Measure and cocycle rigidity for certain nonuniformly hyperbolic actions of higher-rank abelian groups

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July 2010, 4(3): 487-515. doi: 10.3934/jmd.2010.4.487

Measure and cocycle rigidity for certain nonuniformly hyperbolic actions of higher-rank abelian groups

Anatole Katok ^1, and Federico Rodriguez Hertz ^2,

1.	Department of Mathematics, The Pennsylvania State University, University Park, PA 16802
2.	IMERL-Facultad de Ingeniería, Universidad de la República, ulio Herrera y Reissig 565, CC 30, 11300 Montevideo, Uruguay

Received January 2010 Revised August 2010 Published October 2010

Abstract
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We prove absolute continuity of "high-entropy'' hyperbolic invariant measures for smooth actions of higher-rank abelian groups assuming that there are no proportional Lyapunov exponents. For actions on tori and infranilmanifolds the existence of an absolutely continuous invariant measure of this kind is obtained for actions whose elements are homotopic to those of an action by hyperbolic automorphisms with no multiple or proportional Lyapunov exponents. In the latter case a form of rigidity is proved for certain natural classes of cocycles over the action.

Keywords: entropy, nonuniform hyperbolicity, measure rigidity, synchronizing time change., Lyapunov metric.

Mathematics Subject Classification: 37C40, 37D25, 37C8.

Citation: Anatole Katok, Federico Rodriguez Hertz. Measure and cocycle rigidity for certain nonuniformly hyperbolic actions of higher-rank abelian groups. Journal of Modern Dynamics, 2010, 4 (3) : 487-515. doi: 10.3934/jmd.2010.4.487

References:

[1]	L. Barreira and Y. Pesin, "Lyapunov Exponents and Smooth Ergodic Theory," University Lecture Series, 23, AMS, Providence, R.I., 2002.
[2]	L. Barreira and Y. Pesin, "Nonuniform Hyperbolicity: Dynamics of Systems with Nonzero Lyapunov Exponents," Encyclopedia of Mathematics and Its Applications, 115, Cambridge University Press, 2007.
[3]	D. Damjanovic and A. Katok, Local Rigidity of Partially Hyperbolic Actions. II. The geometric method and restrictions of Weyl chamber flows on $SL(n, R)$/$\Gamma$,, , ().
[4]	R. de la Llave, Smooth conjugacy and SRB measures for uniformly and non-uniformly hyperbolic systems, Comm. Math. Phys., 150 (1992), 289-320. doi: doi:10.1007/BF02096662.
[5]	M. Einsiedler and E. Lindenstrauss, Rigidity properties of $\mathbbZ^d$-actions on tori and solenoids, Electron. Res. Announc. Amer. Math. Soc., 9 (2003), 99-110. doi: doi:10.1090/S1079-6762-03-00117-3.
[6]	H. Hu, Some ergodic properties of commuting diffeomorphisms, Ergodic Theory Dynam. Systems, 13 (1993), 73-100. doi: doi:10.1017/S0143385700007215.
[7]	B. Kalinin and A. Katok, Invariant measures for actions of higher-rank abelian groups, Proc. Symp. Pure Math, 69 (2001), 593-637.
[8]	B. Kalinin and A. Katok, Measure rigidity beyond uniform hyperbolicity: Invariant measures for Cartan actions on tori, Journal of Modern Dynamics, 1 (2007), 123-146.
[9]	B. Kalinin, A. Katok and F. Rodriguez Hertz, New progress in nonuniform measure and cocycle rigidity, Electronic Research Announcements in Mathematical Sciences, 15 (2008), 79-92.
[10]	B. Kalinin, A. Katok and F. Rodriguez Hertz, Nonuniform measure rigidity,, Annals of Mathematics, ().
[11]	A. Katok, V. Nitica and A. Török, Non-Abelian cohomology of abelian Anosov actions, Ergod. Th. & Dynam. Syst., 20 (2000), 259-288. doi: doi:10.1017/S0143385700000122.
[12]	A. Katok and V. Nitica, "Differentiable Rigidity of Higher-Rank Abelian Group Actions I. Introduction and Cocycle Problem,", Cambridge University Press, ().
[13]	A. Katok and F. Rodriguez Hertz, Uniqueness of large invariant measures for $\Z^k$ actions with Cartan homotopy data, Journal of Modern Dynamics, 1 (2007), 287-300.
[14]	A. Katok and R. J. Spatzier, First cohomology of Anosov actions of higher-rank abelian groups and applications to rigidity, Publ. Math. IHES, 79 (1994), 131-156. doi: doi:10.1007/BF02698888.
[15]	A. Katok and R. J. Spatzier, Subelliptic estimates of polynomial differential operators and applications to rigidity of abelian actions, Math. Res. Letters, 1 (1994), 193-202.
[16]	F. Ledrappier and J.-S. Xie, Vanishing transverse entropy in smooth ergodic theory,, preprint., ().
[17]	F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms. I. Characterization of measures satisfying Pesin's entropy formula, Annals of Mathematics, 122 (1985), 509-539. doi: doi:10.2307/1971328.
[18]	F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms. Part II: Relations between entropy, exponents and dimension, Annals of Mathematics, 122 (1985), 540-574. doi: doi:10.2307/1971329.
[19]	F. Rodriguez Hertz, M. Rodriguez Hertz, A. Tahzibi and R. Ures, A criterion for ergodicity of non-uniformly hyperbolic diffeomorphisms, Electronic Research Announcements in Mathematical Sciences, 14 (2007), 74-81.

