The following dichotomy for affine actions on the torus , , 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.
Funding source: National Science Foundation
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.1 Convexity estimates
For all non-negative numbers with and we have
For all non-negative numbers s, we have
(i) One way to show interpolation estimates in the scale of norms is to derive them from the existence of smoothing operators and from the norm inequalities for the smoothing operators. This is done in  for spaces where , which includes the case of . Another elementary proof for interpolation without going through smoothing operators can be found in .
(ii) An immediate corollary of the interpolation estimates is the following fact:
where and are Lipschitz constants for f and g, respectively. ∎
In the proof we shorten the notation to .
(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 denote partial derivation in one of the basis directions and let denote coordinate functions of g. Since
we can apply part (ii) of the previous proposition to :
Here we invoked part (ii) of the previous proposition again to obtain the last line of estimates above. Since for the -norm we have
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 , where denotes . Then , where denote coordinate functions of g. This implies (by using Proposition A.1 (ii)) the following estimate for the first derivatives:
For the -norm we have
which together with the estimates above implies the claim. ∎
Let and assume that
is invertible and if we write then
for all .
For 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 which satisfy interpolation estimates. Thus the claim follows from Proposition A.1 (i) and [9, Lemma 2.3.6]. ∎
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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