Abstract
We show $\Bbb C^\infty$ local rigidity for $\mathbb{Z}^k$ $(k\ge 2)$ higher rank partially hyperbolic actions by toral automorphisms, using a generalization of the KAM (Kolmogorov-Arnold-Moser) iterative scheme. We also prove the existence of irreducible genuinely partially hyperbolic higher rank actions on any torus $\mathbb{T}^N$ for any even $N\ge 6$.
-
[Benveniste]
E. J. Benveniste, "Rigidity of isometric lattice actions on compact Riemannian manifolds," Geom. Funct. Anal., vol. 10, iss. 3, pp. 516-542, 2000.@article {Benveniste, MRKEY = {1779610},
AUTHOR = {Benveniste, E. J.},
TITLE = {Rigidity of isometric lattice actions on compact {R}iemannian manifolds},
JOURNAL = {Geom. Funct. Anal.},
FJOURNAL = {Geometric and Functional Analysis},
VOLUME = {10},
YEAR = {2000},
NUMBER = {3},
PAGES = {516--542},
ISSN = {1016-443X},
CODEN = {GFANFB},
MRCLASS = {37C85 (22E40 53C24)},
MRNUMBER = {2001j:37055},
MRREVIEWER = {A. I. Danilenko},
DOI = {10.1007/PL00001627},
ZBLNUMBER = {0970.22012},
} -
[BM] P. E. Blanksby and H. L. Montgomery, "Algebraic integers near the unit circle," Acta Arith., vol. 18, pp. 355-369, 1971.
@article {BM, MRKEY = {0296021},
AUTHOR = {Blanksby, P. E. and Montgomery, H. L.},
TITLE = {Algebraic integers near the unit circle},
JOURNAL = {Acta Arith.},
FJOURNAL = {Polska Akademia Nauk. Instytut Matematyczny. Acta Arithmetica},
VOLUME = {18},
YEAR = {1971},
PAGES = {355--369},
ISSN = {0065-1036},
MRCLASS = {10F25},
MRNUMBER = {45 \#5082},
MRREVIEWER = {L. Carlitz},
ZBLNUMBER = {0221.12003},
} -
[BP] M. Brin and Y. Pesin, "Partially hyperbolic dynamical systems," Math. USSR-Izv., vol. 8, pp. 177-218, 1974.
@article{BP,
author={Brin, M. and Pesin, Y.},
TITLE={Partially hyperbolic dynamical systems},
JOURNAL={Math. USSR-Izv.},
VOLUME={8},
YEAR={1974},
PAGES={177--218},
} -
[camacho] C. Camacho and A. Lins Neto, Geometric Theory of Foliations, Boston, MA: Birkhäuser, 1985.
@book {camacho, MRKEY = {824240},
AUTHOR = {Camacho, C{é}sar and Lins Neto, Alcides},
TITLE = {Geometric Theory of Foliations},
PUBLISHER = {Birkhäuser},
ADDRESS = {Boston, MA},
YEAR = {1985},
PAGES = {vi+205},
ISBN = {0-8176-3139-9},
MRCLASS = {57R30 (58F18)},
MRNUMBER = {87a:57029},
ZBLNUMBER = {0568.57002},
} -
[D-JMD] D. Damjanović, "Central extensions of simple Lie groups and rigidity of some abelian partially hyperbolic algebraic actions," J. Mod. Dyn., vol. 1, iss. 4, pp. 665-688, 2007.
@article {D-JMD, MRKEY = {2342703},
AUTHOR = {Damjanovi{ć},
Danijela},
TITLE = {Central extensions of simple {L}ie groups and rigidity of some abelian partially hyperbolic algebraic actions},
JOURNAL = {J. Mod. Dyn.},
FJOURNAL = {Journal of Modern Dynamics},
VOLUME = {1},
YEAR = {2007},
NUMBER = {4},
PAGES = {665--688},
ISSN = {1930-5311},
MRCLASS = {37C85 (22F30 37A20 37D30 37D40 53C24)},
MRNUMBER = {2008k:37062},
MRREVIEWER = {David Michael Fisher},
ZBLNUMBER = {1139.37017},
} -
[DK] D. Damjanović and A. Katok, "Local rigidity of actions of higher rank abelian groups and KAM method," Electron. Res. Announc. Amer. Math. Soc., vol. 10, pp. 142-154, 2004.
@article {DK, MRKEY = {2119035},
AUTHOR = {Damjanovi{ć},
Danijela and Katok, Anatole},
TITLE = {Local rigidity of actions of higher rank abelian groups and {KAM} method},
JOURNAL = {Electron. Res. Announc. Amer. Math. Soc.},
FJOURNAL = {Electronic Research Announcements of the American Mathematical Society},
VOLUME = {10},
YEAR = {2004},
PAGES = {142--154},
ISSN = {1079-6762},
MRCLASS = {37C85 (37C15 37J40)},
MRNUMBER = {2005k:37044},
MRREVIEWER = {Manfred Einsiedler},
DOI = {10.1090/S1079-6762-04-00139-8},
ZBLNUMBER = {1068.37015},
} -
[DKWCF] D. Damjanović and A. Katok, "Periodic cycle functionals and cocycle rigidity for certain partially hyperbolic $\Bbb R^k$ actions," Discrete Contin. Dyn. Syst., vol. 13, iss. 4, pp. 985-1005, 2005.
