LOCAL RIGIDITY OF HOMOGENEOUS PARABOLIC ACTIONS: I. A Model Case

We show a weak form of local differentiable rigidity for the rank $2$ abelian 
action of upper unipotents on $SL(2,R)$ $\times$ $SL(2,R)$ $/\Gamma$. Namely, for a 
$2$-parameter family of sufficiently small perturbations of the action, 
satisfying certain transversality conditions, there exists a parameter for 
which the perturbation is smoothly conjugate to the action up to an 
automorphism of the acting group. This weak form of rigidity for the parabolic 
action in question is optimal since the action lives in a family of dynamically 
different actions. The method of proof is based on a KAM-type iteration and we 
discuss in the paper several other potential applications of our approach.


INTRODUCTION
1.1. KAM/Harmonic analysis method in differentiable rigidity. In this paper we develop an approach for proving local differentiable rigidity of higher rank abelian groups, i.e. Z k × R l , k + l ≥ 2, based on KAM-type iteration scheme, that was first introduced in [5,6]. In those papers local differentiable rigidity was proved for Z k , k ≥ 2 actions by partially hyperbolic automorphisms of a torus. Here we deal with certain non-hyperbolic actions of R 2 and R 3 , namely actions of certain unipotent subgroups of semisimple Lie groups on homogeneous spaces by translations. From the point of view of general classification of dynamical systems those actions are parabolic [10]. Before proceeding to specifics we will describe in general terms the version of the method adapted to the use for parabolic actions.
First, an important feature of the parabolic case, that sets it apart from all hyperbolic and many partially hyperbolic actions, comes from the fact that among homogeneous actions on semi-simple groups unipotent ones are not stable. Unipotent abelian subalgebras of a given dimension k of a semisimple Lie algebra g have positive codimension d in the variety of kdimensional abelian subalgebras of g. Thus among the non-homogeneous perturbations of a unipotent R k action α those conjugate to a (maybe different) unipotent action are expected to have positive codimension, greater or equal than d. A natural way to find such perturbations is to consider d-parametric familiesα(λ) of perturbed R k actions and look for an action 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 higher rank hyperbolic actions in [11]), we will look for conditional rigidity. Let as point out that in the most traditional KAM results dealing with perturbations of translations or linear flows on the torus (elliptic systems in the terminology of [10]) one also has to consider parametric families with the exception of the low-dimensional cases (dimension one for maps and two for flows) when rotation number serves as a modulus of conjugacy.
What is described below is a somewhat idealized scheme that highlights essential features of the method. While dealing with concrete problems, including those considered in this paper, certain steps may be combined and order of other steps may be reversed.
Step 1. Conjugacy equation is formally linearized at the target. Solution of the linearized equation is attempted by finding an inverse on a proper subspace of data.
Step 2. Obstructions for solving the linearized conjugacy equation for a particular element of the action are found.
There are infinitely many obstructions for solving the linearized (cohomological) equations for an individual generator of the action. This is a crucial difference with standard KAM for translations where obstruction is unique. In some cases (including those considered in this paper) those obstructions are invariant distributions for the action element. This has to do with parabolicity of our examples. If some hyperbolicity is present, as in [5,6], the linearized equations are twisted in some way and obstructions have a somewhat different form.
Step 5. a) Linearized equation is solved. This involves glueing of solutions constructed 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 of regularity in the chosen collection of norms in the Fréchet spaces, such as C r or Sobolev norms.
Step 6. The perturbation can be split into two terms due to the commutation relations: one for which the linearized equations are satisfied, and the other "quadratically small" with tame estimates. This step also uses harmonic analysis.
Step 7. Conjugacy provided by the solution of the linearized equation transforms the (modified in the conditional case) perturbed action into an action quadratically close to the target with a fixed loss of regularity in the estimates.
Step 8. The process is iterated and the product of conjugacies converges to a conjugacy between the (modified) original and perturbed action (with adjusted parameters if necessary).
The last two steps are completely independent of the specific ways previous steps are performed and depend only on the conclusions reached at those steps.
Since we deal with the higher rank actions the basic iteration scheme is somewhat similar to that applied by Moser [14] to commuting rotations of the circle (which is the basis of the scheme used later in [6]).

