Smooth Livšic regularity for piecewise expanding maps
Authors:
Matthew Nicol and Tomas Persson
Journal:
Proc. Amer. Math. Soc. 140 (2012), 905-914
MSC (2010):
Primary 37D50, 37A20; Secondary 37A25
DOI:
https://doi.org/10.1090/S0002-9939-2011-10949-3
Published electronically:
July 11, 2011
MathSciNet review:
2869074
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: We consider the regularity of measurable solutions to the cohomological equation














- 1. J. Aaronson and M. Denker. Local limit theorems for partial sums of stationary sequences generated by Gibbs-Markov maps, Stochast. Dynam. 1 (2001), 193-237. MR 1840194 (2002h:37014)
- 2. J. Aaronson, M. Denker, O. Sarig and R. Zweimüller, Aperiodicity of cocycles and stochastic properties of non-Markov maps, Stoch. Dyn. 4 (2004), 31-62. MR 2069366 (2005e:37015)
- 3. V. Baladi, Positive transfer operators and decay of correlations, Advanced Series in Nonlinear Dynamics 16, World Scientific Publishing, River Edge, NJ, 2000. MR 1793194 (2001k:37035)
- 4. H. Bruin, M. Holland and M. Nicol. Livšic regularity for Markov systems, Ergod. Th. and Dynam. Sys. 25 (2005), 1739-1765. MR 2183291 (2006j:37036)
- 5. R. de la Llave, J. M. Marco and E. Moriyon, Canonical perturbation theory of Anosov systems and regularity results for the Livsic cohomological equation, Annals of Math. (2) 123 (1986), 537-611. MR 840722 (88h:58091)
- 6. D. Fried, The flat-trace asymptotics of a uniform system of contractions, Ergodic Theory Dynam. Systems 15 (1995), no. 6, 1061-1073. MR 1366308 (97c:58124)
- 7. S. Gouëzel, Regularity of coboundaries for nonuniformly expanding maps, Proceedings of the American Mathematical Society 134:2 (2006), 391-401. MR 2176007 (2006g:37006)
- 8. F. Hofbauer and G. Keller, Ergodic properties of invariant measures for piecewise monotonic transformations, Math. Z. 180 (1982), 119-140. MR 656227 (83h:28028)
- 9. F. Hofbauer and G. Keller, Equilibrium states for piecewise monotonic transformations, Ergodic Theory and Dynamical Systems 2 (1982), 23-43. MR 684242 (85f:58069)
- 10. O. Jenkinson. Smooth cocycle rigidity for expanding maps and an application to Mostow rigidity, Math. Proc. Camb. Phil. Soc. 132 (2002), 439-452. MR 1891682 (2002k:37027)
- 11. H. Keynes and D. Newton. Ergodic measures for non-abelian compact group extensions. Compositio Math. 32 (1976), 53-70. MR 0400188 (53:4023)
- 12. C. Liverani, Decay of correlations for piecewise expanding maps, journal of Statistical Physics 78 (1995), 1111-1129. MR 1315241 (96d:58077)
- 13. A. N. Livšic. Cohomology of dynamical systems, Mathematics of the USSR Izvestija 6(6) (1972), 1278-1301. MR 0334287 (48:12606)
- 14. A. N. Livšic. Homology properties of Y-systems, Math. Notes 10 (1971), 758-763. MR 0293669 (45:2746)
- 15. M. Nicol and A. Scott, Livšic theorems and stable ergodicity for group extensions of hyperbolic systems with discontinuities, Ergodic Theory and Dynamical Systems 23 (2003), 1867-1889. MR 2032492 (2004k:37051)
- 16. M. Nicol and M. Pollicott, Measurable cocycle rigidity for some noncompact groups, Bull. Lond. Math. Soc. 31 (1999), 592-600. MR 1703845 (2000k:37004)
- 17. M. Nicol and M. Pollicott, Livšic theorems for semisimple Lie groups, Erg. Th. and Dyn. Sys. 21 (2001), 1501-1509. MR 1855844 (2002h:37048)
- 18. M. Nicol and A. Scott, Livšic theorem and stable ergodicity for group extensions of hyperbolic systems with discontinuities, Ergod. Th. and Dyn. Sys. 23 (2003), 1867-1889. MR 2032492 (2004k:37051)
- 19. M. Noorani, Ergodicity and weak-mixing of homogeneous extensions of measure-preserving transformations with applications to Markov shifts, Monatsh. Math. 123 (1997), 149-170. MR 1430502 (98g:28023)
- 20. W. Parry and M. Pollicott, Zeta functions and closed orbits for hyperbolic systems, Astérisque (Soc. Math. France), 187-188 (1990), 1-268. MR 1085356 (92f:58141)
- 21. W. Parry and M. Pollicott. The Livsic cocycle equation for compact Lie group extensions of hyperbolic systems. J. London Math. Soc. (2) 56 (1997), 405-416. MR 1489146 (99d:58109)
- 22. M. Pollicott and C. Walkden, Livsic theorems for connected Lie groups, Trans. Amer. Math. Soc. 353 (2001), 2879-2895. MR 1828477 (2002c:37045)
- 23. M. Pollicott and M. Yuri, Regularity of solutions to the measurable Livsic equation, Trans. Amer. Math. Soc. 351 (1999), 559-568. MR 1621702 (99d:58110)
- 24. A. Scott, Livšic theorems and the stable ergodicity of compact group extensions of systems with some hyperbolicity, thesis, University of Surrey (2003).
- 25. C. Walkden, Livsic theorems for hyperbolic flows, Trans. Amer. Math. Soc. 352 (2000), 1299-1313. MR 1637106 (2000j:37036)
- 26. C. Walkden, Livsic regularity theorems for twisted cocycle equations over hyperbolic systems, J. London Math. Soc. 61 (2000), 286-300. MR 1745387 (2001i:37048)
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Additional Information
Matthew Nicol
Affiliation:
Department of Mathematics, University of Houston, Houston, Texas 77204-3008
Email:
nicol@math.uh.edu
Tomas Persson
Affiliation:
Institute of Mathematics, Polish Academy of Sciences, ulica Śniadeckich 8, P.O. Box 21, 00-956 Warszawa, Poland
Address at time of publication:
Centre for Mathematical Sciences, Lund University, Box 118, 22 100 Lund, Sweden
Email:
tomasp@maths.lth.se
DOI:
https://doi.org/10.1090/S0002-9939-2011-10949-3
Received by editor(s):
July 23, 2010
Received by editor(s) in revised form:
December 14, 2010
Published electronically:
July 11, 2011
Additional Notes:
The second author was supported by EC FP6 Marie Curie ToK programme CODY
Communicated by:
Bryna Kra
Article copyright:
© Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.