# American Institute of Mathematical Sciences

August  2017, 37(8): 4379-4390. doi: 10.3934/dcds.2017187

## Livšic theorem for banach rings

 1 Dept. of Math & Computer Science, St. John's University, Queens, NY, USA 2 Deptartment of Mathematics, The Pennsilvania State University, University Park, PA, USA

Received  November 2016 Revised  May 2017 Published  April 2017

The Livšic Theorem for Hölder continuous cocycles with values in Banach rings is proved. We consider a transitive homeomorphism ${\sigma :X\to X}$ that satisfies the Anosov Closing Lemma and a Hölder continuous map ${a:X\to B^\times}$ from a compact metric space $X$ to the set of invertible elements of some Banach ring $B$. The map $a(x)$ is a coboundary with a Hölder continuous transition function if and only if $a(\sigma^{n-1}p)\ldots a(\sigma p)a(p)$ is the identity for each periodic point $p=\sigma^n p$.

Citation: Genady Ya. Grabarnik, Misha Guysinsky. Livšic theorem for banach rings. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4379-4390. doi: 10.3934/dcds.2017187
##### References:
 [1] H. Bercovici and V. Nitica, A Banach algebra version of the Livšic theorem, Discr. Contin. Dyn. Syst., 4 (1998), 523-534. doi: 10.3934/dcds.1998.4.523. Google Scholar [2] H. Federer, Geometric Measure Theory, Springer, New York, 1969. Google Scholar [3] H. Furstenberg and H. Kesten, Products of random matrices, The Annals of Mathematical Statistics, 31 (1960), 457-469. doi: 10.1214/aoms/1177705909. Google Scholar [4] M. Guysinsky, Livšic Theorem for cocycles with values in the group of diffeomorphisms, preprint, 2013.Google Scholar [5] B. Kalinin, Livšic theorem for matrix cocycles, Annals of Mathematics, 173 (2011), 1025-1042. doi: 10.4007/annals.2011.173.2.11. Google Scholar [6] B. Kalinin and V. Sadovskaya, Periodic approximation of Lyapunov exponents for Banach cocycles, arXiv: 1608.05758.Google Scholar [7] A. Karlsson and G. A. Margulis, A multiplicative ergodic theorem and nonpositively curved spaces, Communications in Mathematical Physics, 208 (1999), 107-123. doi: 10.1007/s002200050750. Google Scholar [8] A. Katok and B. Hasseblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511809187. Google Scholar [9] J. F. C. Kingman, The ergodic theory of subadditive stochastic processes, J. Roy. Statist. Soc. Ser. B, 30 (1968), 499-510. Google Scholar [10] A. Livšic, Certain properties of the homology of Y-systems, Math. Zametki, 10 (1971), 555-564. Google Scholar [11] A. Livšic, Cohomology of dynamical systems, Izv. Akad. Nauk SSSR Ser. Mat., 36 (1972), 1296-1320. Google Scholar [12] R. de la Llave and A. Windsor, Livšic theorems for non-commutative groups including groups of diffeomorphisms.and invariant geometric structures, Ergodic Theory Dynam. Systems, 30 (2010), 1055-1100. doi: 10.1017/S014338570900039X. Google Scholar [13] M. A. Naimark, Normed Rings, Translated from the first Russian edition by Leo F. Boron P. Noordhoff N. V. , Groningen, 1964. Google Scholar [14] V. Nitica and A. Torok, Cohomology of dynamical systems and rigidity of partially hyperbolic actions of higher-rank lattices, Duke Math. J., 79 (1995), 751-810. doi: 10.1215/S0012-7094-95-07920-4. Google Scholar [15] W. Parry, The Livšic periodic point theorem for non-abelian cocycles, Ergodic Theory Dynam. Systems, 19 (1999), 687-701. doi: 10.1017/S0143385799146789. Google Scholar [16] M. Pollicott and C. P. Walkden, Livšic theorems for connected Lie groups, Trans. Amer. Math. Soc., 353 (2001), 2879-2895. doi: 10.1090/S0002-9947-01-02708-8. Google Scholar [17] K. Schmidt, Remarks on Livšic theory for nonabelian cocycles, Ergodic Theory Dynam. Systems, 19 (1999), 703-721. doi: 10.1017/S0143385799146790. Google Scholar [18] S. J. Schreiber, On growth rates of subadditive functions for semiflows, J. Differential Equations, 148 (1998), 334-350. doi: 10.1006/jdeq.1998.3471. Google Scholar [19] L. Zhu, Livšic theorem for cocycles with value in GL(N, $\mathbb{Q}_p$), Ph. D. thesis, The Pennsylvania State University, (2012), 1-54. Google Scholar