Conjugacy problem and linearization. Now we describe
Step 1 in complete generality. Let G = G 1 × G 2 × · · · × G m , where G j (j = 1, . . . , m) are simple Lie groups. Let X = G/Γ where Γ is an irreducible cocompact lattice in G. Let g denote the Lie algebra of G and g j Lie algebras of G j . Let U 1 , . . . , U d be some commuting elements of g. We consider an action α : R d × X → X by left translations on X whose generating vector fields are U 1 , . . . , U d , namely U k = ∂α(t 1 ,t 2 ,x) ∂t k | (0,0) . A smooth perturbationα of the action α is generated by commuting vector fieldsŨ k = U k + W k (k = 1, . . . , d) on X which are small perturbations of U k '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: where h 1 = exp H and Vect ∞ (X) denotes the space of C ∞ vector fields on X.
Obviously M • L = 0. Denote by L → M the non-linear sequence of operators defined by (1.1) and(1.2). The "conjugacy problem" or the "perturbation problem" is the question of whether the sequence L → M is exact on some small open sets.
Linearizing the sequence L → M at H 0 = 0 and at (U 1 , . . . , U d ) ∈ Vect ∞ (X) d we obtain the linearized sequence L → M as follows: The following Proposition is immediate from the fact that M is simply a linearization of the commutation condition and that we are interested in perturbations in the space of commutative actions. It will be used later.
This shows that the image of M is quadratically small with respect to W, so having a tame splitting (see Section 2 for definition of "tame") for the L → M sequence is the same as having an approximate solution to the linearized version of the conjugacy problem.
Finally, one may try to reduce the sequence L → M to a simpler sequence, for functions rather than vector fields: where by L → M we denote the following sequence of operators: Now the general strategy of approaching the local perturbation problem for a general R k action can be roughly summarized as: Note. All the splittings above as well as the exactness should be tame, see Section 2 for exact definitions.
The paper is dedicated to proving the first statement that L → M splits and the two implications in (1.7) for certain actions under some additional conditions: certain small modifications and smaller splitting spaces. These small modifications will be simply coordinate changes in the acting group and considering smaller spaces is necessary because actions we deal with have non-trivial first cohomology, see Step 4 of the general scheme. The generality in which these statements are proved increases from left to right.
• The splitting of L → M is proved for certain type of examples (the Main example for the next section) via the splitting construction in Section 3. • The first implication in (1.7) is discussed in the Section 4 and relies very much on the fact that the actions we consider here are upper unipotent, but applies to all the examples we describe in this paper. • The second implication in (1.7) is proved through the KAM-type iteration scheme which is very general, in Section 6 and Section 7.
1.3. Description of some unipotent actions. The choice of examples below is not accidental. They represent exactly those unipotent homogeneous actions of R k , k ≥ 2 for which Step 2 of the general scheme has been performed 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 example below are carried out in Sections 3 and 4. Only the first part (tame estimates and splitting for functions) is specific for that example. Reduction from vector fields to functions (Section 4) is applicable to all three cases and is valid in even more general setting. The last two steps are carried out in Sections 6 and 7 and, as we already pointed out, are based only on the conclusions of the previous steps and are hence of general character. We will discuss remaining steps (5b and 6) for examples 1.3.2 and 1.3.3 as well as other applications of our method, both in progress and potential, in Section 9.
The Lie algebra g j has a basis U j + , U j 0 and U j − for j = 1, 2, where: Our model parabolic action is the action by left translations on X generated by commuting unipotent elements We denote the pair of vector fields (U 1 + , U 2 + ) generating the action α RR by U and the pair ( , Γ be an irreducible cocompact lattice in G, X = G/Γ. g 1 = sl(2, R), and g 2 = sl(2, C). The Lie algebra g 1 is generated by U 1 + , U 1 − and U 1 0 as in the example above, while g 2 is generated by Our example is the action α RC of R 3 on X, by left translations, generated by commuting unipotent elements . ′ We denote the triple of vector fields (U 1 + , U 2 + , (U 2 + ) ′ ) generating the action α RC by U, and the triple ( Example on SL(2, C). Let G = SL(2, C), Γ be an irreducible cocompact lattice in G, X = G/Γ. In this case m = 1, g = g 1 = sl(2, C), and g 1 is generated by U 1 Our third example is the action α C of R 2 on X by left translations, generated by commuting unipotent elements If the first Betti number of Γ is not zero there are additional obstructions in the cohomology problem and they correspond to certain specific classes of perturbations of action α C .
1.4. Cocycle rigidity of parabolic (unipotent) actions. David Mieszkowski [12] studied real valued cocycles over the actions of unipotent subgroups described above. He proved vanishing of the obstructions and constructed smooth solutions in all three cases, obtained tame estimates for both product cases (actions α RR in the main example 1.3.1 and α RC in Example 1.3.2) and for the former also gave a partial proof for a splitting of an almost cocycle 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 the description of obstructions and solution of cohomological equations for horocycle 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) factor is essential. Solutions are constructed for the unipotent element from that factor and estimates are carried out. Additional generator(s) serve an auxiliary purpose to guarantee the vanishing of obstructions for the solution of the cohomological equations for the first factor. In fact, specific choices of the second factor in these two cases has to do with the fact that those are the only cases where irreducible lattices exist in the product. Example 1.3.3 on SL(2, C)/Γ is completely different as we will remark later in Section 9.1.
The space L 2 (SL(2, R)/Γ) is decomposed into the direct sum of irreducible representation spaces for SL(2, R); representations of principal, complementary and discrete series are treated separately; each carries obstructions for solutions of the cohomological equation Thus, not only general perturbations but already time changes of the horocycle flow are highly unstable (infinitely many moduli of smooth conjugacy) in contrast with the Diophantine elliptic case.
For an R 2 action generated by unipotents exp U 1 and exp U 2 the basic cocycle condition has the form (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 with tame estimates for an appropriate family of Sobolev norms.
But for G = SL(2, C) the analytic description of the irreducible representations is used.
Mieczkowski obtained tame estimates with respect to standard Sobolev norms for both product cases.
But for the SL(2, C) case Mieczkowski only obtained tame estimates with respect 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 out the program outlined in Section 1.1 for the action α RR in the main example of Section 1.3.1.
We use the following norms for a familyŨ(λ) = (Ũ 1 (λ),Ũ 2 (λ)) of pairs of vector fields on X: • For a fixed λ, Ũ (λ) r denotes the maximum of the C r norms of U 1 (λ) andŨ 2 (λ). • Ũ (λ) r,(m) is the maximum of the C r norms of all the partial derivatives ofŨ in the λ variable of order less or equal to m. • For the "averages" map Φ(Ũ) : R 2 → R 2 , the notation Φ(Ũ) (r) stands for the usual C r norm.
THEOREM 1 (Main Theorem). Let U = (U 1 + , U 2 + ) be as in the main example 1.3.1 and α RR be the R 2 action generated by U (see (1.8)).
The work on examples 1.3.2 and 1.3.3 is in progress and appropriate local rigidity 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 the form which can be used in even greater generality than the remaining two examples.
2. SOME ANALYTIC TOOLS 2.1. Graded Fréchet spaces and tame operators. Definition 1. By a graded Fréchet space we mean a Fréchet space X with the collection of norms · r (r ∈ N 0 ) such that x r ≤ x r+k for every r, k ≥ 0, for every x ∈ X. Definition 2. A map L : X → Y between two graded Fréchet spaces is r 0tame if for every x ∈ X and for every r ≥ 0 we have that Lx r ≤ C r x r+r 0 Definition 3. Tame maps L : X → Y and M : Y → Z between graded Fréchet spaces are said to form a sequence of maps if ML = 0. We denote such a sequence of maps by L → M. If ImL = KerM we call the sequence exact. We call the sequence L → M r 0 -tamely exact if there exists an r 0 -tame map L ′ from KerM to X such that LL ′ = Id on KerM.

