have ‖λn+1 − λn‖ ≤ Kntr0 εn + Errn+1(t, r). But we can obtain abetter estimate for ‖λn+1 − λn‖.

Namely, since ‖Φn+1 − Φn‖0 ≤ Errn+1(t, r), it easily follows that‖Φn(λn+1)‖ ≤ Errn+1(t, r). Now by using Taylor expansion of Φn about λn

we have:

(6.13)‖λn+1 − λn‖ ≤ C‖Φn(λn+1)‖ + CKn(Kntr0 εn + Errn+1(t, r))2

≤ CErrn+1(t, r) + CKn(Kntr0 εn + Errn+1(t, r))2

where we used the assumption that Φn ∈ O, which is a small neighbor-hood of Id in the C1 norm, so both the derivative of Φn and its inverse havea universal bound (depending only on the constant R of the Lemma 2), welabelled this constant here C. �

7. CONVERGENCE

In this section we assume α to be any of the actions αRR, αRC or αC in thethree examples in Section 1.3. Assume that Proposition 4 holds for α (withthe obvious replacement of R2 by R3 in the case of αRC). In what followswe set an iterative scheme and show the convergence of the process to asmooth conjugacy between the initial perturbation and α up to a coordinatechange.

Recall that R is the constant from Lemma 2, and that C is the constantsuch that differentiable maps in the C1 neighborhood of Id of size C arediffeomorphisms.

For a start let ε0 < min{R/2, C2}. Later in the estimates, the initialbound for ε0 will be made even smaller depending on the constants whichappear in the estimates of the Proposition 4.

Recall that r0 in Proposition 4 is a fixed positive number which dependsonly on the dimension of X and spectral gap, i.e. on the lattice Γ. Set r =8r0 + 4.

Let α0(λ) be a family of perturbations of α, λ ∈ B(0, ε0) = {x ∈ R2|‖x‖ ≤ε0}, such that for its generating commuting d-tuple of vector fields U0(λ) =U + W0(λ) the following hold:

(10) ‖W0(λ)‖0 ≤ ε0

(20) If Φ0 := Φ(W0), then ‖Φ0 − Id‖(1) <1

2C ε0 (where C is the constant

in Lemma 2). Then Φ0 ∈ O and by Lemma 2 it has a zero λ0 such that‖λ0‖ < ε0/2.


(30) W0(λ) is C2 in λ and ‖W0(λ)‖0,(2) ≤ K0

(40) ‖W0(λ)‖r0+r ≤ δr,0 := ε−10

(50) By defining t0 = ε− 1

3r0+1

0 it is now easy to check that

tr00 ε

1− 1r+r0

0 δ1

r+r0r,0 < C

By applying Proposition 4 one performs the first step of iteration andconstructs the new family of perturbations generated by the d-tuple of com-muting vector fields U1(λ) = U + W1(λ).

Assume that at the n-th step of iteration, Un(λ) = U + Wn(λ) is a gener-ating d-tuple of vector fields for a perturbation αn(λ) of α satisfying the as-sumptions of Proposition 4 and let Un+1(λ) = U + Wn+1(λ) be the pertur-bation obtained after the Proposition 4 is applied to Un(λ) = U + Wn(λ).This gives an infinite sequence of perturbations providing that Proposition4 can be applied at every step.

Now we show that by choosing ε0 possibly even smaller, and by choos-ing tn and δr,n at every step in a convenient way, the Proposition 4 indeedapplies at every step, and moreover the infinite sequence of actions ob-tained in this way converges as well as the sequence of conjugacies hn

obtained by successive application of Proposition 4. Naturally, the proofproceeds by induction. We will fix the value of parameter r at r = 8r0 + 4throughout the induction process. The value of t will depend on n as fol-lows:

εn = ε(4/3)n

0 , tn = ε− 1

3r0+1

n , δr,n = ε−1n .

