July  1998, 4(3): 523-534. doi: 10.3934/dcds.1998.4.523

A Banach algebra version of the Livsic theorem


Department of Mathematics, Indiana University, Bloomington, IN 47405, United States


Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, RO-70700 Bucharest, Romania

Received  May 1997 Revised  August 1997 Published  April 1998

The goal of this paper is to study Banach algebra valued cocycles over Anosov actions. The study of cohomological equations over Anosov diffeomorphisms and flows was started in two influential papers by Livsic ([L1], [L2]), and experienced later a tremendous development. This paper is a continuation of [NT1] and [NT2]. We show here that the techniques used to study cocycles with values in Lie groups, and with values in diffeomorphism groups, can be adapted to Banach algebra valued cocycles. Results of this nature are necessary for the study of extensions of Anosov group actions on infinite dimensional manifolds.
Citation: Hari Bercovici, Viorel Niţică. A Banach algebra version of the Livsic theorem. Discrete and Continuous Dynamical Systems, 1998, 4 (3) : 523-534. doi: 10.3934/dcds.1998.4.523

Genady Ya. Grabarnik, Misha Guysinsky. Livšic theorem for banach rings. Discrete and Continuous Dynamical Systems, 2017, 37 (8) : 4379-4390. doi: 10.3934/dcds.2017187


H. Bercovici, V. Niţică. Cohomology of higher rank abelian Anosov actions for Banach algebra valued cocycles. Conference Publications, 2001, 2001 (Special) : 50-55. doi: 10.3934/proc.2001.2001.50


Cecilia González-Tokman, Anthony Quas. A concise proof of the multiplicative ergodic theorem on Banach spaces. Journal of Modern Dynamics, 2015, 9: 237-255. doi: 10.3934/jmd.2015.9.237


Alex Blumenthal. A volume-based approach to the multiplicative ergodic theorem on Banach spaces. Discrete and Continuous Dynamical Systems, 2016, 36 (5) : 2377-2403. doi: 10.3934/dcds.2016.36.2377


Robert I. McLachlan, Ander Murua. The Lie algebra of classical mechanics. Journal of Computational Dynamics, 2019, 6 (2) : 345-360. doi: 10.3934/jcd.2019017


Richard H. Cushman, Jędrzej Śniatycki. On Lie algebra actions. Discrete and Continuous Dynamical Systems - S, 2020, 13 (4) : 1115-1129. doi: 10.3934/dcdss.2020066


Paul Breiding, Türkü Özlüm Çelik, Timothy Duff, Alexander Heaton, Aida Maraj, Anna-Laura Sattelberger, Lorenzo Venturello, Oǧuzhan Yürük. Nonlinear algebra and applications. Numerical Algebra, Control and Optimization, 2021  doi: 10.3934/naco.2021045


Neşet Deniz Turgay. On the mod p Steenrod algebra and the Leibniz-Hopf algebra. Electronic Research Archive, 2020, 28 (2) : 951-959. doi: 10.3934/era.2020050


Mark Pollicott. Local Hölder regularity of densities and Livsic theorems for non-uniformly hyperbolic diffeomorphisms. Discrete and Continuous Dynamical Systems, 2005, 13 (5) : 1247-1256. doi: 10.3934/dcds.2005.13.1247


Heinz-Jürgen Flad, Gohar Harutyunyan. Ellipticity of quantum mechanical Hamiltonians in the edge algebra. Conference Publications, 2011, 2011 (Special) : 420-429. doi: 10.3934/proc.2011.2011.420


Franz W. Kamber and Peter W. Michor. Completing Lie algebra actions to Lie group actions. Electronic Research Announcements, 2004, 10: 1-10.


Viktor Levandovskyy, Gerhard Pfister, Valery G. Romanovski. Evaluating cyclicity of cubic systems with algorithms of computational algebra. Communications on Pure and Applied Analysis, 2012, 11 (5) : 2023-2035. doi: 10.3934/cpaa.2012.11.2023


Chris Bernhardt. Vertex maps for trees: Algebra and periods of periodic orbits. Discrete and Continuous Dynamical Systems, 2006, 14 (3) : 399-408. doi: 10.3934/dcds.2006.14.399


Sonja Cox, Arnulf Jentzen, Ryan Kurniawan, Primož Pušnik. On the mild Itô formula in Banach spaces. Discrete and Continuous Dynamical Systems - B, 2018, 23 (6) : 2217-2243. doi: 10.3934/dcdsb.2018232


Pengliang Xu, Xiaomin Tang. Graded post-Lie algebra structures and homogeneous Rota-Baxter operators on the Schrödinger-Virasoro algebra. Electronic Research Archive, 2021, 29 (4) : 2771-2789. doi: 10.3934/era.2021013


Yuri Latushkin, Valerian Yurov. Stability estimates for semigroups on Banach spaces. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 5203-5216. doi: 10.3934/dcds.2013.33.5203


Goro Akagi, Mitsuharu Ôtani. Evolution equations and subdifferentials in Banach spaces. Conference Publications, 2003, 2003 (Special) : 11-20. doi: 10.3934/proc.2003.2003.11


José Gómez-Torrecillas, F. J. Lobillo, Gabriel Navarro. Convolutional codes with a matrix-algebra word-ambient. Advances in Mathematics of Communications, 2016, 10 (1) : 29-43. doi: 10.3934/amc.2016.10.29


Oǧul Esen, Hasan Gümral. Geometry of plasma dynamics II: Lie algebra of Hamiltonian vector fields. Journal of Geometric Mechanics, 2012, 4 (3) : 239-269. doi: 10.3934/jgm.2012.4.239


A. S. Dzhumadil'daev. Jordan elements and Left-Center of a Free Leibniz algebra. Electronic Research Announcements, 2011, 18: 31-49. doi: 10.3934/era.2011.18.31

2021 Impact Factor: 1.588


  • PDF downloads (89)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]