Fréchet spaces and gradings to be used in the proof of Theorem 1.
The space X is compact, thus spaces of smooth functions and smooth vector fields are graded Fréchet spaces. We will use the following notation: (1) C ∞ (X) denotes the graded Fréchet space of C ∞ functions with grading given either by the usual C r norms or by the Sobolev norms. We will always specify which grading we use, and the chosen grading will be denoted by · r . Similarly C ∞ (X) d and C ∞ (X) d×d denote d-tuples and d × d matrices, respectively, of C ∞ -functions on X.
(2) Vect ∞ (X) denotes the graded Fréchet space of vector fields on X of class C ∞ with grading given by the component-wise norms (C r or Sobolev). These we also denote by · r . (3) Vect ∞ (X) d and Vect ∞ (X) d×d denote d-tuples, and d × d matrices of vector fields on X, respectively. The grading on this space is given by norms which we again denote by · r and which are again just maxima over the corresponding coordinate wise norms.

Smoothing operators and some norm inequalities.
The space X is compact, thus spaces of smooth functions and smooth vector fields are graded Fréchet spaces. There exists a collection of smoothing operators S t : C ∞ (X) → C ∞ (X), t > 0, such that the following holds: Smoothing operators on C ∞ (X) clearly induce smoothing operators on Vect ∞ (X) d , Vect ∞ (X) d×d via smoothing operators applied to coordinate maps.
It is easy to see that averages of F with respect to the Haar measure on X, in various directions in the tangent space do not affect the properties of smoothing operators listed above, so without loss of generality we may assume that S t are such that averages of The following inequality implies the statement of Proposition 1. It will also be used later in Section 6 . Let F, G ∈ Vect ∞ (X). Then for r ≥ 0 Also, existence of smoothing operators on a graded space induces the usual interpolation inequalities for the norms which make up the grading. This has been demonstrated in several papers, for example [16] and in a more general set-up in [9].

