## Global rigidity of higher rank Anosov actions on tori and nilmanifolds

HTML articles powered by AMS MathViewer

- by David Fisher, Boris Kalinin and Ralf Spatzier; with an appendix by James F. Davis
- J. Amer. Math. Soc.
**26**(2013), 167-198 - DOI: https://doi.org/10.1090/S0894-0347-2012-00751-6
- Published electronically: August 20, 2012
- PDF | Request permission

## Abstract:

We show that sufficiently irreducible Anosov actions of higher rank abelian groups on tori and nilmanifolds are $C^{\infty }$-conjugate to affine actions.## References

- Auslander, L.; Green, L.; Hahn, F.
*Flows on homogeneous spaces.*With the assistance of L. Markus and W. Massey, and an appendix by L. Greenberg. Annals of Mathematics Studies, No. 53, Princeton University Press, Princeton, N.J., 1963. - Luis Barreira and Yakov Pesin,
*Nonuniform hyperbolicity*, Encyclopedia of Mathematics and its Applications, vol. 115, Cambridge University Press, Cambridge, 2007. Dynamics of systems with nonzero Lyapunov exponents. MR**2348606**, DOI 10.1017/CBO9781107326026 - P. E. Blanksby and H. L. Montgomery,
*Algebraic integers near the unit circle*, Acta Arith.**18**(1971), 355–369. MR**296021**, DOI 10.4064/aa-18-1-355-369 - Danijela Damjanović and Anatole Katok,
*Local rigidity of partially hyperbolic actions I. KAM method and $\Bbb Z^k$ actions on the torus*, Ann. of Math. (2)**172**(2010), no. 3, 1805–1858. MR**2726100**, DOI 10.4007/annals.2010.172.1805 - James F. Davis and Paul Kirk,
*Lecture notes in algebraic topology*, Graduate Studies in Mathematics, vol. 35, American Mathematical Society, Providence, RI, 2001. MR**1841974**, DOI 10.1090/gsm/035 - Davis, J. F.; Petrosyan, N.
*Manifolds and Poincaré complexes*, in preparation. - F. T. Farrell and L. E. Jones,
*Anosov diffeomorphisms constructed from $\pi _{1}\,\textrm {Diff}\,(S^{n})$*, Topology**17**(1978), no. 3, 273–282. MR**508890**, DOI 10.1016/0040-9383(78)90031-9 - F. T. Farrell, A. Gogolev.
*Anosov diffeomorphisms constructed from*$\pi _k ({\mathrm {Diff}}(S^n))$, preprint. - David Fisher and Gregory Margulis,
*Almost isometric actions, property (T), and local rigidity*, Invent. Math.**162**(2005), no. 1, 19–80. MR**2198325**, DOI 10.1007/s00222-004-0437-5 - David Fisher and Gregory Margulis,
*Local rigidity of affine actions of higher rank groups and lattices*, Ann. of Math. (2)**170**(2009), no. 1, 67–122. MR**2521112**, DOI 10.4007/annals.2009.170.67 - David Fisher, Boris Kalinin, and Ralf Spatzier,
*Totally nonsymplectic Anosov actions on tori and nilmanifolds*, Geom. Topol.**15**(2011), no. 1, 191–216. MR**2776843**, DOI 10.2140/gt.2011.15.191 - John Franks,
*Anosov diffeomorphisms on tori*, Trans. Amer. Math. Soc.**145**(1969), 117–124. MR**253352**, DOI 10.1090/S0002-9947-1969-0253352-7 - Andrey Gogolev,
*Diffeomorphisms Hölder conjugate to Anosov diffeomorphisms*, Ergodic Theory Dynam. Systems**30**(2010), no. 2, 441–456. MR**2599887**, DOI 10.1017/S0143385709000169 - Gorodnik, A.; Spatzier, R.
*Exponential Mixing of Nilmanifold Automorphisms*, preprint in preparation. - Gorodnik, A.; Spatzier, R.
*Mixing Properties of Commuting Nilmanifold Automorphisms*, preprint in preparation. - Ben Green and Terence Tao,
*The quantitative behaviour of polynomial orbits on nilmanifolds*, Ann. of Math. (2)**175**(2012), no. 2, 465–540. MR**2877065**, DOI 10.4007/annals.2012.175.2.2 - Morris W. Hirsch and Barry Mazur,
*Smoothings of piecewise linear manifolds*, Annals of Mathematics Studies, No. 80, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974. MR**0415630** - M. W. Hirsch, C. C. Pugh, and M. Shub,
*Invariant manifolds*, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977. MR**0501173**, DOI 10.1007/BFb0092042 - Lars Hörmander,
*The analysis of linear partial differential operators. I*, Classics in Mathematics, Springer-Verlag, Berlin, 2003. Distribution theory and Fourier analysis; Reprint of the second (1990) edition [Springer, Berlin; MR1065993 (91m:35001a)]. MR**1996773**, DOI 10.1007/978-3-642-61497-2 - Boris Kalinin and Anatole Katok,
*Invariant measures for actions of higher rank abelian groups*, Smooth ergodic theory and its applications (Seattle, WA, 1999) Proc. Sympos. Pure Math., vol. 69, Amer. Math. Soc., Providence, RI, 2001, pp. 593–637. MR**1858547**, DOI 10.1090/pspum/069/1858547 - Boris Kalinin and Anatole Katok,
*Measure rigidity beyond uniform hyperbolicity: invariant measures for Cartan actions on tori*, J. Mod. Dyn.**1**(2007), no. 1, 123–146. MR**2261075**, DOI 10.3934/jmd.2007.1.123 - Boris Kalinin and Victoria Sadovskaya,
