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.

[2]

H. Federer, Geometric Measure Theory, Springer, New York, 1969.

[3]

H. Furstenberg and H. Kesten, Products of random matrices, The Annals of Mathematical Statistics, 31 (1960), 457-469. doi: 10.1214/aoms/1177705909.

[4]

M. Guysinsky, Livšic Theorem for cocycles with values in the group of diffeomorphisms, preprint, 2013.

[5]

B. Kalinin, Livšic theorem for matrix cocycles, Annals of Mathematics, 173 (2011), 1025-1042. doi: 10.4007/annals.2011.173.2.11.

[6]

B. Kalinin and V. Sadovskaya, Periodic approximation of Lyapunov exponents for Banach cocycles, arXiv: 1608.05758.

[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.

[8]

A. Katok and B. Hasseblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511809187.

[9]

J. F. C. Kingman, The ergodic theory of subadditive stochastic processes, J. Roy. Statist. Soc. Ser. B, 30 (1968), 499-510.

[10]

A. Livšic, Certain properties of the homology of Y-systems, Math. Zametki, 10 (1971), 555-564.

[11]

A. Livšic, Cohomology of dynamical systems, Izv. Akad. Nauk SSSR Ser. Mat., 36 (1972), 1296-1320.

[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.

[13]

M. A. Naimark, Normed Rings, Translated from the first Russian edition by Leo F. Boron P. Noordhoff N. V. , Groningen, 1964.

[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.

[15]

W. Parry, The Livšic periodic point theorem for non-abelian cocycles, Ergodic Theory Dynam. Systems, 19 (1999), 687-701. doi: 10.1017/S0143385799146789.

[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.

[17]

K. Schmidt, Remarks on Livšic theory for nonabelian cocycles, Ergodic Theory Dynam. Systems, 19 (1999), 703-721. doi: 10.1017/S0143385799146790.

[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.

[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.

show all references

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.

[2]

H. Federer, Geometric Measure Theory, Springer, New York, 1969.

[3]

H. Furstenberg and H. Kesten, Products of random matrices, The Annals of Mathematical Statistics, 31 (1960), 457-469. doi: 10.1214/aoms/1177705909.

[4]

M. Guysinsky, Livšic Theorem for cocycles with values in the group of diffeomorphisms, preprint, 2013.

[5]

B. Kalinin, Livšic theorem for matrix cocycles, Annals of Mathematics, 173 (2011), 1025-1042. doi: 10.4007/annals.2011.173.2.11.

[6]

B. Kalinin and V. Sadovskaya, Periodic approximation of Lyapunov exponents for Banach cocycles, arXiv: 1608.05758.

[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.

[8]

A. Katok and B. Hasseblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511809187.

[9]

J. F. C. Kingman, The ergodic theory of subadditive stochastic processes, J. Roy. Statist. Soc. Ser. B, 30 (1968), 499-510.

[10]

A. Livšic, Certain properties of the homology of Y-systems, Math. Zametki, 10 (1971), 555-564.

[11]

A. Livšic, Cohomology of dynamical systems, Izv. Akad. Nauk SSSR Ser. Mat., 36 (1972), 1296-1320.

[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.

[13]

M. A. Naimark, Normed Rings, Translated from the first Russian edition by Leo F. Boron P. Noordhoff N. V. , Groningen, 1964.

[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.

[15]

W. Parry, The Livšic periodic point theorem for non-abelian cocycles, Ergodic Theory Dynam. Systems, 19 (1999), 687-701. doi: 10.1017/S0143385799146789.

[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.

[17]

K. Schmidt, Remarks on Livšic theory for nonabelian cocycles, Ergodic Theory Dynam. Systems, 19 (1999), 703-721. doi: 10.1017/S0143385799146790.

[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.

[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.

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