Condition (4) of Proposition 4 is easiest to check, for all n we have:

‖Wn‖r,n+1 ≤ Crt2r0n δr,n = Crε

− 2+3r03r0+1

n < ε−4/3n = δr,n+1.

Also, condition (5) follows immediately given the assumptions on r andε0:

tr0n ε

1− 1r+r0

n δ1

r+r0r,n ≤ ε

1− 13r0+1− 2

r+r0n ≤ C

Next is the estimate for Errn+1(tn, r):

Errn+1(tn, r) ≤

C(ε2n + ε

43 +( 2

3−2r0+1r0+r )

n + ε43 +(− 7

3 +2 r3r0+1 )

n + ε43 +( 2

3−2r0

3r0+1− 2r0+r )

n + ε43 +( 5

3−2r0

3r0+1− 3r0+r )

n )

From this it easily follows that the assumption r = 8r0 + 4 on r suffices for

(7.1) Errn+1(tn, r) < ε43n = εn+1

This will hold for any constant in place of C, the only difference would bea possibly smaller upper bound for ε0.


Now by (e) of Proposition 4

Kn+1(tn, r) = (1+ Ctr0n ε

1− 1r+r0

n δ1

r+r0r,n )Kn = (1+ Cε

1− r03r0+1− 2

r+r0n )Kn < (1+ ε

12n)Kn

by our choice of r and for ε0 sufficiently small. Then due to rapid conver-

gence of εn there exists a constant K, such that for each n, Kn = K0 ∏n−1k=1 (1 +

(√

ε0)k4/3) ≤ K. The constant K varies with the upper bound for ε0, but a

large upper bound is sufficient to make K independent of all other con-stants in the scheme, for example ε0 < 1/2 suffices.

Thus by (e) of Proposition 4 for all n:

‖Wn(λ)‖0,(2) ≤ Kn < K

By (d) of the Proposition 4:

‖Φn+1 − Φn‖(1) ≤ Knε1− r0

3r0+1

n + Err(n + 1) < ε23n

This implies that for every n, ‖Φn − Id‖(1) ≤ ε0, in particular if ε0 < R/2where R is the constant in Lemma 2, then Φn ∈ O so the condition (2) ofProposition 4 holds for all n.

Now from (d) of Proposition 4

2K‖λn+1 − λn‖ ≤

CKErrn+1(tn, r) + CK4ε2− 2r0

3r0+1

n + 2CK2ε73−

r03r0+1

n + CK2(Errn+1(tn, r))2< ε

43n

Consequently λn converges to some λ ∈ B, such that for all n:

2K‖λ − λn‖ ≤ 2εn+1

Combining this with (7.1), for all n:

(7.2) ‖Wn(λ)‖0 ≤ εn

Now for H we have an estimate in the C1 norm:

‖Hn‖1 ≤ Ct2r0n ‖Wn(λn)‖1 ≤ Cε

− 13r0+1

n ε1− 1

r+r0n δ

1r+r0r,n ≤ Cε

1− r+7r0+2

(r+r0)(3r0+1

n

Thus for all n:

(7.3) ‖Hn‖1 ≤ ε1/2n

Recall that hn := exp Hn. Let Hn = hn ◦ · · · ◦ h0. Since the gener-ators of the action αn+1(λ) = H−1

n ◦ α0(λ) ◦ Hn are by the constructionU + Wn+1(λ) and they satisfy the estimate (7.2) for all n, it follows that thesequence of actions αn(λ) converges to the action α in C0 topology.

The maps Tn acting on the generators U + Wn(λ) are only coordinatechanges which due to (f) of Proposition 4 accumulate to an ε0-small coordi-nate change of α but do not change the action α.

This cumulative coordinate change is the map π which appears in thestatement of Theorem 1.


By (7.3), Hn converges in C1 topology to a C1 conjugacy H between α0(λ)and α.