FIRST COHOMOLOGY AND TAME SPLITTING FOR FUNCTIONS
In this section we show that the exact sequence of maps , has an r 0 -tame splitting. The constant r 0 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 cohomological equation, i.e. showed that the sequence M → L is r 0 -tamely exact where r 0 depends only on the lattice Γ. In the process he uses explicit obstructions to the solutions of cohomology problem for a single action generator U k , k = 1, 2. The obstructions come from work of Flaminio and Forni [8]. It is important for our application to remark that the space of cohomological obstructions for each U k is found to be finite dimensional in each irreducible representation (this follows immediately from [8]). More precisely, depending on the kind of irreducible representation the space of obstructions is either one or two dimensional.
Below we give a general construction of a tame splitting for the product examples, in the situation where for one action generator there is a finite dimensional space of obstructions for the cohomology problem in each irreducible representation.

Remark 2.
Only in this section the notation · r is used for Sobolev norms.
In the rest of the paper · r stands for C r norms.
Sobolev norm of f ∈ W s (H) is then given by: imply global Sobolev estimate: This reduces the proof of the existence of tame splitting in Corollary 1 below to proving the existence of a tame splitting in each irreducible H µ,θ in such a way that the constants involved in tame estimates can be chosen so that they do not depend on (µ, θ).
When SL(2, R) × SL(2, R)/Γ is compact the above direct integral decompositions into irreducible representations reduces to infinite discrete sums.
We expect that the Theorem below will be used for other actions, in particular for α RC from Example 1.3.2, so we formulate it and prove it in some generality. THEOREM 2. Let G 1 and G 2 be Lie groups and let H µ and H θ be irreducible unitary representations of G 1 and G 2 , respectively. Let H = H µ ⊗ H θ and assume . . , m. Further, assume that there exists s 0 ≥ 0, σ ≥ 0 independent of the pair µ, θ, such that the following hold for k = 1, 2 and any s sufficiently large: (  (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 α RR from the main example was based on a specific and explicit construction of projections in each irreducible representation which used detailed descriptions of the representations. This proof completed and corrected attempted proof of Theorem 18 in [12] that in fact contains an error because it uses only one of the two obstructions that appear in any irreducible representation of SL(2, R) of the principal series.
The shorter and more general proof presented below was inspired by our discussions with Livio Flaminio.
By applying a linear map on the m-tuple (γ 1 , . . . , γ m ) we can obtain an m-tuple of maps (γ 1 , . . . ,γ m ) satisfying D j (γ i ) = δ ij . The norm of the linear map is bounded above by the maximum of the norms of Thus without loss of generality we may assume that we have γ 1 , . . . , γ m ∈ W s (H µ ) such that for any i = 1, . . . , m, γ i s ≤ C (where C is a constant multiple of the maximum of the norms of D i 's on W s (H)), such that Let V s 1 be the intersection of kernels ofD i 's in W s (H). The following is the central ingredient in the proof of the Theorem.
To check commutativity we use the fact that U 2 commutes withD i : This concludes the proof of the claim. Now for f ∈ W s (H), we may define P f := f − R f . Then from the properties of R obtained in the claim we have that P : W s (H) → V s 1 and P f s ≤ C f s , and P commutes with U 2 as well.
Given U 2 f − U 1 g = φ, by assumption (1) and the properties of R we get U 2 R 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 Since for sufficiently large s, U 2 h − g ∈ W s 0 , by assumptions (1), (2) and (3) The assumptions of Theorem 2 are checked using the results of [8] similarly to the way the same results were used by Mieczkowski in his proof of vanishing of the first cohomology over the action α RR modulo averages [12]. Below we summarize conclusions of [8] adapted to the product case in the form which is sufficient for the purpose of the current paper.  The tame splitting of the L → M sequence is now an easy corollary of the previous two theorems: Since by Theorem 3 the constants in the estimates do not depend on the representation H µ × H θ , it is clear that after defining L ′ and M ′ on all of H µ ⊗ H θ by gluing L ′ and M ′ in each irreducible representation, the tame estimates in Sobolev norms for L ′ and M ′ remain true. From these the tame estimates in the C r norms follow in a standard way with the loss of r 0 derivatives where r 0 depends only on the dimension of X and the lattice Γ.

