Livsic theorems for hyperbolic flows
Author:
C. P. Walkden
Journal:
Trans. Amer. Math. Soc. 352 (2000), 12991313
MSC (1991):
Primary 58F15; Secondary 22E99
Published electronically:
September 17, 1999
MathSciNet review:
1637106
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We consider Hölder cocycle equations with values in certain Lie groups over a hyperbolic flow. We extend Livsic's results that measurable solutions to such equations must, in fact, be Hölder continuous.
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C. P. Walkden, Stable ergodicity of skew products of onedimensional hyperbolic flows, Discrete and Continuous Dynamical Systems, to appear.
 [Bo]
 R. Bowen, Symbolic dynamics for hyperbolic flows, Amer. J. Math. 95 (1973), 429459. MR 49:4041
 [BR]
 R. Bowen and D. Ruelle, The ergodic theory of Axiom flows, Invent. Math. 29 (1975), 181202. MR 52:1786
 [CFS]
 I. P. Cornfeld, S. V. Fomin, and Ya. G. Sinai, Ergodic Theory, Springer, Berlin, 1982. MR 87f:28019
 [GK]
 V. Guillemin and D. Kazhdan, On the cohomology of certain dynamical systems, Topology 19 (1980), 291299. MR 81j:58067
 [HK]
 S. Hurder and A. Katok, Differentiability, rigidity and GodbillonVey classes for Anosov flows, Publ. Math., I.H.E.S. 72 (1990), 561. MR 92b:58179
 [Jo]
 J. L. Journé, On a regularity problem occuring in connection with Anosov diffeomorphisms, Comm. Math. Phys. 106 (1986), 345352. MR 88b:58103
 [KH]
 A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopædia of Math., vol. 54, C.U.P., Cambridge, 1995. MR 96c:58055
 [Li1]
 A. N. Liv\v{s}ic, Homology properties of Ysystems, Math. Notes 10 (1971), 758763.
 [Li2]
 A. N. Liv\v{s}ic, Cohomology of dynamical systems, Math. U.S.S.R., Izv. 36 (1972), 12781301. MR 48:12606
 [Ll1]
 R. de la Llave, Smooth conjugacy and SRB measures for uniformly and nonuniformly hyperbolic systems, Comm. Math. Phys. 150 (1992), 289320. MR 94a:58153
 [Ll2]
 R. de la Llave, Analytic regularity of solutions of Livsic's cohomology equation and some applications to analytic conjugacy of hyperbolic dynamical systems, Ergodic Theory Dynam. Systems 17 (1997), 649662. MR 98d:58136
 [LMM]
 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. 123 (1986), 537611. MR 88h:58091
 [NP]
 M. Nicol and M. Pollicott, Measurable cocycle rigidity for some noncompact groups, preprint, UMIST and Manchester, 1997.
 [NT1]
 V. Ni\c{t}ic\u{a} and A. Török, Regularity results for the solutions of the Livsic cohomology equation with values in diffeomorphism groups, Ergodic Theory Dynam. Systems 16 (1996), 325333. MR 97m:58151
 [NT2]
 V. Ni\c{t}ic\u{a} and A. Török, Regularity of the transfer map for cohomologous cocycles, Ergodic Theory Dynam. Systems 18 (1998), 11871209. CMP 99:03
 [Pa1]
 W. Parry, Skew products of shifts with a compact Lie group, J. London Math. Soc. 56 (1997), 395404. CMP 98:06
 [Pa2]
 W. Parry, The Liv\v{s}ic periodic point theorem for two nonabelian cocycles, preprint, Warwick, 1997.
 [PP1]
 W. Parry and M. Pollicott, Zeta Functions and the Periodic Orbit Structure of Hyperbolic Dynamics, Astérique, vol. 187188, Société Mathématique de France, 1990. MR 92f:58141
 [PP2]
 W. Parry and M. Pollicott, The Livsic cocycle equation for compact Lie group extensions of hyperbolic systems, J. London Math. Soc. 56 (1997), 405416. CMP 98:06
 [Ra]
 M. Ratner, Markov partitions for Anosov flows on dimensional manifolds, Israel J. Math. 15 (1973), 92114. MR 49:4042
 [Sh]
 M. Shub, Global Stability of Dynamical Systems, Springer, Berlin, 1987. MR 87m:58086
 [Wa1]
 C. P. Walkden, Liv\v{s}ic regularity theorems for twisted cocycle equations over hyperbolic systems, J. London Math. Soc., to appear.
 [Wa2]
 C. P. Walkden, Stable ergodicity of skew products of onedimensional hyperbolic flows, Discrete and Continuous Dynamical Systems, to appear.
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Additional Information
C. P. Walkden
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, U.K.
Address at time of publication:
Department of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, U.K.
Email:
cwalkden@ma.man.ac.uk
DOI:
http://dx.doi.org/10.1090/S0002994799024289
PII:
S 00029947(99)024289
Received by editor(s):
October 14, 1997
Published electronically:
September 17, 1999
Additional Notes:
Parts of this paper formed parts of a Ph.D.\ thesis written at Warwick University. Research supported by EPSRC Grant 94004020.
Article copyright:
© Copyright 1999
American Mathematical Society