That the conjugacy H is of class C∞ is obtained via interpolation inequal-ities exactly as in [6] (end of section 5.4), which is in turn the same as in [14].This completes the proof of Theorem 1.

8. FEW WORDS ON POSSIBILITIES OF OBTAINING THE MAIN RESULT VIA

SOME VARIANT OF IMPLICIT FUNCTION THEOREM

The sequences of operators described in Section 1.2 are similar to the se-quences considered in [7] for discrete group actions. In the case of a smoothaction of a finitely generated finitely presented group, tame splitting of thelinearized sequence at one point implies tame splitting of the linearizedsequence in a neighborhood of that point (as in [1] for example) and thisimplies that the non-linear sequence is tamely exact by a general theoremof Hamilton [9], which is labeled in [9] as Nash-Moser theorem for exactsequences. This then implies full local rigidity for the action. Now for con-tinuous group actions this does not help since the first cohomology overthe action (even when it is very well understood) is not trivial, so one can-not have a tame splitting for the linearized sequence, except on a possiblysmaller space, namely on the trivial cohomology class. But there is no rea-son why this class would be invariant under conjugation (in fact, it is not),therefore Hamilton’s theorem does not apply.

Another possibility is to try to use one of the (generalized) implicit func-tion theorems. For example the one due to Zehnder in [16]. The set-up inthis direction would be as follows.

Let U = (U1, . . . , Ud) be the d-tuple of commuting generating vectorfields for the action α in one of the cases described in Section 1.3. Let V =(V1, . . . , Vd) be the d-tuple of ’opposite’ directions (see Section 1.3). Let

W : Rd → Vect∞(X)d be a continuously differentiable family such that foreach λ, d-tuple of vector fields U + W(λ) are pairwise commutative, let

λ ∈ Rd, H ∈ Vect∞(X), π ∈ Md×d (d × d matrices) and let UT denote thetranspose of the d-tuple U. Let W0 : Rd → Vect∞(X)d be the initial familydefined by W0(λ) = (U1 + λ1V1, . . . , Ud + λdVd). Define the following map:

(8.1) F(W, (λ, H, π)) = (exp H)∗(U + W(λ) − πUT) − U

This map is well defined for all H in a sufficiently small neighborhood of 0.Clearly F(W0, (0, 0, 0)) = 0.

(*) There exists neighborhood O of U and neighborhood U of (0, 0, 0) in

Rd × Vect∞(X) × Md×d and a continuous map f : O → U such that for allW ∈ O: F(W, f (W)) = 0.

If (*) holds then this implies that for the action α of Rd on X which isgenerated by d-tuple of vector fields U + W(λ) − πU with W sufficiently


small, there exists a transformation π there exists a parameter λ and thediffeomorphism h = exp H which conjugates the action generated by U +W(λ) − πU to U.

The main condition for obtaining (*) via the generalized implicit functiontheorem of Zehnder is that the derivative of F at (W0, (0, 0, 0)) in the secondvariable is approximately invertible in a neighborhood of (W0, (0, 0, 0)) andexactly invertible at (W0, (0, 0, 0)). This derivative operator which needs tobe inverted is [H, U] + λ · V + πUT. The contents of the Section 4 and Sec-tion 5 show that this derivative restricted to the space λ = 0 has an approxi-mate right inverse at (W0, (0, 0, 0)), which is in fact never exact (due to non-trivial relations in the acting group), and this approximate right inverse canbe constructed only for the data which has zero averages in V directions. Ifthere were no extra relations in the acting group, i.e. if there were no com-mutativity condition for the data, there would be no problem: the averagesof the data in the V directions would be precisely the value of λ and aftersubtracting the averages from the data one is in the situation when λ = 0as well as the averages in the V directions for the data, so by our resultsthere would be an approximate inverse. This would be similar to the ap-proach for small perturbations of constant Diophantine vector fields on toriof dimension greater than 2. Small constant modifications of the data accu-mulate then to a constant modification of the initial perturbations, modulowhich one has smooth conjugacy to the initial action. But there is a catch inthe higher rank case. The catch is that such a modification of a commutingd-tuple of vector fields by a constant for the examples in this paper givesa non-commuting d-tuple of vector fields, and for a non-commuting dataour results give no approximate inverse for the derivative operator. Com-mutativity is crucial for the existence of the approximate solution. Thusthe classical approach in this situation would lead us inevitably out of the

space of Rd actions. This is the main reason why Zehnder’s generalizedimplicit function theorem does not apply in this case.