TAME SPLITTING FOR VECTOR FIELDS
As mentioned in the introduction, due to Proposition 1 the existence of tame splitting of the L → M-sequence in (1.4) implies the existence of an approximate right inverse for the linearization L of our initial conjugacy problem and this is the necessary ingredient in any attempt to solve a conjugacy problem 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 α RR from main example, where A is a specifically defined functional on Vect ∞ (X) d . To do this 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 problems. Those simpler problems inductively reduce to existence of the tame splitting for functions i.e. for the sequence L → M to which the results of the previous section apply.
We give a unified proof for all three examples described in Section 1.3 that splitting of functions implies splitting for vector fields, in anticipation of forthcoming tame splitting results for functions for actions α RC and α C from Examples 1.3.2 and 1.3.3. Furthermore, the scheme of this proof can be adapted to even more general situations of unipotent homogeneous actions.
Reader should be warned though that uniformity comes at a price of certain clarity since in fact in the formulas written in a generic way m equals either one or two, and d equals two or three. Rewriting the calculations for each specific case will make them more compact and easier to visualize.
Let G j (j = 1, 2) be either SL(2, R) or SL(2, C) and consider one of the actions described in Section 1.
We will use generic notation α for the actions α RR , α RC and α C . Generators of α are as follows: (1) Main example 1.3.1: From the definition of U k it follows that [U k , U j + ] = 0 for any j, k, and Then from (4.1) we have: This completely describes the linear operator L : H → (L U 1 H, . . . , L U d H) with respect to the basis given by U j + , U j 0 , U j − , j = 1, m. It is clear from this description that j-th block of L k is given by: and let L be the following map: Then the j-th block of L is given by: By denoting and using the notationL for the operator Operator M is defined to be the operator which acts by M j in each direction g j . Let (4.14) and define C ′ and D ′ to act by C ′ j and D ′ j on each g j . 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.
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 andĀ is the coordinate-wise average map.
Proof: First notice that it follows from (4.8) and (4.15) that the exact sequence M → L is 3s-tame. By assumption there exist operators L ′ and M ′ such that (4.18) LL Recall that from (4.8), L j =L + C j and from (4.15) M j =M + D j . Now define operators L ′ j by: Operator M ′ j makes sense on the image of M, which consists of g j -valued d × d skew-symmetric matrices. The following is obtained immediately from definitions above: and M ′ are assumed to be r-tame, and C j and D j are obviously 0-tame, it follows immediately that L ′ and M ′ are 3r-tame.
Using the assumption (4.18), and (4.11), it follows that (4.24) L j L ′ j + M ′ j M j = I on the kernel of the map −Ā j + C j C ′ jĀ j + D ′ j D jĀj , whereĀ j denotes the map which assigns to a given vector field the triple of averages in the three directions U j + , U j 0 and U j − . Maps C ′ j and D ′ j are defined in (4.16). Thus by denoting we have the required splitting on KerA: is defined to be equal to A j in each direction g j , j = 1, . . . , m, just as L ′ and M ′ are defined to be L ′ j and M ′ j in each g j .
Corollary 1 asserts that assumptions of Proposition 2 hold for the action α RR of the main example. This gives the principal ingredient for carrying out the iteration scheme.

Remark 5.
While it is obvious from the definition that KerA has finite codimension, 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 of parameters.

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 that part (ii) of this step in our setting involves conjugating the unipotent action inside the group G since unipotent homogeneous actions in our cases are all conjugate to the standard ones. Thus standard perturbations produce actions isomorphic to the original ones.
Part (iii) corresponds to taking parametric families in the Main Theorem with the number of parameters exactly equal to the codimension of unipotent homogeneous actions among all homogeneous actions of R d .
Recall that It is immediate from definitions that C j C ′ j + D ′ j D j = I. This implies In fact, because of the definition of ψ kl (4.2) it is not difficult to see that D ′ DA(W + ) involves the averages of W in all the directions U l for which ψ kl = 0 and that CC ′ A(W − ) involves all the averages of W in the direction of V l whenever ψ kl = 0. In particular, in the notation of Theorem 1 we have: On the other hand, the following proposition shows that D ′ DA(W + ) can be assumed to be quadratically small by making a good choice of generating vector fields for the action.
where W A ∈ KerA, W e 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) W A r ≤ C(1 + W 0 ) W r , r ≥ 0; (4) W e ≤ C W 2 0 . Proof: Let k = l and a kl denote the average of the component ofW k in the direction of U l . Let Then it is obvious from the definition of B that the estimate (1) holds for the norm of T − I. LetŨ = TŪ = U + TW ′ , let W = TW ′ and W" = (T − I)W ′ . Then W = W ′ + W". This implies: From (1): W" r = (T − I)W ′ r ≤ C W 0 W r for r ≥ 0. This implies: As for the operators D ′ DA and C ′ CA the following is immediate from the definition of W: by denoting W e = D ′ DA(W" + ) + C ′ CA(W" + ) we obtain where W e is constant and satisfies W e ≤ C W 2 0 .

ITERATIVE STEP
The following is an immediate corollary of the classical implicit function theorem.

LEMMA 2. There exists an open ball
To simplify notations in this section we will denote the action α RR of the main example simply by α. Recall that Φ denotes the mapping which takes a 2-parameter family of perturbations of α generated by U 1 + W 1 (λ) and U 2 + W 2 (λ) where U i := U i + (i = 1, 2), to the pair of real numbers (µ 1 , µ 2 ) where µ i is the average of W i (λ) in the U i − direction with respect to the Haar measure, for i = 1 or 2.
Recall that Φ (r) denotes the usual C r norm of Φ as a map from R 2 to R 2 , and that W 0,(r) stands the supremum of the C r norms of W in the λ variable over X. As before, we reserve the notation W(λ) r for the usual C r 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 the iterative step of the construction of conjugacy. This is done for convenience of the reader since these same notations are used in the convergence proof in the next section. What is in fact proved here is that, given a family of perturbations of the action α satisfying a certain set of conditions, one constructs a conjugacy such that the new family of actions satisfies another set of conditions. The letter C is used generically to denote various constants that depend only on the actionα; indices such as r, indicate additional dependence of the corresponding parameter. PROPOSITION 4. There exist constants r 0 andC such that the following holds:

be a family of perturbations of U such that for all λ in a fixed closed ball B, r a natural number and t a positive real number such that:
(1) W n (λ) 0 ≤ ε n < 1; (2) Φ n := Φ(W n ) ∈ O has a zero at λ n ; (3) W n (λ) is C 2 in λ and W n (λ) 0,(2) ≤ K n ; (4) W n (λ) r 0 +r ≤ δ r,n ; Then there exist a linear map T n : R 2 → R 2 and a C ∞ vector-field H n on X, such that for h n := exp H n ,Ũ n+1 (λ) : (1) ≤ K n t r 0 ε n + Err n+1 (t, r). If Φ n+1 is in O, then it has a zero at λ n+1 ∈ B which satisfies λ n+1 − λ n ≤ CErr n+1 (t, r) + CK n (K n t r 0 ε n + Err n+1 (t, r)) 2 (e) W n+1 (λ) is C 2 in λ and Remark 6. The proof of this Proposition relies only on Proposition 3 and Corollary 2 which hold for the main example, and is otherwise general.
Proof: Within this proof we use the notationW n (λ) :=Ũ n (λ) − U (instead of W n (λ)) for the perturbation in the n-th step, in order to be consistent with the notation of Proposition 3.
From Proposition 3 it follows that for all λ and all s ≥ 0 we have: s Due to loss of regularity which appears in estimates of Corollary 2 (recall that the operators which appear in Corollary 2 are r 0 -tame) it is customary to use the smoothing operators described in Section 2.1. Then we have: By the choice of smoothing operators if W n A (λ) ∈ KerA, then S n t W A (λ) ∈ KerA. Thus we can apply Corollary 2 to S t W n A (λ) to obtain And for all s ≥ 0 and r ≥ 0: is a constant (depending on λ) and (6.6) W n e (λ) ≤ C W n (λ) 2 0 Now by the assumption (2), there exists λ n such that Φ n (λ n ) = 0. Then at λ n we have M(W n A (λ n )) = M(W n (λ n )) + M(W n e (λ n )) From Corollary 2, the operator M ′ is r 0 -tame and M is 0-tame. Therefore, using Lemma 1, for any r > 0, and estimates (6.1) and (6.2), the following estimates hold: Define H n := H n (λ n ). Let h n = exp H n . Due to (6.4), the map h n is close to the identity to the order of the initial perturbation. So by choosing sufficiently small perturbations initially, one can make sure that h n is a diffeomorphism on X. For this purpose, by the estimate (6.4) and interpolation estimates we have r,n <C, whereC is a fixed constant. This is assumption (5).
We show now that this family of actions, with parameter λ satisfies the statements of the Theorem.
Namely, since Φ n+1 − Φ n 0 ≤ Err n+1 (t, r), it easily follows that Φ n (λ n+1 ) ≤ Err n+1 (t, r). Now by using Taylor expansion of Φ n about λ n we have: where we used the assumption that Φ n ∈ O, which is a small neighborhood of Id in the C 1 norm, so both the derivative of Φ n and its inverse have a universal bound (depending only on the constant R of the Lemma 2), we labelled this constant here C.

CONVERGENCE
In this section we assume α to be any of the actions α RR , α RC or α C in the three examples in Section 1.3. Assume that Proposition 4 holds for α (with the obvious replacement of R 2 by R 3 in the case of α RC ). In what follows we set an iterative scheme and show the convergence of the process to a smooth conjugacy between the initial perturbation and α up to a coordinate change.
Recall that R is the constant from Lemma 2, and thatC is the constant such that differentiable maps in the C 1 neighborhood of Id of sizeC are diffeomorphisms.
For a start let ε 0 < min{R/2,C 2 }. Later in the estimates, the initial bound for ε 0 will be made even smaller depending on the constants which appear in the estimates of the Proposition 4.
Recall that r 0 in Proposition 4 is a fixed positive number which depends only on the dimension of X and spectral gap, i.e. on the lattice Γ. Setr = 8r 0 + 4.
That the conjugacy H is of class C ∞ is obtained via interpolation inequalities 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.

