Livsic theorems for hyperbolic flows

Author:
C. P. Walkden

Journal:
Trans. Amer. Math. Soc. **352** (2000), 1299-1313

MSC (1991):
Primary 58F15; Secondary 22E99

Published electronically:
September 17, 1999

MathSciNet review:
1637106

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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.

**[Bo]**Rufus Bowen,*Symbolic dynamics for hyperbolic flows*, Amer. J. Math.**95**(1973), 429–460. MR**0339281****[BR]**Rufus Bowen and David Ruelle,*The ergodic theory of Axiom A flows*, Invent. Math.**29**(1975), no. 3, 181–202. MR**0380889****[CFS]**I. P. Cornfeld, S. V. Fomin, and Ya. G. Sinaĭ,*Ergodic theory*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 245, Springer-Verlag, New York, 1982. Translated from the Russian by A. B. Sosinskiĭ. MR**832433****[GK]**V. Guillemin and D. Kazhdan,*On the cohomology of certain dynamical systems*, Topology**19**(1980), no. 3, 291–299. MR**579578**, 10.1016/0040-9383(80)90014-2**[HK]**S. Hurder and A. Katok,*Differentiability, rigidity and Godbillon-Vey classes for Anosov flows*, Inst. Hautes Études Sci. Publ. Math.**72**(1990), 5–61 (1991). MR**1087392****[Jo]**Jean-Lin Journé,*On a regularity problem occurring in connection with Anosov diffeomorphisms*, Comm. Math. Phys.**106**(1986), no. 2, 345–351. MR**855316****[KH]**Anatole Katok and Boris Hasselblatt,*Introduction to the modern theory of dynamical systems*, Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge University Press, Cambridge, 1995. With a supplementary chapter by Katok and Leonardo Mendoza. MR**1326374****[Li1]**A. N. Liv\v{s}ic,*Homology properties of Y-systems*, Math. Notes**10**(1971), 758-763.**[Li2]**A. N. Livšic,*Cohomology of dynamical systems*, Izv. Akad. Nauk SSSR Ser. Mat.**36**(1972), 1296–1320 (Russian). MR**0334287****[Ll1]**R. de la Llave,*Smooth conjugacy and S-R-B measures for uniformly and non-uniformly hyperbolic systems*, Comm. Math. Phys.**150**(1992), no. 2, 289–320. MR**1194019****[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), no. 3, 649–662. MR**1452186**, 10.1017/S0143385797079212**[LMM]**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. of Math. (2)**123**(1986), no. 3, 537–611. MR**840722**, 10.2307/1971334**[NP]**M. Nicol and M. Pollicott,*Measurable cocycle rigidity for some non-compact groups*, preprint, UMIST and Manchester, 1997.**[NT1]**Viorel Niţică and Andrei Török,*Regularity results for the solutions of the Livsic cohomology equation with values in diffeomorphism groups*, Ergodic Theory Dynam. Systems**16**(1996), no. 2, 325–333. MR**1389627****[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), 1187-1209. CMP**99:03****[Pa1]**W. Parry,*Skew products of shifts with a compact Lie group*, J. London Math. Soc.**56**(1997), 395-404. CMP**98:06****[Pa2]**W. Parry,*The Liv\v{s}ic periodic point theorem for two non-abelian cocycles*, preprint, Warwick, 1997.**[PP1]**William Parry and Mark Pollicott,*Zeta functions and the periodic orbit structure of hyperbolic dynamics*, Astérisque**187-188**(1990), 268 (English, with French summary). MR**1085356****[PP2]**W. Parry and M. Pollicott,*The Livsic cocycle equation for compact Lie group extensions of hyperbolic systems*, J. London Math. Soc.**56**(1997), 405-416. CMP**98:06****[Ra]**M. Ratner,*Markov partitions for Anosov flows on 𝑛-dimensional manifolds*, Israel J. Math.**15**(1973), 92–114. MR**0339282****[Sh]**Michael Shub,*Global stability of dynamical systems*, Springer-Verlag, New York, 1987. With the collaboration of Albert Fathi and Rémi Langevin; Translated from the French by Joseph Christy. MR**869255****[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 one-dimensional 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/S0002-9947-99-02428-9

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