@article {DKWCF, MRKEY = {2166714},
AUTHOR = {Damjanovi{ć},
Danijela and Katok, Anatole},
TITLE = {Periodic cycle functionals and cocycle rigidity for certain partially hyperbolic {$\Bbb R\sp k$} actions},
JOURNAL = {Discrete Contin. Dyn. Syst.},
FJOURNAL = {Discrete and Continuous Dynamical Systems. Series A},
VOLUME = {13},
YEAR = {2005},
NUMBER = {4},
PAGES = {985--1005},
ISSN = {1078-0947},
MRCLASS = {37D20 (22F05 37A20 37C40 37C85)},
MRNUMBER = {2006g:37046},
MRREVIEWER = {Viorel Ni{\c{t}}ic{\u{a}}},
DOI = {10.3934/dcds.2005.13.985},
ZBLNUMBER = {1109.37029},
} -
[DK-CR] D. Damjanović and A. Katok, "Local rigidity of restrictions of Weyl chamber flows," C. R. Math. Acad. Sci. Paris, vol. 344, iss. 8, pp. 503-508, 2007.
@article {DK-CR, MRKEY = {2324486},
AUTHOR = {Damjanovi{ć},
Danijela and Katok, Anatole},
TITLE = {Local rigidity of restrictions of {W}eyl chamber flows},
JOURNAL = {C. R. Math. Acad. Sci. Paris},
FJOURNAL = {Comptes Rendus Mathématique. Académie des Sciences. Paris},
VOLUME = {344},
YEAR = {2007},
NUMBER = {8},
PAGES = {503--508},
ISSN = {1631-073X},
MRCLASS = {37C85 (22F30 37A20 37D30 37D40)},
MRNUMBER = {2008k:37061},
MRREVIEWER = {Dmitry Y. Kleinbock},
DOI = {10.1016/j.crma.2007.03.009},
ZBLNUMBER = {1113.37015},
} -
[DK-PHII] D. Damjanović and A. Katok, Local rigidity of partially hyperbolic actions. II. The geometric method and restrictions of Weyl chamber flows on $\mathop{ SL}(n,\mathbb{R})/\Gamma$.
@misc{DK-PHII,
author = {Damjanovi{ć},
Danijela and Katok, Anatole},
TITLE = { Local rigidity of partially hyperbolic actions. II. The geometric method and restrictions of Weyl chamber flows on $\mathop{\rm SL}(n,\mathbb{R})/\Gamma$},
NOTE={preprint},
} -
[DK-uni] D. Damjanović and A. Katok, Local rigidity of certain homogeneous parabolic actions I: A model case, 2010.
@misc{DK-uni,
author = {Damjanovi{ć},
Danijela and Katok, Anatole},
TITLE = {Local rigidity of certain homogeneous parabolic actions I: A model case},
NOTE={preprint},
YEAR={2010},
} -
[EKL] M. Einsiedler, A. Katok, and E. Lindenstrauss, "Invariant measures and the set of exceptions to Littlewood’s conjecture," Ann. of Math., vol. 164, iss. 2, pp. 513-560, 2006.
@article {EKL, MRKEY = {2247967},
AUTHOR = {Einsiedler, Manfred and Katok, Anatole and Lindenstrauss, Elon},
TITLE = {Invariant measures and the set of exceptions to {L}ittlewood's conjecture},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {164},
YEAR = {2006},
NUMBER = {2},
PAGES = {513--560},
ISSN = {0003-486X},
CODEN = {ANMAAH},
MRCLASS = {22F10 (11J83 28A80 28D15 37A15)},
MRNUMBER = {2007j:22032},
MRREVIEWER = {Thomas Ward},
DOI = {10.4007/annals.2006.164.513},
ZBLNUMBER = {1109.22004},
} -
[EinsiedlerTFisher] M. Einsiedler and T. Fisher, "Differentiable rigidity for hyperbolic toral actions," Israel J. Math., vol. 157, pp. 347-377, 2007.
@article{EinsiedlerTFisher,
author={Einsiedler, Manfred and Fisher, T.},
TITLE={Differentiable rigidity for hyperbolic toral actions},
JOURNAL={Israel J. Math.},
VOLUME={157},
YEAR={2007},
PAGES={347--377},
MRNUMBER={2342454},
ZBLNUMBER={1112.37022},
} -
[Fishersurvey] D. Fisher, "Local rigidity of group actions: past, present, future," in Dynamics, Ergodic Theory, and Geometry, Cambridge: Cambridge Univ. Press, 2007, vol. 54, pp. 45-97.