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 sequences considered in [7] for discrete group actions. In the case of a smooth action of a finitely generated finitely presented group, tame splitting of the linearized sequence at one point implies tame splitting of the linearized sequence in a neighborhood of that point (as in [1] for example) and this implies that the non-linear sequence is tamely exact by a general theorem of Hamilton [9], which is labeled in [9] as Nash-Moser theorem for exact sequences. This then implies full local rigidity for the action. Now for continuous group actions this does not help since the first cohomology over the action (even when it is very well understood) is not trivial, so one cannot have a tame splitting for the linearized sequence, except on a possibly smaller space, namely on the trivial cohomology class. But there is no reason 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 function theorems. For example the one due to Zehnder in [16]. The set-up in this direction would be as follows.
Let U = (U 1 , . . . , U d ) be the d-tuple of commuting generating vector fields for the action α in one of the cases described in Section 1.3. Let V = (V 1 , . . . , V d ) be the d-tuple of 'opposite' directions (see Section 1.3). Let W : R d → Vect ∞ (X) d be a continuously differentiable family such that for each λ, d-tuple of vector fields U + W(λ) are pairwise commutative, let λ ∈ R d , H ∈ Vect ∞ (X), π ∈ M d×d (d × d matrices) and let U T denote the transpose of the d-tuple U.
. Define the following map: This map is well defined for all H in a sufficiently small neighborhood of 0. Clearly F (W 0 , (0, 0, 0)) = 0.
(*) There exists neighborhood O of U and neighborhood U of (0, 0, 0) in R d × Vect ∞ (X) × M d×d and a continuous map f : O → U such that for all W ∈ O: F (W, f (W)) = 0.
If (*) holds then this implies that for the actionα of R d on X which is generated 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 the diffeomorphism h = exp H which conjugates the action generated by U + W(λ) − πU to U.
The main condition for obtaining (*) via the generalized implicit function theorem of Zehnder is that the derivative of F at (W 0 , (0, 0, 0)) in the second variable is approximately invertible in a neighborhood of (W 0 , (0, 0, 0)) and exactly invertible at (W 0 , (0, 0, 0)). This derivative operator which needs to be inverted is [H, U] + λ · V + πU T . The contents of the Section 4 and Section 5 show that this derivative restricted to the space λ = 0 has an approximate right inverse at (W 0 , (0, 0, 0)), which is in fact never exact (due to nontrivial relations in the acting group), and this approximate right inverse can be constructed only for the data which has zero averages in V directions. If there were no extra relations in the acting group, i.e. if there were no commutativity condition for the data, there would be no problem: the averages of the data in the V directions would be precisely the value of λ and after subtracting the averages from the data one is in the situation when λ = 0 as well as the averages in the V directions for the data, so by our results there would be an approximate inverse. This would be similar to the approach for small perturbations of constant Diophantine vector fields on tori of dimension greater than 2. Small constant modifications of the data accumulate then to a constant modification of the initial perturbations, modulo which one has smooth conjugacy to the initial action. But there is a catch in the higher rank case. The catch is that such a modification of a commuting d-tuple of vector fields by a constant for the examples in this paper gives a non-commuting d-tuple of vector fields, and for a non-commuting data our results give no approximate inverse for the derivative operator. Commutativity is crucial for the existence of the approximate solution. Thus the classical approach in this situation would lead us inevitably out of the space of R d actions. This is the main reason why Zehnder's generalized implicit 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 next step in our program is proving rigidity for the unipotent actions α RC and α C from Examples 1.3.2 and 1.3.3 from Section 1.3. As we pointed out in that section, remaining steps are 5b (tame estimates) for the scalar equation and 6 (tame splitting). 9.1.1. SL(2, C) case. In Mieczkowski's approach, the cohomological equation for Example 1.3.3 is solved using analytic description of irreducible representation of SL(2, C) in certain spaces of C ∞ functions on the complex plane and identifying obstructions within each irreducible subspace. For an individual one-parameter subgroup of the action α C in addition to the obstruction "at infinity" there is a whole continuous family of invariant distributions and their simultaneous vanishing for one-cocycles is achieved by a proper version of the higher rank trick.
Mieszkowski does not obtain tame estimates for solutions in full Sobolev or C r 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 not linearly. For Sobolev norms associated to those embeddings he obtains tame estimates that are sufficient to conclude that the solutions are C ∞ via elliptic regularity but with the loss of half of the derivatives. The one missing direction generates a compact subgroup that commutes with the R 2 unipotent homogeneous action α C . There is a general argument suggested by L. Flaminio that allows to obtain tame estimates in the centralizer directioná priori.
Splitting however represents an additional problem since the space of invariant distributions for one-parameter subgroups of α C is infinite dimensional within each irreducible space for SL(2, C). Those obstructions have a form of integrals over lines parallel to a certain direction for a function of two variables with a certain behavior at infinity that guarantees existence of such integrals. While there is no natural projection to the space of cocycles (that in this case correspond to the functions whose integrals along the lines in question vanish) there is a natural strategy for constructing a tame splitting by distributing the value of the obstructions along certain intervals using smooth kernels. This is a work in progress.