9. COMMENTS ON UNIPOTENT AND OTHER ACTIONS

9.1. Actions on SL(2, C)/Γ and SL(2, R)× SL(2, C)/Γ. An immediate nextstep in our program is proving rigidity for the unipotent actions αRC andαC from Examples 1.3.2 and 1.3.3 from Section 1.3. As we pointed out inthat section, remaining steps are 5b (tame estimates) for the scalar equationand 6 (tame splitting).

9.1.1. SL(2, C) case. In Mieczkowski’s approach, the cohomological equa-tion for Example 1.3.3 is solved using analytic description of irreduciblerepresentation of SL(2, C) in certain spaces of C∞ functions on the complexplane and identifying obstructions within each irreducible subspace. Foran individual one-parameter subgroup of the action αC in addition to the


obstruction “at infinity” there is a whole continuous family of invariant dis-tributions and their simultaneous vanishing for one-cocycles is achieved bya proper version of the higher rank trick.

Mieszkowski does not obtain tame estimates for solutions in full Sobolevor Cr norms. Instead he considers two embeddings of SL(2, R) into SL(2, C)whose Lie algebras generate full Lie algebra sl(2, C) via brackets, but notlinearly. For Sobolev norms associated to those embeddings he obtainstame estimates that are sufficient to conclude that the solutions are C∞ viaelliptic regularity but with the loss of half of the derivatives. The one miss-ing direction generates a compact subgroup that commutes with the R2

unipotent homogeneous action αC. There is a general argument suggestedby L. Flaminio that allows to obtain tame estimates in the centralizer direc-tion a priori.

Splitting however represents an additional problem since the space of in-variant distributions for one-parameter subgroups of αC is infinite dimen-sional within each irreducible space for SL(2, C). Those obstructions havea form of integrals over lines parallel to a certain direction for a function oftwo variables with a certain behavior at infinity that guarantees existenceof such integrals. While there is no natural projection to the space of co-cycles (that in this case correspond to the functions whose integrals alongthe lines in question vanish) there is a natural strategy for constructing atame splitting by distributing the value of the obstructions along certainintervals using smooth kernels. This is a work in progress.

It is also worth pointing out a peculiar feature of the SL(2, C) case. Forexactly one of the principal series representations (with parameter values(0, 2)) the higher rank trick does not eliminate the obstruction at infinityfor vanishing of a cocycle. The multiplicity d of this representation in thedecomposition of L2(SL(2, C)/Γ) into irreducible subspaces is equal to thefirst Betti number of the group Γ. If this number is not zero, there are twopossibilities: (i) to consider 2 + d-parametric families of actions near theunipotent action αC and find an action conjugate to αC up to an automor-phism of R2 or (ii) to start with a standard d-dimensional family of timechanges for the action αC generated by harmonic forms with varying co-homology, and than consider a 2-parametric family of perturbations of αC

and look for a parameter value that produces an action conjugate to an el-ement of this d-dimensional family. Once the problem of tame splitting isresolved, both of these approaches can be carried out.

9.1.2. SR(2, R)× SL(2, C) case. For the R3 action αRC in Example 1.3.2Mieczkowski proves in [12] that the sequence L → M is tamely exact sothat Step 5b follows in this case. Once tame splitting for SL(2, C) is provedTheorem 2 will imply tame splitting for this case.