@incollection {Fishersurvey, MRKEY = {2369442},
AUTHOR = {Fisher, David},
TITLE = {Local rigidity of group actions: past, present, future},
BOOKTITLE = {Dynamics, Ergodic Theory, and Geometry},
SERIES = {Math. Sci. Res. Inst. Publ.},
VOLUME = {54},
PAGES = {45--97},
PUBLISHER = {Cambridge Univ. Press},
ADDRESS = {Cambridge},
YEAR = {2007},
MRCLASS = {37C85 (22F50 28D15 37D20)},
MRNUMBER = {2009g:37021},
MRREVIEWER = {Raul Quiroga},
ZBLNUMBER = {1144.22008},
} -
[Fisher] D. Fisher, First cohomology and local rigidity of group actions.
@misc{Fisher,
author = {Fisher, David},
TITLE={First cohomology and local rigidity of group actions},
NOTE={{\it Ann. of Math.},
to appear},
} -
[FisherMargulis] D. Fisher and G. Margulis, "Local rigidity of affine actions of higher rank groups and lattices," Ann. of Math., vol. 170, iss. 1, pp. 67-122, 2009.
@article {FisherMargulis, MRKEY = {2521112},
AUTHOR = {Fisher, David and Margulis, Gregory},
TITLE = {Local rigidity of affine actions of higher rank groups and lattices},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {170},
YEAR = {2009},
NUMBER = {1},
PAGES = {67--122},
ISSN = {0003-486X},
CODEN = {ANMAAH},
MRCLASS = {22Fxx},
MRNUMBER = {2521112},
DOI = {10.4007/annals.2009.170.67},
ZBLNUMBER = {1186.22015},
} -
[GK] M. Guysinsky and A. Katok, "Normal forms and invariant geometric structures for dynamical systems with invariant contracting foliations," Math. Res. Lett., vol. 5, iss. 1-2, pp. 149-163, 1998.
@article {GK, MRKEY = {1618331},
AUTHOR = {Guysinsky, M. and Katok, A.},
TITLE = {Normal forms and invariant geometric structures for dynamical systems with invariant contracting foliations},
JOURNAL = {Math. Res. Lett.},
FJOURNAL = {Mathematical Research Letters},
VOLUME = {5},
YEAR = {1998},
NUMBER = {1-2},
PAGES = {149--163},
ISSN = {1073-2780},
MRCLASS = {58F36 (58F15)},
MRNUMBER = {99b:58215},
MRREVIEWER = {Nantian Qian},
ZBLNUMBER = {0988.37063},
} -
[Hamilton] R. S. Hamilton, "The inverse function theorem of Nash and Moser," Bull. Amer. Math. Soc., vol. 7, iss. 1, pp. 65-222, 1982.
@article {Hamilton, MRKEY = {656198},
AUTHOR = {Hamilton, Richard S.},
TITLE = {The inverse function theorem of {N}ash and {M}oser},
JOURNAL = {Bull. Amer. Math. Soc.},
FJOURNAL = {American Mathematical Society. Bulletin. New Series},
VOLUME = {7},
YEAR = {1982},
NUMBER = {1},
PAGES = {65--222},
ISSN = {0273-0979},
CODEN = {BAMOAD},
MRCLASS = {58C15 (46G05 58D15)},
MRNUMBER = {83j:58014},
MRREVIEWER = {Peter Michor},
DOI = {10.1090/S0273-0979-1982-15004-2},
ZBLNUMBER = {0499.58003},
} -
[HPS] M. W. Hirsch, C. C. Pugh, and M. Shub, Invariant Manifolds, New York: Springer-Verlag, 1977, vol. 583.
@book {HPS, MRKEY = {0501173},
AUTHOR = {Hirsch, M. W. and Pugh, C. C. and Shub, M.},
TITLE = {Invariant Manifolds},
SERIES = {Lecture Notes in Math.},
VOLUME={583},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1977},
PAGES = {ii+149},
MRCLASS = {58F15 (58F10)},
MRNUMBER = {58 \#18595},
MRREVIEWER = {M. C. Irwin},
ZBLNUMBER = {0355.58009},
} -
[Hurder] S. Hurder, "Affine Anosov actions," Michigan Math. J., vol. 40, iss. 3, pp. 561-575, 1993.
@article {Hurder, MRKEY = {1236179},
AUTHOR = {Hurder, Steven},
TITLE = {Affine {A}nosov actions},
JOURNAL = {Michigan Math. J.},
FJOURNAL = {The Michigan Mathematical Journal},
VOLUME = {40},
YEAR = {1993},
NUMBER = {3},
PAGES = {561--575},
ISSN = {0026-2285},
MRCLASS = {58F15 (22E40 58F11)},
MRNUMBER = {94g:58165},
MRREVIEWER = {Alexander Starkov},
DOI = {10.1307/mmj/1029004838},
ZBLNUMBER = {0809.54035},
} -
[KH] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge: Cambridge Univ. Press, 1995, vol. 54.
