# On local rigidity of partially hyperbolic affine ℤk actions

• Danijela Damjanović and Bassam Fayad

## Abstract

The following dichotomy for affine k actions on the torus 𝕋d, k,d, is shown to hold: (i) The linear part of the action has no rank-one factors, and then the affine action is locally rigid. (ii) The linear part of the action has a rank-one factor, and then the affine action is locally rigid in a probabilistic sense if and only if the rank-one factors are trivial. Local rigidity in a probabilistic sense means that rigidity holds for a set of full measure of translation vectors in the rank-one factors.

Award Identifier / Grant number: DMS-0758555

Funding statement: The first author was supported by NSF grant DMS-0758555. The second author was supported by ANR-15-CE40-0001 and by the project BRNUH.

## A Appendix

In this appendix, we give references and proofs for the estimates used in the proofs of Lemma 3.3 and Proposition 3.2.

### Proposition A.1.

Let f,gClip,(I,Td,R).

1. For all non-negative numbers a1,a2,s1,s2 with a1+a2=1 and s1a1+s2a2=s we have

flip(I),sCs1,s2flip(I),s1a1flip(I),s2a2.
2. For all non-negative numbers s, we have

fglip(I),sCs(flip(I),sglip(I),0+flip(I),0glip(I),s).

### Proof.

(i) One way to show interpolation estimates in the scale of Clip,s norms is to derive them from the existence of smoothing operators and from the norm inequalities for the smoothing operators. This is done in [24] for spaces Cα,s where 0<α1, which includes the case of Clip,s. Another elementary proof for interpolation without going through smoothing operators can be found in [5].

(ii) An immediate corollary of the interpolation estimates is the following fact:

flip(I),iglip(I),jC(flip(I),kglip(I),l+flip(I),mglip(I),n)

if (i,j) lies on the line segment joining (k,l) and (m,n) (see [9, Corollary 2.2.2]). Statement (ii) follows from this by using the product rule on derivatives (see [9, Corollary 2.2.3]) and the following inequality:

Lip(fg)=supxy|(fg)(x)-(fg)(y)||x-y|
sup(|f(x)-f(y)||g(x)||x-y|+|g(x)-g(y)||f(y)||x-y|)
Lfg0+f0Lg,

where Lf and Lg are Lipschitz constants for f and g, respectively. ∎

### Proposition A.2.

Let f,gClip,(I,Td+1,Rd+1).

1. h(x)=f(x+g(x))-f(x) verifies

hlip(I),sCs(flip(I),0glip(I),s+1+flip(I),s+1glip(I),0).
2. k(x)=f(x+g(x))-f(x)-Dfg(x) verifies

ksCs(flip(I),0glip(I),s+2+flip(I),s+2glip(I),0).

### Proof.

In the proof we shorten the notation lip(I),s to lip,s.

(i) It suffices to prove the estimates for the coordinate functions of f, so in what follows we assume that f denotes a coordinate function of f. Let Di1 denote partial derivation in one of the basis directions and let gj denote coordinate functions of g. Since

Di1h=Di1(f(x+g(x))-f(x))=jDj1fDi1gj,

we can apply part (ii) of the previous proposition to Dj1fDi1gj:

D1hlip,sCmaxjDj1fDi1gjlip,s
Csmaxj(Dj1flip,sDi1gjlip,0+Dj1flip,0Di1gjlip,s)
Csmaxj(flip,s+1gjlip,1+flip,1gjlip,s+1)
Cs(flip,s+2glip,0+flip,0glip,s+2).

Here we invoked part (ii) of the previous proposition again to obtain the last line of estimates above. Since for the lip,0-norm we have

hlip,0=f(x+g(x))-f(x)lip,0Lfg0flip,0glip,0,

the claim follows.

(ii) Again by reducing to coordinate functions we look at one coordinate function of k and f (which we denote by k and f as well), so we have k=f(x+g(x))-f-iDi1fgi, where Di1 denotes /xi. Then Dj1k=-iDj1Di1fgi, where gi denote coordinate functions of g. This implies (by using Proposition A.1 (ii)) the following estimate for the first derivatives:

Dj1klip,siDj1Di1fgilip,s
Cs(Dj1Di1flip,sgilip,0+Dj1Di1flip,0glip,s)
Cs(flip,s+2gilip,0+flip,2glip,s)
Cs(flip,s+2glip,0+flip,0glip,s+2).

For the lip,0-norm we have

klip,0Lfg0+maxi{Di1fgilip,0}Cflip,1glip,0

which together with the estimates above implies the claim. ∎

### Proposition A.3.

Let hClip,(I,Td+1,Rd+1) and assume that

hlip(I),112.

Then

f:𝕋d+1𝕋d+1,xH(x)=x+h(x)

is invertible and if we write H-1(x)=x+h¯(x) then

h¯lip(I),sCshlip(I),s

for all sN.

### Proof.

For Cs norms this is proved for example in [9, Lemma 2.3.6]. The proof uses induction and interpolation estimates, and it is general to the extent that it applies to any sequence of norms on C which satisfy interpolation estimates. Thus the claim follows from Proposition A.1 (i) and [9, Lemma 2.3.6]. ∎

## Acknowledgements

We are grateful to Artur Avila, Hakan Eliasson, Anatole Katok and Raphae’l Krikorian for fruitful discussions and suggestions. We gratefully acknowledge the suggestions of the referee that helped us make several improvements to the initial version of this paper.

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