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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

 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

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.
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.  Google Scholar [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. Google Scholar [3] D. Damjanovic and A. Katok, Local Rigidity of Partially Hyperbolic Actions. II. The geometric method and restrictions of Weyl chamber flows on $SL(n, R)$/$\Gamma$,, , ().   Google Scholar [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.  Google Scholar [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.  Google Scholar [6] H. Hu, Some ergodic properties of commuting diffeomorphisms, Ergodic Theory Dynam. Systems, 13 (1993), 73-100. doi: doi:10.1017/S0143385700007215.  Google Scholar [7] B. Kalinin and A. Katok, Invariant measures for actions of higher-rank abelian groups, Proc. Symp. Pure Math, 69 (2001), 593-637.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] B. Kalinin, A. Katok and F. Rodriguez Hertz, Nonuniform measure rigidity,, Annals of Mathematics, ().   Google Scholar [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.  Google Scholar [12] A. Katok and V. Nitica, "Differentiable Rigidity of Higher-Rank Abelian Group Actions I. Introduction and Cocycle Problem,", Cambridge University Press, ().   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [16] F. Ledrappier and J.-S. Xie, Vanishing transverse entropy in smooth ergodic theory,, preprint., ().   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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.  Google Scholar [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. Google Scholar [3] D. Damjanovic and A. Katok, Local Rigidity of Partially Hyperbolic Actions. II. The geometric method and restrictions of Weyl chamber flows on $SL(n, R)$/$\Gamma$,, , ().   Google Scholar [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.  Google Scholar [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.  Google Scholar [6] H. Hu, Some ergodic properties of commuting diffeomorphisms, Ergodic Theory Dynam. Systems, 13 (1993), 73-100. doi: doi:10.1017/S0143385700007215.  Google Scholar [7] B. Kalinin and A. Katok, Invariant measures for actions of higher-rank abelian groups, Proc. Symp. Pure Math, 69 (2001), 593-637.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] B. Kalinin, A. Katok and F. Rodriguez Hertz, Nonuniform measure rigidity,, Annals of Mathematics, ().   Google Scholar [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.  Google Scholar [12] A. Katok and V. Nitica, "Differentiable Rigidity of Higher-Rank Abelian Group Actions I. Introduction and Cocycle Problem,", Cambridge University Press, ().   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [16] F. Ledrappier and J.-S. Xie, Vanishing transverse entropy in smooth ergodic theory,, preprint., ().   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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