@book {KH, MRKEY = {1326374},
AUTHOR = {Katok, Anatole and Hasselblatt, Boris},
TITLE = {Introduction to the Modern Theory of Dynamical Systems},
SERIES = {Encyclopedia of Math. Appl.},
VOLUME = {54},
PUBLISHER = {Cambridge Univ. Press},
ADDRESS = {Cambridge},
YEAR = {1995},
PAGES = {xviii+802},
ISBN = {0-521-34187-6},
MRCLASS = {58Fxx (34Cxx 34Dxx 58-01 58F11 58F15)},
MRNUMBER = {96c:58055},
MRREVIEWER = {Edoh Amiran},
ZBLNUMBER = {0878.58020},
} -
[AKSK] A. Katok and S. Katok, "Higher cohomology for abelian groups of toral automorphisms," Ergodic Theory Dynam. Systems, vol. 15, iss. 3, pp. 569-592, 1995.
@article {AKSK, MRKEY = {1336707},
AUTHOR = {Katok, Anatole and Katok, Svetlana},
TITLE = {Higher cohomology for abelian groups of toral automorphisms},
JOURNAL = {Ergodic Theory Dynam. Systems},
FJOURNAL = {Ergodic Theory and Dynamical Systems},
VOLUME = {15},
YEAR = {1995},
NUMBER = {3},
PAGES = {569--592},
ISSN = {0143-3857},
MRCLASS = {58F15 (22E40)},
MRNUMBER = {96k:58167},
MRREVIEWER = {Alexander Starkov},
DOI = {10.1017/S0143385700008531},
ZBLNUMBER = {0851.57039},
} -
[KK] A. Katok and S. Katok, "Higher cohomology for abelian groups of toral automorphisms. II. The partially hyperbolic case, and corrigendum," Ergodic Theory Dynam. Systems, vol. 25, iss. 6, pp. 1909-1917, 2005.
@article {KK, MRKEY = {2183300},
AUTHOR = {Katok, Anatole and Katok, Svetlana},
TITLE = {Higher cohomology for abelian groups of toral automorphisms. {II}. {T}he partially hyperbolic case, and corrigendum},
JOURNAL = {Ergodic Theory Dynam. Systems},
FJOURNAL = {Ergodic Theory and Dynamical Systems},
VOLUME = {25},
YEAR = {2005},
NUMBER = {6},
PAGES = {1909--1917},
ISSN = {0143-3857},
MRCLASS = {37D20 (22E40 37C85)},
MRNUMBER = {2006g:37047},
MRREVIEWER = {Viorel Ni{\c{t}}ic{\u{a}}},
DOI = {10.1017/S0143385705000271},
ZBLNUMBER={1109.57023},
} -
[KKS] A. Katok, S. Katok, and K. Schmidt, "Rigidity of measurable structure for ${\Bbb Z}^d$-actions by automorphisms of a torus," Comment. Math. Helv., vol. 77, iss. 4, pp. 718-745, 2002.
@article {KKS, MRKEY = {1949111},
AUTHOR = {Katok, Anatole and Katok, Svetlana and Schmidt, Klaus},
TITLE = {Rigidity of measurable structure for {${\Bbb Z}\sp d$}-actions by automorphisms of a torus},
JOURNAL = {Comment. Math. Helv.},
FJOURNAL = {Commentarii Mathematici Helvetici},
VOLUME = {77},
YEAR = {2002},
NUMBER = {4},
PAGES = {718--745},
ISSN = {0010-2571},
CODEN = {COMHAX},
MRCLASS = {37A35 (11K55 37A15 37A45 37C85 37D30)},
MRNUMBER = {2003h:37007},
MRREVIEWER = {David Michael Fisher},
DOI = {10.1007/PL00012439},
ZBLNUMBER = {1035.37005},
} -
[KKo] A. Katok and A. Kononenko, "Cocycles’ stability for partially hyperbolic systems," Math. Res. Lett., vol. 3, iss. 2, pp. 191-210, 1996.
@article {KKo, MRKEY = {1386840},
AUTHOR = {Katok, Anatole and Kononenko, A.},
TITLE = {Cocycles' stability for partially hyperbolic systems},
JOURNAL = {Math. Res. Lett.},
FJOURNAL = {Mathematical Research Letters},
VOLUME = {3},
YEAR = {1996},
NUMBER = {2},
PAGES = {191--210},
ISSN = {1073-2780},
MRCLASS = {58F15},
MRNUMBER = {97d:58152},
MRREVIEWER = {Edoh Amiran},
ZBLNUMBER={0853.58082},
} -
[KatokLewis] A. Katok and J. Lewis, "Local rigidity for certain groups of toral automorphisms," Israel J. Math., vol. 75, iss. 2-3, pp. 203-241, 1991.
