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Statistical and deterministic dynamics of maps with memory
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 |
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 $p=\sigma^n p$.
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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