*Global rigidity for totally nonsymplectic Anosov $\Bbb Z^k$ actions*, Geom. Topol.**10**(2006), 929–954. MR**2240907**, DOI 10.2140/gt.2006.10.929 - Boris Kalinin and Victoria Sadovskaya,
*On the classification of resonance-free Anosov $\Bbb Z^k$ actions*, Michigan Math. J.**55**(2007), no. 3, 651–670. MR**2372620**, DOI 10.1307/mmj/1197056461 - Boris Kalinin and Ralf Spatzier,
*On the classification of Cartan actions*, Geom. Funct. Anal.**17**(2007), no. 2, 468–490. MR**2322492**, DOI 10.1007/s00039-007-0602-2 - Anatole Katok and Boris Hasselblatt,
*Introduction to the modern theory of dynamical systems*, Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge University Press, Cambridge, 1995. With a supplementary chapter by Katok and Leonardo Mendoza. MR**1326374**, DOI 10.1017/CBO9780511809187 - A. Katok and J. Lewis,
*Local rigidity for certain groups of toral automorphisms*, Israel J. Math.**75**(1991), no. 2-3, 203–241. MR**1164591**, DOI 10.1007/BF02776025 - Anatole Katok and Ralf J. Spatzier,
*First cohomology of Anosov actions of higher rank abelian groups and applications to rigidity*, Inst. Hautes Études Sci. Publ. Math.**79**(1994), 131–156. MR**1307298**, DOI 10.1007/BF02698888 - Yitzhak Katznelson,
*An introduction to harmonic analysis*, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR**0248482** - Robion C. Kirby and Laurence C. Siebenmann,
*Foundational essays on topological manifolds, smoothings, and triangulations*, Annals of Mathematics Studies, No. 88, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1977. With notes by John Milnor and Michael Atiyah. MR**0645390**, DOI 10.1515/9781400881505 - D. A. Lind,
*Dynamical properties of quasihyperbolic toral automorphisms*, Ergodic Theory Dynam. Systems**2**(1982), no. 1, 49–68. MR**684244**, DOI 10.1017/s0143385700009573 - Anthony Manning,
*There are no new Anosov diffeomorphisms on tori*, Amer. J. Math.**96**(1974), 422–429. MR**358865**, DOI 10.2307/2373551 - Gregory A. Margulis and Nantian Qian,
*Rigidity of weakly hyperbolic actions of higher real rank semisimple Lie groups and their lattices*, Ergodic Theory Dynam. Systems**21**(2001), no. 1, 121–164. MR**1826664**, DOI 10.1017/S0143385701001109 - M. S. Raghunathan,
*Discrete subgroups of Lie groups*, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68, Springer-Verlag, New York-Heidelberg, 1972. MR**0507234**, DOI 10.1007/978-3-642-86426-1 - Federico Rodriguez Hertz,
*Global rigidity of certain abelian actions by toral automorphisms*, J. Mod. Dyn.**1**(2007), no. 3, 425–442. MR**2318497**, DOI 10.3934/jmd.2007.1.425 - Jeffrey Rauch and Michael Taylor,
*Regularity of functions smooth along foliations, and elliptic regularity*, J. Funct. Anal.**225**(2005), no. 1, 74–93. MR**2149919**, DOI 10.1016/j.jfa.2005.03.018 - Sebastian J. Schreiber,
*On growth rates of subadditive functions for semiflows*, J. Differential Equations**148**(1998), no. 2, 334–350. MR**1643183**, DOI 10.1006/jdeq.1998.3471 - A. N. Starkov,
*The first cohomology group, mixing, and minimal sets of the commutative group of algebraic actions on a torus*, J. Math. Sci. (New York)**95**(1999), no. 5, 2576–2582. Dynamical systems. 7. MR**1712745**, DOI 10.1007/BF02169057 - C. T. C. Wall,
*Surgery on compact manifolds*, London Mathematical Society Monographs, No. 1, Academic Press, London-New York, 1970. MR**0431216** - Peter Walters,
*Conjugacy properties of affine transformations of nilmanifolds*, Math. Systems Theory**4**(1970), 327–333. MR**414830**, DOI 10.1007/BF01704076

## Bibliographic Information

**David Fisher**- Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405
- MR Author ID: 684089
- Email: fisherdm@indiana.edu
**Boris Kalinin**- Affiliation: Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802
- MR Author ID: 603534
- Email: bvk102@psu.edu
**Ralf Spatzier**- Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
- Email: spatzier@umich.edu
**James F. Davis**- Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405
- MR Author ID: 194576
- Email: jfdavis@indiana.edu
- Received by editor(s): October 3, 2011
- Received by editor(s) in revised form: November 17, 2011, and May 17, 2012
- Published electronically: August 20, 2012
- Additional Notes: The authors were supported in part by NSF grants DMS-0643546, DMS-1101150 and DMS-0906085

The contributing author was supported in part by NSF grant DMS-1210991 - © Copyright 2012
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: J. Amer. Math. Soc.
**26**(2013), 167-198 - MSC (2010): Primary 37C15, 37C85, 37D20, 53C24; Secondary 42B05
- DOI: https://doi.org/10.1090/S0894-0347-2012-00751-6
- MathSciNet review: 2983009