In this paper, we prove a version of Livšic theorem for a class of matrix cocycles over a $C^2$ Axiom A flow. As a by-product, an approximative theorem on Lyapunov exponents is also obtained which assets that Lyapunov exponents of a given ergodic measure can be approximated by those of periodic measures.
| [1] |
Arnold L. Random dynamical systems. 1998 Berlin: Springer Monographs in Mathematics. Springer-Verlag
|
| [2] |
Avila A, Kocsard A, Liu X. Livšic theorem for diffeomorphism cocycles. Geom. Funct. Anal.. 2018, 28 4 943-964
|
| [3] |
Bowen R. Periodic orbits for hyperbolic flows. Amer. J. Math.. 1972, 94 1-30
|
| [4] |
Backes L, Poletti M. A Livšic theorem for matrix cocycles over non-uniformly hyperbolic systems. J. Dynam. Different. Equat.. 2019, 31 4 1825-1838
|
| [5] |
Butler C. Measurable rigidity of the cohomological equation for linear cocycles over hyperbolic systems. Israel J. Math.. 2018, 227 1 27-61
|
| [6] |
Cao, Y.,Zou, R. Livšic theorem for matrix cocycles over nonuniformly hyperbolic systems. Stoch. Dyn. 19 , no. 2, 1950010, 12 pp (2019)
|
| [7] |
Cao, Y.,Zou, R. Livšic.: theorems for Banach cocycles: existence and regularity J. Funct. Anal. 280 , no. 5, 108889, (2021)
|
| [8] |
Grabarnik, G. Ya., Guysinsky, M. Livšic theorem for banach rings. Discrete Contin. Dyn. Syst. 37 , no. 8, 4379–4390 (2017)
|
| [9] |
Hirsch M, Palis J, Pugh C, Shub M. Neighborhoods of hyperbolic sets. Invent. Math.. 1969, 9 121-134
|
| [10] |
Katok, A., Hasselblatt, B. (1995) Introduction to the modern theory of dynamical systems. Encyclopedia of Mathematics and its Applications, 54. Cambridge University Press, Cambridge
|
| [11] |
Katok A, Kononenko A. Cocycles’ stability for partially hyperbolic systems. Math. Res. Lett.. 1996, 3 2 191-210
|
| [12] |
Kalinin B. Livšic theorem for matrix cocycles. Ann. Math.. 2011, 173 2 1025-1042
|
| [13] |
Kalinin B. Holonomies and cohomology for cocycles over partially hyperbolic diffeomorphisms. Discrete Contin. Dyn. Syst.. 2016, 36 1 245-259
|
| [14] |
Kocsard A, Potrie R. Livšic theorem for low-dimensional diffeomorphism cocycles. Comment. Math. Helv.. 2016, 91 1 39-64
|
| [15] |
Liu, P., Qian, M. (1995) Smooth ergodic theory of random dynamical systems. Lecture Notes in Mathematics, Springer-Verlag, Berlin
|
| [16] |
Lian Z, Young L-S. Lyapunov exponents, periodic orbits, and horseshoes for semiflows on Hilbert spaces. J. Amer. Math. Soc.. 2012, 25 3 637-665
|
| [17] |
Lian Z, Young L-S. Lyapunov exponents, periodic orbits and horseshoes for mappings of Hilbert spaces. Ann. Henri Poincaré.. 2011, 12 6 1081-1108
|
| [18] |
Livšic AN. Homology properties of Y-systems. Mat. Zametki. 1971, 10 758-763
|
| [19] |
Livšic, A.N.: Cohomology of dynamical systems. Izv. Akad. Nauk SSSR, Ser. Mat. 36 1296–1320 (1972)
|
| [20] |
de la Llave R, Windsor A. Livšic theorems for non-commutative groups including diffeomorphism groups and results on the existence of conformal structures for Anosov systems. Ergodic Theory Dynam. Syst.. 2010, 30 4 1055-1100
|
| [21] |
Navas, A., Ponce, M.: A Livšic type theorem for germs of analytic diffeomorphisms. al structures for Anosov systems. Nonlinearity 26 , no. 1, 297–305 (2013)
|
| [22] |
Pugh C, Shub M. The $\Omega $-stability theorem for flows. Invent. Math.. 1970, 11 150-158
|
| [23] |
Pollicott M, Walkden CP. Livšic theorems for connected Lie groups. Trans. Amer. Math. Soc.. 2001, 353 7 2879-2895
|
| [24] |
Schreiber S. On growth rates of subadditive functions for semiflows. J. Different. Equat.. 1998, 148 2 334-350
|
| [25] |
Wilkinson A. The cohomological equation for partially hyperbolic diffeomorphisms. Astérisque No.. 2013, 358 75-165
|
| [26] |
Walkden CP. Livšic theorems for hyperbolic flows. Trans. Amer. Math. Soc.. 2000, 352 3 1299-1313
|
Funding
National Natural Science Foundation of China(11725105)
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