show all references

References:

[1]	L. Barreira and Y. Pesin, "Lyapunov Exponents and Smooth Ergodic Theory," University Lecture Series, 23, AMS, Providence, R.I., 2002.
[2]	L. Barreira and Y. Pesin, "Nonuniform Hyperbolicity: Dynamics of Systems with Nonzero Lyapunov Exponents," Encyclopedia of Mathematics and Its Applications, 115, Cambridge University Press, 2007.
[3]	D. Damjanovic and A. Katok, Local Rigidity of Partially Hyperbolic Actions. II. The geometric method and restrictions of Weyl chamber flows on $SL(n, R)$/$\Gamma$,, , ().
[4]	R. de la Llave, Smooth conjugacy and SRB measures for uniformly and non-uniformly hyperbolic systems, Comm. Math. Phys., 150 (1992), 289-320. doi: doi:10.1007/BF02096662.
[5]	M. Einsiedler and E. Lindenstrauss, Rigidity properties of $\mathbbZ^d$-actions on tori and solenoids, Electron. Res. Announc. Amer. Math. Soc., 9 (2003), 99-110. doi: doi:10.1090/S1079-6762-03-00117-3.
[6]	H. Hu, Some ergodic properties of commuting diffeomorphisms, Ergodic Theory Dynam. Systems, 13 (1993), 73-100. doi: doi:10.1017/S0143385700007215.
[7]	B. Kalinin and A. Katok, Invariant measures for actions of higher-rank abelian groups, Proc. Symp. Pure Math, 69 (2001), 593-637.
[8]	B. Kalinin and A. Katok, Measure rigidity beyond uniform hyperbolicity: Invariant measures for Cartan actions on tori, Journal of Modern Dynamics, 1 (2007), 123-146.
[9]	B. Kalinin, A. Katok and F. Rodriguez Hertz, New progress in nonuniform measure and cocycle rigidity, Electronic Research Announcements in Mathematical Sciences, 15 (2008), 79-92.
[10]	B. Kalinin, A. Katok and F. Rodriguez Hertz, Nonuniform measure rigidity,, Annals of Mathematics, ().
[11]	A. Katok, V. Nitica and A. Török, Non-Abelian cohomology of abelian Anosov actions, Ergod. Th. & Dynam. Syst., 20 (2000), 259-288. doi: doi:10.1017/S0143385700000122.
[12]	A. Katok and V. Nitica, "Differentiable Rigidity of Higher-Rank Abelian Group Actions I. Introduction and Cocycle Problem,", Cambridge University Press, ().
[13]	A. Katok and F. Rodriguez Hertz, Uniqueness of large invariant measures for $\Z^k$ actions with Cartan homotopy data, Journal of Modern Dynamics, 1 (2007), 287-300.
[14]	A. Katok and R. J. Spatzier, First cohomology of Anosov actions of higher-rank abelian groups and applications to rigidity, Publ. Math. IHES, 79 (1994), 131-156. doi: doi:10.1007/BF02698888.
[15]	A. Katok and R. J. Spatzier, Subelliptic estimates of polynomial differential operators and applications to rigidity of abelian actions, Math. Res. Letters, 1 (1994), 193-202.
[16]	F. Ledrappier and J.-S. Xie, Vanishing transverse entropy in smooth ergodic theory,, preprint., ().
[17]	F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms. I. Characterization of measures satisfying Pesin's entropy formula, Annals of Mathematics, 122 (1985), 509-539. doi: doi:10.2307/1971328.
[18]	F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms. Part II: Relations between entropy, exponents and dimension, Annals of Mathematics, 122 (1985), 540-574. doi: doi:10.2307/1971329.
[19]	F. Rodriguez Hertz, M. Rodriguez Hertz, A. Tahzibi and R. Ures, A criterion for ergodicity of non-uniformly hyperbolic diffeomorphisms, Electronic Research Announcements in Mathematical Sciences, 14 (2007), 74-81.

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