Livšic theorems and stable ergodicity for group extensions of hyperbolic systems with discontinuities

Ergodic Theory and Dynamical Systems

Volumn 23, Issue 6, 2003, Pages 1867-1889

  Nicol, Matthew ^a   Scott, Andrew ^a  

^a UNIVERSITY OF SURREY   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 7544223716     PISSN: 01433857     EISSN: None     Source Type: Journal    
DOI: 10.1017/S014338570300021X     Document Type: Article

References (26)

1. A. Boyarsky and P. Góra. Laws of chaos: invariant measures and dynamical systems in one dimension. Probability and Its Applications. Birkhäuser, Basel, 1997.
2. R. de la Llave, J. M. Marco and R. Moriyón. Canonical perturbation theory of Anosov systems and regularity results for the Livšic cohomology equation. Ann. Math. 123 (1986), 537-611.
3. M. J. Field. Generating sets for compact semisimple Lie groups. Proc. Amer. Math. Soc. 127(11) (1999), 3361-3365.
4. M. J. Field and W. Parry. Stable ergodicity of skew extensions by compact Lie groups. Topology 38 (1999), 167-187.
5. B. Hunt, T. Sauer and J. Yorke. Prevalence: a translation invariant 'almost every' on infinite-dimensional spaces. Bull. Amer. Math. Soc. 27(2) (1992), 217-238.
6. O. Jenkinson. Smooth cocycle rigidity for expanding maps and an application to Mostow rigidity. Math. Proc. Camb. Phil. Soc. 132 (2002), 439-452.
7. A. Katok. Cocycles, cohomology and combinatorial constructions in ergodic theory. Proc. Symp. Pure Math. 69 (2001), 107-173.
8. A. Katok, J.-M. Strelcyn with the collaboration of F. Ledrappier and F. Przytycki. Invariant Manifolds, Entropy and Billiards; Smooth Maps with Singularities (Lecture Notes in Mathematics, 1222). Springer, New York, 1986.
9. H. B. Keynes and D. Newton. Ergodic measures for nonabelian compact group extensions. Composition Math. 32 (1976), 53-70.
10. T. Krüger and S. Troubetskoy. Markov partitions for non-uniformly hyperbolic systems with singularities. Ergod. Th. & Dynam. Sys. 12 (1992), 487-508.
11. M. Kuranishi. Two element generations on semisimple Lie groups. Kodai. Math. Sem. Rep. 5(6)(1949), 9-10.
12. C. Liverani and M. Wojtkowski. Ergodicity in Hamiltonian systems. Dynamics Reported, Vol. 4. Eds. C. K. R. T. Jones, U. Kirchgraber and H. O. Walters. Springer, Berlin, 1994, pp. 131-200.
13. A. N. Livšic. Cohomology of dynamical systems. Math. USSR Izv. 6(6) (1972), 1278-1301.
14. A. N. Livšic. Homology properties of Y-Systems. Math. Notes 10 (1971), 758-763.
15. M. Nicol and M. Pollicott. Measurable cocycle rigidity for some non-compact groups. Bull. Lond. Math. Soc. 31 (1999), 592-600.
16. M. Nicol and M. Pollicott. Livsic's theorem for semisimple lie groups. Ergod. Th. & Dynam. Sys. 21 (2001), 1501-1509.
17. V. Nitica and A. Török. Regularity of the transfer map for cohomologous cocycles. Ergod. Th. & Dynam. Sys. 18 (1998), 1187-1209.
18. M. S. M. Noorani. Ergodicity and weak-mixing of homogeneous extensions of measure preserving transformations with application to Markov shifts. Mon. Math. 123(2) (1997), 149-170.
19. W. Parry. Skew-products of shifts with a compact Lie group. J. Lond. Math. Soc. 56(2) (1997), 400-404.
20. W. Parry and M. Pollicott. The Livsic cocyle equation for compact Lie group extensions of hyperbolic systems. J. Lond. Math. Soc. 56(2) (1997), 405-416.
21. W. Parry and M. Pollicott. Stability of mixing for toral extensions of hyperbolic systems. Proc. Steklov Inst. 216 (1997), 354-363.
22. Y. Pesin. Lyapunov characteristic exponents and smooth ergodic theory. Russian Math. Surveys 32(4) (1977), 55-114.
23. M. Pollicott and C. Walkden. Livšic theorems for connected lie groups. Trans. Amer. Math. Soc. 353(7) (2001), 2879-2895.
24. M. Pollicott and M. Yuri. Regularity of solutions to the measurable Livšic equation. Trans. Amer. Math. Soc. 351(2) (1999), 559-568.
25. D. Rudolph. Classifying the isometric extensions of Bernoulli shifts. J. d'Analyse Math. 34 (1978), 36-60.
26. P. Walters. Equilibrium states for β-transformations and related transformations. Math. Z. 159 (1978), 65-88.