To summarize, the only ingredient left to prove the counterpart of The-orem 1 for the actions in Examples 1.3.2 and 1.3.3 is a splitting of a pairof scalar functions ( f , g) within an irreducible representation of SL(2, C)


into a coboundary (LU1h,LU2

h), and the remainder that can be estimatedtamely through the norms of LU2

f −LU1g, where

U1 =

(

0 10 0

)

, U2 =

(

0 i0 0

)

.

9.2. Other unipotent actions. In recent paper [15] Felipe Ramirez consid-ered the following generalization of our Main example 1.3.1: he assumesthat SL(2, R) × SL(2, R) (or, more generally, its finite cover) embeds intoa noncompact simple Lie group G with finite and considers the action ofthe upper-diagonal unipotents in SL(2, R)× SL(2, R) on G/Γ. Theorem B’in [15] assets that any C∞ cocycle over that action is C∞ cohomologousto a constant cocycle. If one can add tame estimates and tame splittingto Ramirez’ result our scheme becomes applicable. Tame estimates in theSL(2, R)× SL(2, R) direction follow the same way as in the present paper.One can also obtain tame estimates in the centralizer direction by the ar-gument suggested by L. Flaminio. Once tame estimates are obtained tamesplitting for functions follows from Theorem 2. However, deduction of thesplitting for vector fields from splitting for functions does not directly fol-low from Proposition 2 and needs to be reworked.

These observations lead to new cases of rigidity that will be discussed indetail in a subsequent paper joint with Flaminio.

Not surprisingly, not all higher rank abelian unipotent actions on homo-geneous spaces of semisimple groups can be treated with the use of rep-resentation theory for SL(2, R) or SL(2, C) where we have good insightsinto the structure of invariant distributions and solutions of cohomologi-cal equations. A simple example, where cohomological equations are notwell understood, is the action by the R2 subgroup of SL(3, R) generatedby Id + e12 and Id + e13, where eij is the matrix with 1 in the place (i, j) andzeroes elsewhere.

9.3. Some partially hyperbolic actions. While partial hyperbolicity in dy-namics in its most general form simply means that the linearized system isuniformly hyperbolic in some directions and non-hyperbolic in others, thegenuine partially hyperbolic paradigm strives to explore situations wherehyperbolicity is somehow prevalent. As examples of this approach to localdifferentiable rigidity one may consider the results based on the “geometricmethod” and applications of algebraic K-theory started in our work [3, 4, 2]and recently extended by Zhenqi Wang [17, 18].

Approach developed in this paper is applicable to certain actions wherepartial hyperbolicity is present but could not be considered prevalent sincein particular some elements of the action are non-hyperbolic. We postponedetailed description of the relevant situations to a future paper and nowmention only another “model” example.


Mixed example on SL(2, R)× SL(2, R). Consider the R2 action β on SL(2, R)×SL(2, R)/Γ generated by commuting vector fields U1

+ and U20 as in the

Main example 1.3.1. The first generator is unipotent, hence the correspond-ing one-parameter subgroup acts parabolically. The second generator ispartially hyperbolic with isometric action in the neutral directions. Mi-esczkowski showed vanishing of the obstructions and solved the scalar co-homological equation in this case, see [12, Theorem 2]; in his thesis [13]he obtained necessary tame estimates for the diagonal action on SL(2, R)parallel to the estimates in [8] for unipotent actions.

The only difference with our model unipotent action αRR in the MainExample 1.3.1 is in the structure of linearized conjugacy equations. Due tothe presence of a partially hyperbolic generator some of the equations aretwisted (compare with the discrete time partially hyperbolic case in [6]).Accordingly, considerations of Section 4 do not apply directly and have tobe modified. This modifications is fairly routine, albeit somewhat tedious.The twisted equation related to the partially hyperbolic generator alwayshas a solution but obstructions appear in the bootstrap regularity and theirvanishing again involves invoking a version of Higher rank trick. Thoseobstructions are not exactly invariant distributions but have similar nature(again compare with [6]). An interesting difference with our model unipo-tent case is that one only needs one-parametric families of perturbationssince homogeneous actions conjugate to β up to a an automorphism of theacting group have codimension one among homogeneous R2 actions onSL(2, R)× SL(2, R)/Γ.