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##### References:
 [1] H. Bercovici and V. Nitica, A Banach algebra version of the Livšic theorem, Discr. Contin. Dyn. Syst., 4 (1998), 523-534. doi: 10.3934/dcds.1998.4.523. Google Scholar [2] H. Federer, Geometric Measure Theory, Springer, New York, 1969. Google Scholar [3] H. Furstenberg and H. Kesten, Products of random matrices, The Annals of Mathematical Statistics, 31 (1960), 457-469. doi: 10.1214/aoms/1177705909. Google Scholar [4] M. Guysinsky, Livšic Theorem for cocycles with values in the group of diffeomorphisms, preprint, 2013.Google Scholar [5] B. Kalinin, Livšic theorem for matrix cocycles, Annals of Mathematics, 173 (2011), 1025-1042. doi: 10.4007/annals.2011.173.2.11. Google Scholar [6] B. Kalinin and V. Sadovskaya, Periodic approximation of Lyapunov exponents for Banach cocycles, arXiv: 1608.05758.Google Scholar [7] A. Karlsson and G. A. Margulis, A multiplicative ergodic theorem and nonpositively curved spaces, Communications in Mathematical Physics, 208 (1999), 107-123. doi: 10.1007/s002200050750. Google Scholar [8] A. Katok and B. Hasseblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511809187. Google Scholar [9] J. F. C. Kingman, The ergodic theory of subadditive stochastic processes, J. Roy. Statist. Soc. Ser. B, 30 (1968), 499-510. Google Scholar [10] A. Livšic, Certain properties of the homology of Y-systems, Math. Zametki, 10 (1971), 555-564. Google Scholar [11] A. Livšic, Cohomology of dynamical systems, Izv. Akad. Nauk SSSR Ser. Mat., 36 (1972), 1296-1320. Google Scholar [12] R. de la Llave and A. Windsor, Livšic theorems for non-commutative groups including groups of diffeomorphisms.and invariant geometric structures, Ergodic Theory Dynam. Systems, 30 (2010), 1055-1100. doi: 10.1017/S014338570900039X. Google Scholar [13] M. A. Naimark, Normed Rings, Translated from the first Russian edition by Leo F. Boron P. Noordhoff N. V. , Groningen, 1964. Google Scholar [14] V. Nitica and A. Torok, Cohomology of dynamical systems and rigidity of partially hyperbolic actions of higher-rank lattices, Duke Math. J., 79 (1995), 751-810. doi: 10.1215/S0012-7094-95-07920-4. Google Scholar [15] W. Parry, The Livšic periodic point theorem for non-abelian cocycles, Ergodic Theory Dynam. Systems, 19 (1999), 687-701. doi: 10.1017/S0143385799146789. Google Scholar [16] M. Pollicott and C. P. Walkden, Livšic theorems for connected Lie groups, Trans. Amer. Math. Soc., 353 (2001), 2879-2895. doi: 10.1090/S0002-9947-01-02708-8. Google Scholar [17] K. Schmidt, Remarks on Livšic theory for nonabelian cocycles, Ergodic Theory Dynam. Systems, 19 (1999), 703-721. doi: 10.1017/S0143385799146790. Google Scholar [18] S. J. Schreiber, On growth rates of subadditive functions for semiflows, J. Differential Equations, 148 (1998), 334-350. doi: 10.1006/jdeq.1998.3471. Google Scholar [19] L. Zhu, Livšic theorem for cocycles with value in GL(N, $\mathbb{Q}_p$), Ph. D. thesis, The Pennsylvania State University, (2012), 1-54. Google Scholar
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