@article {KatokLewis, MRKEY = {1164591},
AUTHOR = {Katok, Anatole and Lewis, J.},
TITLE = {Local rigidity for certain groups of toral automorphisms},
JOURNAL = {Israel J. Math.},
FJOURNAL = {Israel Journal of Mathematics},
VOLUME = {75},
YEAR = {1991},
NUMBER = {2-3},
PAGES = {203--241},
ISSN = {0021-2172},
CODEN = {ISJMAP},
MRCLASS = {58F10 (22A30 22D40 22E40 58F15 58F30)},
MRNUMBER = {93g:58076},
MRREVIEWER = {Steven E. Hurder},
DOI = {10.1007/BF02776025},
ZBLNUMBER = {0785.22012},
} -
[KL] A. Katok and J. Lewis, "Global rigidity results for lattice actions on tori and new examples of volume-preserving actions," Israel J. Math., vol. 93, pp. 253-280, 1996.
@article {KL, MRKEY = {1380646},
AUTHOR = {Katok, Anatole and Lewis, J.},
TITLE = {Global rigidity results for lattice actions on tori and new examples of volume-preserving actions},
JOURNAL = {Israel J. Math.},
FJOURNAL = {Israel Journal of Mathematics},
VOLUME = {93},
YEAR = {1996},
PAGES = {253--280},
ISSN = {0021-2172},
CODEN = {ISJMAP},
MRCLASS = {22E40 (58F15)},
MRNUMBER = {96k:22021},
MRREVIEWER = {Scot Adams},
DOI = {10.1007/BF02761106},
ZBLNUMBER = {0857.57038},
} -
[KNT] A. Katok, V. Nicticua, and A. Török, "Non-abelian cohomology of abelian Anosov actions," Ergodic Theory Dynam. Systems, vol. 20, iss. 1, pp. 259-288, 2000.
@article {KNT, MRKEY = {1747022},
AUTHOR = {Katok, Anatole and Ni{\c{t}}ic{\u{a}},
Viorel and T{ö}r{ö}k, Andrei},
TITLE = {Non-abelian cohomology of abelian {A}nosov actions},
JOURNAL = {Ergodic Theory Dynam. Systems},
FJOURNAL = {Ergodic Theory and Dynamical Systems},
VOLUME = {20},
YEAR = {2000},
NUMBER = {1},
PAGES = {259--288},
ISSN = {0143-3857},
MRCLASS = {37D20 (22F05 37A20 37C85)},
MRNUMBER = {2001f:37034},
MRREVIEWER = {Dave Witte Morris},
DOI = {10.1017/S0143385700000122},
ZBLNUMBER = {0977.57042},
} -
[KSfirst] A. Katok and R. J. Spatzier, "First cohomology of Anosov actions of higher rank abelian groups and applications to rigidity," Inst. Hautes Études Sci. Publ. Math., iss. 79, pp. 131-156, 1994.
@article {KSfirst, MRKEY = {1307298},
AUTHOR = {Katok, Anatole and Spatzier, Ralf J.},
TITLE = {First cohomology of {A}nosov actions of higher rank abelian groups and applications to rigidity},
JOURNAL = {Inst. Hautes Études Sci. Publ. Math.},
FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
NUMBER = {79},
YEAR = {1994},
PAGES = {131--156},
ISSN = {0073-8301},
CODEN = {PMIHA6},
MRCLASS = {58F15 (22E40)},
MRNUMBER = {96c:58132},
MRREVIEWER = {Alexander Starkov},
URL = {http://www.numdam.org/item?id=PMIHES_1994__79__131_0},
ZBLNUMBER = {0819.58027},
} -
[KSpartial] A. Katok and R. J. Spatzier, "Subelliptic estimates of polynomial differential operators and applications to rigidity of abelian actions," Math. Res. Lett., vol. 1, iss. 2, pp. 193-202, 1994.
@article {KSpartial, MRKEY = {1266758},
AUTHOR = {Katok, Anatole and Spatzier, Ralf J.},
TITLE = {Subelliptic estimates of polynomial differential operators and applications to rigidity of abelian actions},
JOURNAL = {Math. Res. Lett.},
FJOURNAL = {Mathematical Research Letters},
VOLUME = {1},
YEAR = {1994},
NUMBER = {2},
PAGES = {193--202},
ISSN = {1073-2780},
MRCLASS = {35H05 (35D10 57S20 58F18)},
MRNUMBER = {95b:35042},
MRREVIEWER = {Chu Li Fu},
ZBLNUMBER = {0836.57026},
} -
[KS] A. Katok and R. J. Spatzier, "Differential rigidity of Anosov actions of higher rank abelian groups and algebraic lattice actions," Tr. Mat. Inst. Steklova, vol. 216, iss. Din. Sist. i Smezhnye Vopr., pp. 292-319, 1997.
