Abstract
We consider an ergodic invariant measure $\mu$ for a smooth action $\alpha$ of $\mathbb{Z}^k$, $k\ge 2$, on a $(k+1)$-dimensional manifold or for a locally free smooth action of $\mathbb{R}^k$, $k\ge 2$, on a $(2k+1)$-dimensional manifold. We prove that if $\mu$ is hyperbolic with the Lyapunov hyperplanes in general position and if one element in $\mathbb{Z}^k$ has positive entropy, then $\mu$ is absolutely continuous. The main ingredient is absolute continuity of conditional measures on Lyapunov foliations which holds for a more general class of smooth actions of higher rank abelian groups.
-
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@book {BP, MRKEY = {1862379},
AUTHOR = {Barreira, Luis and Pesin, Yakov B.},
TITLE = {Lyapunov Exponents and Smooth Ergodic Theory},
SERIES = {Univ. Lecture Ser.},
VOLUME = {23},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {2002},
PAGES = {xii+151},
ISBN = {0-8218-2921-1},
MRCLASS = {37D25 (34D08 37C40 37D10)},
MRNUMBER = {1862379},
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ZBLNUMBER = {01682963},
} -
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@book {BPbook, MRKEY = {2348606},
AUTHOR = {Barreira, Luis and Pesin, Yakov B.},
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SERIES = {Encyclopedia Math. Appl.},
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PAGES = {xiv+513},
ISBN = {978-0-521-83258-8; 0-521-83258-6},
MRCLASS = {37D25 (34D08 37A25 37C40 37D10)},
MRNUMBER = {2348606},
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ZBLNUMBER = {1144.37002},
} -
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D. Damjanovic and A. Katok, "Local Rigidity of Partially Hyperbolic Actions I. KAM method and $\mathbb{Z}^k$ actions on the torus," Ann. of Math., vol. 172, pp. 1805-1858, 2010.
@article{DK-KAM,
author={Damjanovic, D. and Katok, A.},
TITLE={Local Rigidity of Partially Hyperbolic Actions {I. KAM} method and {$\mathbb{Z}^k$} actions on the torus},
JOURNAL={Ann. of Math.},
VOLUME={172},
PAGES={1805--1858},
YEAR={2010},
MRNUMBER={2726100},
ZBLNUMBER={1209.37017},
DOI={10.4007/annals.2010.172.1805},
} -
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@misc{DK-II,
author={Damjanovic, D. and Katok, A.},
TITLE={Local Rigidity of Partially Hyperbolic Actions. {II}. The geometric method and restrictions of Weyl chamber flows on ${SL}(n, {R})/{\Gamma}$ },
URL={www.math.psu.edu/katok$\_$a/papers.html},
} -
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AUTHOR = {Kalinin, Boris and Katok, Anatole},
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BOOKTITLE = {Smooth Ergodic Theory and its Applications},
VENUE={{S}eattle, {WA},
1999},
SERIES = {Proc. Sympos. Pure Math.},
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ADDRESS = {Providence, RI},
YEAR = {2001},
MRCLASS = {37C85 (37A15 37A25 37C40 37D20 37D25)},
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NOTE={errata {\it J. Mod. Dyn.} {\bf 4} (2010), 207--209},
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NUMBER = {1},
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PAGES = {117--140},
SERIES={Astérisque},
VOLUME={49},
PUBLISHER = {Soc. Math. France},
ADDRESS = {Paris},
YEAR = {1977},
MRCLASS = {28A65 (58F15)},
MRNUMBER = {0492181},
MRREVIEWER = {D. Newton},
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} -
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FJOURNAL = {Discrete and Continuous Dynamical Systems},
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} -
[KaOb] A. Katok, "Two problems in measure rigidity," in Geometric Group Theory, Symplectic Geometry and Hyperbolic Dynamics.
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NOTE={Oberwolfach reports 33-2006, MFO, Oberwolfach, 5-6},
} -
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@book {KH, MRKEY = {1326374},
AUTHOR = {Katok, Anatole and Hasselblatt, Boris},
TITLE = {Introduction to the Modern Theory of Dynamical Systems},
SERIES = {Encyclopedia 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 = {1326374},
MRREVIEWER = {Edoh Amiran},
ZBLNUMBER = {0878.58020},
} -
[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.