Once tame estimates are obtained for the twisted equation, tame splittingis obtained by the construction from the proof of Theorem 2. After thatconsiderations from Sections 6 and 7 apply directly.

Detailed discussion of this and other partially hyperbolic cases that canbe treated with variations of our method will appear in a subsequent paper.

REFERENCES

[1] E.J.Benveniste, Rigidity of isometric lattice actions on compact Riemannian mani-folds,GAFA, 10, (2000), 516-542.

[2] D. Damjanovic, Central extensions of simple Lie groups and rigidity of some abelianpartially hyperbolic algebraic actions, J. Modern Dyn. 1, (2007), 665–688.

[3] D. Damjanovic and A. Katok, Local rigidity of restrictions of Weyl chamber flows,Comptes Rendus Mathematique 344, no 8, , 503-508.

[4] D. Damjanovic and A. Katok, Local Rigidity of Partially Hyperbolic Actions. II.The geometric method and restrictions of Weyl chamber flows on SL(n, R)/Γ,http://www.math.psu.edu/katok a/papers.html/.

[5] D. Damjanovic and A. Katok, Local rigidity of actions of higher rank abelian groupsand KAM method, ERA–AMS, 10 (2004), 142–154.

[6] D. Damjanovic and A. Katok, Local Rigidity of Partially Hyperbolic Actions of Zk

and Rk, k ≥ 2. I. KAM method and actions on the Torus, Annals of Mathematics, toappear


[7] D. Fisher, First cohomology and local rigidity of group actions, Annals of Mathematics,to appear.

[8] L. Flaminio and G. Forni, Invariant Distributions and Time Averages for HorocycleFlows. Duke J. of Math, 119, (2003), No 3, 465–526.

[9] R.H.Hamilton, The inverse function theorem of Nash and Moser. Bull. Amer. Math. Soc.(N.S.) 7, (1982), no. 1, 65–222.

[10] B. Hasselblatt and A. Katok Principal structures, in Handbook in Dynamical Systems,vol 1A Elsevier, 2002, 1–203.

[11] A. Katok and R. Spatzier, Differential rigidity of Anosov actions of higher rank abeliangroups and algebraic lattice actions, Proc. Steklov Inst. Math. 216 (1997), 287-314.

[12] D. Mieczkowski, The first cohomology of parabolic actions for some higher-rankabelian groups and representation theory, Journal of Modern Dynamics, 1, no. 1,(2007), 61–92.

[13] D. Mieczkowski, The Cohomological Equation and Representation Theory. Ph. D.Thesis, The Pennsylvania State University, 2006.

[14] J. Moser On commuting circle mappings and simoultaneous Diophantine approxima-tions Math. Z. 205, (1990), 105-121.

[15] F. Ramirez Cocycles over higher-rank abelian actions on quotients of semisimple Liegroups, Journal of Modern Dynamics, 3, no. 3, (2009), 335–357.

[16] E. Zehnder, Generalized implicit function theorems with applications to some smalldivisor problems. I. Comm. Pure Appl. Math. 28, (1975)

[17] Z. Wang, Local rigidity of partially hyperbolic actions, preprint.[18] Z. Wang, New cases of differentiable rigidity for partially hyperbolic actions: Sym-

plectic groups and resonance directions, preprint

DEPARTMENT OF MATHEMATICS, RICE UNIVERSITY, 6100 MAIN STR. HOUSTON, TX,77005

DEPARTMENT OF MATHEMATICS, THE PENNSYLVANIA STATE UNIVERSITY, UNIVER-SITY PARK, PA 16802

E-mail address: [email protected], katok [email protected]

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