@article {KS, MRKEY = {1632177},
AUTHOR = {Katok, Anatole and Spatzier, Ralf J.},
TITLE = {Differential rigidity of {A}nosov actions of higher rank abelian groups and algebraic lattice actions},
JOURNAL = {Tr. Mat. Inst. Steklova},
FJOURNAL = {Trudy Matematicheskogo Instituta Imeni V. A. Steklova. Rossiĭskaya Akademiya Nauk},
VOLUME = {216},
YEAR = {1997},
NUMBER = {Din. Sist. i Smezhnye Vopr.},
PAGES = {292--319},
ISSN = {0371-9685},
MRCLASS = {58F15 (22E40 58F10)},
MRNUMBER = {99i:58118},
MRREVIEWER = {Raul Quiroga},
ZBLNUMBER = {0938.37010},
} -
[Katz] Y. Katznelson, "Ergodic automorphisms of $T^{n}$ are Bernoulli shifts," Israel J. Math., vol. 10, pp. 186-195, 1971.
@article {Katz, MRKEY = {0294602},
AUTHOR = {Katznelson, Yitzhak},
TITLE = {Ergodic automorphisms of {$T\sp{n}$} are {B}ernoulli shifts},
JOURNAL = {Israel J. Math.},
FJOURNAL = {Israel Journal of Mathematics},
VOLUME = {10},
YEAR = {1971},
PAGES = {186--195},
ISSN = {0021-2172},
MRCLASS = {28A65},
MRNUMBER = {45 \#3672},
MRREVIEWER = {L. Sucheston},
DOI = {10.1007/BF02771569},
ZBLNUMBER = {0219.28014},
} -
[Kronecker] L. Kronecker, "Zwei Sätze über Gleichungen mit ganzzahligen Coefficienten," J. Reine Angew. Math., vol. 53, 1857.
@article{Kronecker,
author={Kronecker, L.},
TITLE={ Zwei Sätze über Gleichungen mit ganzzahligen Coefficienten},
JOURNAL={J. Reine Angew. Math.},
VOLUME={53},
YEAR={1857},
} -
[L] V. F. Lazutkin, KAM Theory and Semiclassical Approximations to Eigenfunctions, New York: Springer-Verlag, 1993, vol. 24.
@book {L, MRKEY = {1239173},
AUTHOR = {Lazutkin, Vladimir F.},
TITLE = {K{AM} Theory and Semiclassical Approximations to Eigenfunctions},
SERIES = {Ergeb. Math. Grenzgeb.},
VOLUME = {24},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1993},
PAGES = {x+387},
ISBN = {3-540-53389-3},
MRCLASS = {58Fxx (35J05 35P05 58F19 58F27 70H05 81Q20)},
MRNUMBER = {94m:58069},
MRREVIEWER = {Helmut R{ü}ssmann},
ZBLNUMBER = {0814.58001},
} -
[dll] R. de la Llave, "A tutorial on KAM theory," in Smooth Ergodic Theory and its Applications, Providence, RI: Amer. Math. Soc., 2001, vol. 69, pp. 175-292.
@incollection {dll, MRKEY = {1858536},
AUTHOR = {de la Llave, Rafael},
TITLE = {A tutorial on {KAM} theory},
BOOKTITLE = {Smooth Ergodic Theory and its Applications},
VENUE={{S}eattle, {WA},
1999},
SERIES = {Proc. Sympos. Pure Math.},
VOLUME = {69},
PAGES = {175--292},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {2001},
MRCLASS = {37J40 (70H08)},
MRNUMBER = {2002h:37123},
MRREVIEWER = {H. Scott Dumas},
ZBLNUMBER = {1055.37064},
} -
[Mane] R. Mañé, "A proof of the $C^1$ stability conjecture," Inst. Hautes Études Sci. Publ. Math., iss. 66, pp. 161-210, 1988.
@article {Mane, MRKEY = {932138},
AUTHOR = {Ma{ñ}{é},
Ricardo},
TITLE = {A proof of the {$C\sp 1$} stability conjecture},
JOURNAL = {Inst. Hautes Études Sci. Publ. Math.},
FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
NUMBER = {66},
YEAR = {1988},
PAGES = {161--210},
ISSN = {0073-8301},
CODEN = {PMIHA6},
MRCLASS = {58F10 (58F15)},
MRNUMBER = {89e:58090},
MRREVIEWER = {Roberto Moriyon},
ZBLNUMBER = {0678.58022},
} -
[MargulisQian] G. A. Margulis and N. Qian, "Rigidity of weakly hyperbolic actions of higher real rank semisimple Lie groups and their lattices," Ergodic Theory Dynam. Systems, vol. 21, iss. 1, pp. 121-164, 2001.
@article {MargulisQian, MRKEY = {1826664},
AUTHOR = {Margulis, Gregory A. and Qian, Nantian},
TITLE = {Rigidity of weakly hyperbolic actions of higher real rank semisimple {L}ie groups and their lattices},
JOURNAL = {Ergodic Theory Dynam. Systems},
FJOURNAL = {Ergodic Theory and Dynamical Systems},
VOLUME = {21},
YEAR = {2001},
NUMBER = {1},
PAGES = {121--164},
ISSN = {0143-3857},
MRCLASS = {22F05 (22E46 37C85 37D20)},
MRNUMBER = {2003a:22019},
DOI = {10.1017/S0143385701001109},
ZBLNUMBER = {0976.22009},
} -
[Matsumoto] H. Matsumoto, "Sur les sous-groupes arithmétiques des groupes semi-simples déployés," Ann. Sci. École Norm. Sup., vol. 2, pp. 1-62, 1969.