It is also worth pointing out a peculiar feature of the SL(2, C) case. For exactly one of the principal series representations (with parameter values (0, 2)) the higher rank trick does not eliminate the obstruction at infinity for vanishing of a cocycle. The multiplicity d of this representation in the decomposition of L 2 (SL(2, C)/Γ) into irreducible subspaces is equal to the first Betti number of the group Γ. If this number is not zero, there are two possibilities: (i) to consider 2 + d-parametric families of actions near the unipotent action α C and find an action conjugate to α C up to an automorphism of R 2 or (ii) to start with a standard d-dimensional family of time changes for the action α C generated by harmonic forms with varying cohomology, and than consider a 2-parametric family of perturbations of α C and look for a parameter value that produces an action conjugate to an element of this d-dimensional family. Once the problem of tame splitting is resolved, both of these approaches can be carried out. 9.1.2. SR(2, R) × SL(2, C) case. For the R 3 action α RC in Example 1.3.2 Mieczkowski proves in [12] that the sequence L → M is tamely exact so that Step 5b follows in this case. Once tame splitting for SL(2, C) is proved Theorem 2 will imply tame splitting for this case.
To summarize, the only ingredient left to prove the counterpart of Theorem 1 for the actions in Examples 1.3.2 and 1.3.3 is a splitting of a pair of scalar functions ( f , g) within an irreducible representation of SL(2, C) into a coboundary (L U 1 h, L U 2 h), and the remainder that can be estimated tamely through the norms of L U 2 f − L U 1 g, where 9.2. Other unipotent actions. In recent paper [15] Felipe Ramirez considered the following generalization of our Main example 1.3.1: he assumes that SL(2, R) × SL(2, R) (or, more generally, its finite cover) embeds into a noncompact simple Lie group G with finite and considers the action of the 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 ∞ cohomologous to a constant cocycle. If one can add tame estimates and tame splitting to Ramirez' result our scheme becomes applicable. Tame estimates in the SL(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 argument suggested by L. Flaminio. Once tame estimates are obtained tame splitting for functions follows from Theorem 2. However, deduction of the splitting for vector fields from splitting for functions does not directly follow from Proposition 2 and needs to be reworked. These observations lead to new cases of rigidity that will be discussed in detail in a subsequent paper joint with Flaminio.
Not surprisingly, not all higher rank abelian unipotent actions on homogeneous spaces of semisimple groups can be treated with the use of representation theory for SL(2, R) or SL(2, C) where we have good insights into the structure of invariant distributions and solutions of cohomological equations. A simple example, where cohomological equations are not well understood, is the action by the R 2 subgroup of SL(3, R) generated by Id + e 12 and Id + e 13 , where e ij is the matrix with 1 in the place (i, j) and zeroes elsewhere. 9.3. Some partially hyperbolic actions. While partial hyperbolicity in dynamics in its most general form simply means that the linearized system is uniformly hyperbolic in some directions and non-hyperbolic in others, the genuine partially hyperbolic paradigm strives to explore situations where hyperbolicity is somehow prevalent. As examples of this approach to local differentiable rigidity one may consider the results based on the "geometric method" 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 where partial hyperbolicity is present but could not be considered prevalent since in particular some elements of the action are non-hyperbolic. We postpone detailed description of the relevant situations to a future paper and now mention only another "model" example.
Mixed example on SL(2, R) × SL(2, R). Consider the R 2 action β on SL(2, R) × SL(2, R)/Γ generated by commuting vector fields U 1 + and U 2 0 as in the Main example 1.3.1. The first generator is unipotent, hence the corresponding one-parameter subgroup acts parabolically. The second generator is partially hyperbolic with isometric action in the neutral directions. Miesczkowski showed vanishing of the obstructions and solved the scalar cohomological 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 α RR in the Main Example 1.3.1 is in the structure of linearized conjugacy equations. Due to the presence of a partially hyperbolic generator some of the equations are twisted (compare with the discrete time partially hyperbolic case in [6]). Accordingly, considerations of Section 4 do not apply directly and have to be modified. This modifications is fairly routine, albeit somewhat tedious. The twisted equation related to the partially hyperbolic generator always has a solution but obstructions appear in the bootstrap regularity and their vanishing again involves invoking a version of Higher rank trick. Those obstructions are not exactly invariant distributions but have similar nature (again compare with [6]). An interesting difference with our model unipotent case is that one only needs one-parametric families of perturbations since homogeneous actions conjugate to β up to a an automorphism of the acting group have codimension one among homogeneous R 2 actions on SL(2, R) × SL(2, R)/Γ.
Once tame estimates are obtained for the twisted equation, tame splitting is obtained by the construction from the proof of Theorem 2. After that considerations from Sections 6 and 7 apply directly.
Detailed discussion of this and other partially hyperbolic cases that can be treated with variations of our method will appear in a subsequent paper.