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AUTHOR = {Katok, Anatole and Katok, Svetlana and Schmidt, Klaus},
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JOURNAL = {Comment. Math. Helv.},
FJOURNAL = {Commentarii Mathematici Helvetici},
VOLUME = {77},
YEAR = {2002},
NUMBER = {4},
PAGES = {718--745},
ISSN = {0010-2571},
CODEN = {COMHAX},
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ZBLNUMBER = {1035.37005},
} -
[KL]
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 {KL, 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},
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[KN]
A. Katok and V. Nicticua, "Rigidity of higher rank abelian cocycles with values in diffeomorphism groups," Geom. Dedicata, vol. 124, pp. 109-131, 2007.
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AUTHOR = {Katok, Anatole and Ni{\c{t}}ic{\u{a}},
Viorel},
TITLE = {Rigidity of higher rank abelian cocycles with values in diffeomorphism groups},
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YEAR = {2007},
PAGES = {109--131},
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} -
[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},
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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)},
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ZBLNUMBER = {0977.57042},
} -
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AUTHOR = {Katok, Anatole and Rodriguez Hertz, Federico},
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NUMBER = {2},
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NOTE={errata {\it J. Mod. Dyn.} {\bf 4} (2010), 207--209},
} -
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author={Katok, Anatole and Rodriguez Hertz, Federico},
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JOURNAL={J. Mod. Dyn},
VOLUME={4},
YEAR={2010},
PAGES={487--515},
} -
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A. Katok and R. J. Spatzier, "Invariant measures for higher-rank hyperbolic abelian actions," Ergodic Theory Dynam. Systems, vol. 16, pp. 751-778, 1996.
@article {KS3, MRKEY = {1619571},
AUTHOR = {Katok, Anatole and Spatzier, R. J.},
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NOTE={errata {\it Ergodic Theory Dynam. Systems} {\bf 18},
(1998), 503--507; MR1406432},
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DOI = {0.1017/S0143385700009081},
} -
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AUTHOR = {Katok, Anatole and Spatzier, R. J.},
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JOURNAL = {Tr. Mat. Inst. Steklova},
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} -
[S] J. G. Sinaui, "Classical dynamic systems with countably-multiple Lebesgue spectrum. II," Izv. Akad. Nauk SSSR Ser. Mat., vol. 30, pp. 15-68, 1966.
@article {S, MRKEY = {0197684},
AUTHOR = {Sina{\u\i},
Ja. G.},
TITLE = {Classical dynamic systems with countably-multiple {L}ebesgue spectrum. {II}},
JOURNAL = {Izv. Akad. Nauk SSSR Ser. Mat.},
FJOURNAL = {Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya},
NOTE={in {R}ussian; translated in {\it Amer. Math. Soc. Transl. Ser.} {\bf 68} (1968), 64--88},
VOLUME = {30},
YEAR = {1966},
PAGES = {15--68},
ISSN = {0373-2436},
MRCLASS = {28.70 (57.48)},
MRNUMBER = {0197684},
MRREVIEWER = {I. Cuculescu},
} -
[Y]
J. -C. Yoccoz, "Conjugaison différentiable des difféomorphismes du cercle dont le nombre de rotation vérifie une condition diophantienne," Ann. Sci. École Norm. Sup., vol. 17, iss. 3, pp. 333-359, 1984.
@article {Y, MRKEY = {0777374},
AUTHOR = {Yoccoz, J.-C.},
TITLE = {Conjugaison différentiable des difféomorphismes du cercle dont le nombre de rotation vérifie une condition diophantienne},
JOURNAL = {Ann. Sci. École Norm. Sup.},
FJOURNAL = {Annales Scientifiques de l'École Normale Supérieure. Quatrième Série},
VOLUME = {17},
YEAR = {1984},
NUMBER = {3},
PAGES = {333--359},
ISSN = {0012-9593},
CODEN = {ASENAH},
MRCLASS = {58F11},
MRNUMBER = {0777374},
MRREVIEWER = {J. W. Robbin},
URL = {http://www.numdam.org/item?id=ASENS_1984_4_17_3_333_0},
ZBLNUMBER = {0595.57027},
}