@article {Matsumoto, MRKEY = {0240214},
AUTHOR = {Matsumoto, Hideya},
TITLE = {Sur les sous-groupes arithmétiques des groupes semi-simples déployés},
JOURNAL = {Ann. Sci. École Norm. Sup.},
FJOURNAL = {Annales Scientifiques de l'École Normale Supérieure. Quatrième Série},
VOLUME = {2},
YEAR = {1969},
PAGES = {1--62},
ISSN = {0012-9593},
MRCLASS = {20.65 (14.00)},
MRNUMBER = {39 \#1566},
MRREVIEWER = {E. J. Taft},
URL = {http://www.numdam.org/item?id=ASENS_1969_4_2_1_1_0},
ZBLNUMBER = {0261.20025},
} -
[Mie] D. Mieczkowski, "The first cohomology of parabolic actions for some higher-rank abelian groups and representation theory," J. Mod. Dyn., vol. 1, iss. 1, pp. 61-92, 2007.
@article {Mie, MRKEY = {2261072},
AUTHOR = {Mieczkowski, David},
TITLE = {The first cohomology of parabolic actions for some higher-rank abelian groups and representation theory},
JOURNAL = {J. Mod. Dyn.},
FJOURNAL = {Journal of Modern Dynamics},
VOLUME = {1},
YEAR = {2007},
NUMBER = {1},
PAGES = {61--92},
ISSN = {1930-5311},
MRCLASS = {22E40 (22E27 37C85)},
MRNUMBER = {2007i:22015},
MRREVIEWER = {Peter E. Trapa},
ZBLNUMBER = {1119.22009},
} -
[Milnor] J. Milnor, Introduction to Algebraic $K$-Theory, Princeton, N.J.: Princeton Univ. Press, 1971, vol. 72.
@book {Milnor, MRKEY = {0349811},
AUTHOR = {Milnor, John},
TITLE = {Introduction to Algebraic {$K$}-Theory},
SERIES = {Ann. of Math. Studies},
VOLUME={72},
PUBLISHER = {Princeton Univ. Press},
ADDRESS = {Princeton, N.J.},
YEAR = {1971},
PAGES = {xiii+184},
MRCLASS = {18F25 (12A65)},
MRNUMBER = {50 \#2304},
MRREVIEWER = {J.-P. Jouanolou},
ZBLNUMBER = {0237.18005},
} -
[Moser] J. Moser, "On commuting circle mappings and simultaneous Diophantine approximations," Math. Z., vol. 205, iss. 1, pp. 105-121, 1990.
@article {Moser, MRKEY = {1069487},
AUTHOR = {Moser, J{ü}rgen},
TITLE = {On commuting circle mappings and simultaneous {D}iophantine approximations},
JOURNAL = {Math. Z.},
FJOURNAL = {Mathematische Zeitschrift},
VOLUME = {205},
YEAR = {1990},
NUMBER = {1},
PAGES = {105--121},
ISSN = {0025-5874},
CODEN = {MAZEAX},
MRCLASS = {58F11 (11J13 26A18 58F18 58F30)},
MRNUMBER = {92e:58120},
MRREVIEWER = {Autorreferat},
DOI = {10.1007/BF02571227},
ZBLNUMBER = {0689.58031},
} -
[Pesin-PH] Y. B. Pesin, Lectures on Partial Hyperbolicity and Stable Ergodicity, European Mathematical Society (EMS), Zürich, 2004.
@book {Pesin-PH, MRKEY = {2068774},
AUTHOR = {Pesin, Yakov B.},
TITLE = {Lectures on Partial Hyperbolicity and Stable Ergodicity},
SERIES = {Zur. Lect. Adv. Math.},
PUBLISHER = {European Mathematical Society (EMS), Zürich},
YEAR = {2004},
PAGES = {vi+122},
ISBN = {3-03719-003-5},
MRCLASS = {37D30 (37C40 37D10)},
MRNUMBER = {2005j:37039},
MRREVIEWER = {Amie Wilkinson},
DOI = {10.4171/003},
ZBLNUMBER = {1098.37024},
} -
[robbin] J. W. Robbin, "A structural stability theorem," Ann. of Math., vol. 94, pp. 447-493, 1971.
@article {robbin, MRKEY = {0287580},
AUTHOR = {Robbin, J. W.},
TITLE = {A structural stability theorem},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {94},
YEAR = {1971},
PAGES = {447--493},
ISSN = {0003-486X},
MRCLASS = {57.48},
MRNUMBER = {44 \#4783},
MRREVIEWER = {K. R. Meyer},
DOI = {10.2307/1970766},
ZBLNUMBER = {0224.58005},
} -
[RH] F. Rodriguez Hertz, "Stable ergodicity of certain linear automorphisms of the torus," Ann. of Math., vol. 162, iss. 1, pp. 65-107, 2005.
