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On local rigidity of partially hyperbolic affine ℤk actions

  • Danijela Damjanović EMAIL logo 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.

A.1 Convexity estimates

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. ∎

A.2 Composition

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. ∎

A.3 Inversion

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.

References

[1] V. Arnol’d, Small denominators. I. Mapping the circle onto itself, Izv. Akad. Nauk SSSR Ser. Mat. 25 (1961), no. 1, 21–86. Search in Google Scholar

[2] D. Damjanović, Abelian actions with globally hypoelliptic leafwise Laplacian and rigidity, J. Anal. Math. 129 (2016), no. 1, 139–163. 10.1007/s11854-016-0018-8Search in Google Scholar

[3] D. Damjanović and A. Katok, Local rigidity of partially hyperbolic actions. I. KAM method and Zk actions on the torus, Ann. of Math. (2) 172 (2010), no. 3, 1805–1858. 10.4007/annals.2010.172.1805Search in Google Scholar

[4] R. de la Llave, Tutorial on KAM theory, Smooth ergodic theory and its applications (Seattle 1999), Proc. Sympos. Pure Math. 69, American Mathematical Society, Providence (2001), 175–292. 10.1090/pspum/069/1858536Search in Google Scholar

[5] R. de la Llave and R. Obaya, Regularity of the composition operator in spaces of Hölder functions, Discrete Contin. Dyn. Syst. 5 (1999), no. 1, 157–184. 10.3934/dcds.1999.5.157Search in Google Scholar

[6] B. Fayad and K. Khanin, Smooth linearization of commuting circle dieomorphisms, Ann. of Math. (2) 170 (2009), 961–980. 10.4007/annals.2009.170.961Search in Google Scholar

[7] D. Fisher, Local rigidity of group actions: Past, present, future, Dynamics, ergodic theory, and geometry, Math. Sci. Res. Inst. Publ. 54, Cambridge University Press, Cambridge (2007), 45–97. 10.1017/CBO9780511755187.003Search in Google Scholar

[8] D. Fisher and G. Margulis, Local rigidity of affine actions of higher rank groups and lattices, Ann. of Math. (2) 170 (2009), no. 1, 67–122. 10.4007/annals.2009.170.67Search in Google Scholar

[9] R. Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 1, 65–222. 10.1090/S0273-0979-1982-15004-2Search in Google Scholar

[10] S. Hurder, Rigidity for Anosov actions of higher rank lattices, Ann. of Math. (2) 135 (1992), 361–410. 10.2307/2946593Search in Google Scholar

[11] S. Hurder, Affine Anosov actions, Michigan Math. J. 40 (1993), no. 3, 561–575. 10.1307/mmj/1029004838Search in Google Scholar

[12] A. Katok and B. Hasselblatt, Introduction to the modern theory of dynamical systems, Cambridge University Press, Cambridge 1996. 10.1017/CBO9780511809187Search in Google Scholar

[13] A. Katok and E. A. Robinson, Jr., Cocycles, cohomology and combinatorial constructions in ergodic theory, Smooth ergodic theory and its applications (Seattle 1999), Proc. Sympos. Pure Math. 69, American Mathematical Society, Providence (2001), 107–173. 10.1090/pspum/069/1858535Search in Google Scholar

[14] A. Katok and R. Spatzier, Differential rigidity of Anosov actions of higher rank Abelian groups and algebraic lattice actions, Proc. Steklov Inst. Math. 216 (1997), 287–314. Search in Google Scholar

[15] J. Moser, A rapidly convergent iteration method and non-linear partial differential equations. I, Ann. Sc. Norm. Super. Pisa Sci. Fis. Mat. III. Ser. 20 (1966), 265–315. Search in Google Scholar

[16] J. Moser, On commuting circle mappings and simoultaneous Diophantine approximations, Math. Z. 205 (1990), 105–121. 10.1007/BF02571227Search in Google Scholar

[17] V. Nitica and A. Török, Cohomology of dynamical systems and rigidity of partially hyperbolic actions of higher rank lattices, Duke Math. J. 79 (1995), no. 3, 751–810. 10.1215/S0012-7094-95-07920-4Search in Google Scholar

[18] V. Nitica and A. Török, Local rigidity of certain partially hyperbolic actions of product type, Ergodic Theory Dynam. Systems 21 (2001), no. 4, 1213–1237. 10.1017/S0143385701001560Search in Google Scholar

[19] A. S. Pyartli, Diophantine approximations on submanifolds of Euclidean space, Funct. Anal. Appl. 3 (1969), no. 4, 303–306. 10.1007/BF01076316Search in Google Scholar

[20] F. Rodriguez-Hertz and Z. Wang, Global rigidity of higher rank abelian Anosov algebraic actions, Invent. Math. 198 (2014), no. 1, 165–209. 10.1007/s00222-014-0499-ySearch in Google Scholar

[21] A. Starkov, The first cohomology group, mixing, and minimal sets of the commutative group of algebraic actions on a torus, J. Math. Sci. (N. Y.) 95 (1999), no. 5, 2576–2582. 10.1007/BF02169057Search in Google Scholar

[22] A. Török, Rigidity of partially hyperbolic actions of property (T) groups, Discrete Contin. Dyn. Syst. 9 (2003), no. 1, 193–208. 10.3934/dcds.2003.9.193Search in Google Scholar

[23] J.-C. Yoccoz, Conjugaison différentiable des difféomorphismes du cercle dont le nombre de rotation vérifie une condition Diophantienne, Ann. Sci. Éc. Norm. Supér. (4) 17 (1984), 333–361. 10.24033/asens.1475Search in Google Scholar

[24] E. Zehnder, Generalized implicit function theorems with applications to some small divisor problems. I, Comm. Pure Appl. Math. 28 (1975), 91–140. 10.1002/cpa.3160280104Search in Google Scholar

Received: 2013-04-05
Revised: 2016-08-16
Published Online: 2016-11-17
Published in Print: 2019-06-01

© 2019 Walter de Gruyter GmbH, Berlin/Boston

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