@article {RH, MRKEY = {2201693},
AUTHOR = {Rodriguez Hertz, Federico},
TITLE = {Stable ergodicity of certain linear automorphisms of the torus},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {162},
YEAR = {2005},
NUMBER = {1},
PAGES = {65--107},
ISSN = {0003-486X},
CODEN = {ANMAAH},
MRCLASS = {37D30 (37A25 37C40)},
MRNUMBER = {2007d:37043},
MRREVIEWER = {Ian Melbourne},
DOI = {10.4007/annals.2005.162.65},
ZBLNUMBER = {1098.37028},
} -
[Starkov] 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$)$, vol. 95, iss. 5, pp. 2576-2582, 1999.
@article {Starkov, MRKEY = {1712745},
AUTHOR = {Starkov, A. N.},
TITLE = {The first cohomology group, mixing, and minimal sets of the commutative group of algebraic actions on a torus},
JOURNAL = {J. Math. Sci. $($New York$)$},
FJOURNAL = {Journal of Mathematical Sciences (New York)},
VOLUME = {95},
YEAR = {1999},
NUMBER = {5},
PAGES = {2576--2582},
ISSN = {1072-3374},
CODEN = {JMTSEW},
MRCLASS = {37A20 (28D15 37B05 37C85)},
MRNUMBER = {2000m:37005},
MRREVIEWER = {Dmitry Y. Kleinbock},
ZBLNUMBER = {0984.37004},
} -
[Steinberg] R. Steinberg, "Générateurs, relations et revêtements de groupes algébriques," in Colloq. Théorie des Groupes Algébriques, Librairie Universitaire, Louvain, 1962, pp. 113-127.
@incollection {Steinberg, MRKEY = {0153677},
AUTHOR = {Steinberg, Robert},
TITLE = {Générateurs, relations et revêtements de groupes algébriques},
BOOKTITLE = {Colloq. {T}héorie des {G}roupes {A}lgébriques},
VENUE={{B}ruxelles, 1962},
PAGES = {113--127},
PUBLISHER = {Librairie Universitaire, Louvain},
YEAR = {1962},
MRCLASS = {14.50 (20.75)},
MRNUMBER = {27 \#3638},
MRREVIEWER = {T. Ono},
ZBLNUMBER = {0272.20036},
} -
[Veech] W. A. Veech, "Periodic points and invariant pseudomeasures for toral endomorphisms," Ergodic Theory Dynam. Systems, vol. 6, iss. 3, pp. 449-473, 1986.
@article {Veech, MRKEY = {863205},
AUTHOR = {Veech, William A.},
TITLE = {Periodic points and invariant pseudomeasures for toral endomorphisms},
JOURNAL = {Ergodic Theory Dynam. Systems},
FJOURNAL = {Ergodic Theory and Dynamical Systems},
VOLUME = {6},
YEAR = {1986},
NUMBER = {3},
PAGES = {449--473},
ISSN = {0143-3857},
MRCLASS = {58F11 (58F20)},
MRNUMBER = {87m:58096},
MRREVIEWER = {John C. Kieffer},
DOI = {10.1017/S0143385700003606},
ZBLNUMBER = {0616.28009},
} -
[Zehnder] E. Zehnder, "Generalized implicit function theorems with applications to some small divisor problems. I," Comm. Pure Appl. Math., vol. 28, pp. 91-140, 1975.
@article {Zehnder, MRKEY = {0380867},
AUTHOR = {Zehnder, E.},
TITLE = {Generalized implicit function theorems with applications to some small divisor problems. {I}},
JOURNAL = {Comm. Pure Appl. Math.},
FJOURNAL = {Communications on Pure and Applied Mathematics},
VOLUME = {28},
YEAR = {1975},
PAGES = {91--140},
ISSN = {0010-3640},
MRCLASS = {58C15 (58F05)},
MRNUMBER = {52 \#1764},
MRREVIEWER = {J. Moser},
ZBLNUMBER = {0309.58006},
} -
[ZJW1] Z. J. Wang, "Local rigidity of partially hyperbolic actions," J. Mod. Dyn., vol. 4, pp. 271-327, 2010.
@article{ZJW1,
author={Wang, Z. J.},
TITLE={Local rigidity of partially hyperbolic actions},
JOURNAL={J. Mod. Dyn.},
VOLUME={4},
YEAR={2010},
PAGES={271--327},
ZBLNUMBER={pre05790308},
} -
[ZJW2] Z. J. Wang, "New cases of differentiable rigidity for partially hyperbolic actions: symplectic groups and resonance directions," J. Mod. Dyn..
@article{ZJW2,
author={Wang, Z. J.},
TITLE={New cases of differentiable rigidity for partially hyperbolic actions: symplectic groups and resonance directions},
JOURNAL={J. Mod. Dyn.},
NOTE={to appear},
}
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