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Lyapunov exponents of cocycles over non-uniformly hyperbolic systems
Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA |
We consider linear cocycles over non-uniformly hyperbolic dynamical systems. The base system is a diffeomorphism $f$ of a compact manifold $X$ preserving a hyperbolic ergodic probability measure $μ$. The cocycle $\mathcal{A}$ over $f$ is Hölder continuous and takes values in $GL(d, \mathbb{R})$ or, more generally, in the group of invertible bounded linear operators on a Banach space. For a $GL(d, \mathbb{R})$-valued cocycle $\mathcal{A}$ we prove that the Lyapunov exponents of $\mathcal{A}$ with respect to $μ$ can be approximated by the Lyapunov exponents of $\mathcal{A}$ with respect to measures on hyperbolic periodic orbits of $f$. In the infinite-dimensional setting one can define the upper and lower Lyapunov exponents of $\mathcal{A}$ with respect to $μ$, but they cannot always be approximated by the exponents of $\mathcal{A}$ on periodic orbits. We prove that they can be approximated in terms of the norms of the return values of $\mathcal{A}$ on hyperbolic periodic orbits of $f$.
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L. Backes, On the periodic approximation of Lyapunov exponents for semi-invertible cocycles,
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B. Kalinin,
Livšic theorem for matrix cocycles, Annals of Math., 173 (2011), 1025-1042.
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B. Kalinin and V. Sadovskaya,
Cocycles with one exponent over partially hyperbolic systems, Geometriae Dedicata, 167 (2013), 167-188.
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B. Kalinin and V. Sadovskaya, Periodic approximation of Lyapunov exponents for Banach cocycles,
Ergodic Theory and Dynamical Systems, 2017, arXiv: 1608.05757
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A multiplicative ergodic theorem and nonpositively curved spaces, Communications in Mathematical Physics, 208 (1999), 107-123.
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A. Katok,
Lyapunov exponents, entropy and periodic points of diffeomorphisms, Publ. Math. IHES, 51 (1980), 137-173.
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V. Oseledets,
A multiplicative ergodic theorem. Liapunov characteristic numbers for dynamical systems, Trans. Mosc. Math. Soc., 19 (1968), 179-210.
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Ya. Pesin,
Families of invariant manifolds corresponding to nonzero characteristic exponents, Math. USSR-Izv., 40 (1976), 1261-1305.
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Ya. Pesin,
Characteristic Ljapunov exponents and smooth ergodic theory, Russ. Math. Surv., 32 (1977), 55-112,187.
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M. Pollicott,
Lectures on Ergodic Theory and Pesin Theory on Compact Manifolds, Cambridge University Press, 1993.
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Z. Wang and W. Sun,
Lyapunov exponents of hyperbolic measures and hyperbolic periodic orbits, Trans. Amer. Math. Soc., 362 (2010), 4267-4282.
doi: 10.1090/S0002-9947-10-04947-0. |
show all references
References:
[1] |
L. Backes, On the periodic approximation of Lyapunov exponents for semi-invertible cocycles,
Discrete Contin. Dyn. Syst., 37 (2017), 6353–6368, arXiv: 1612.04159.
doi: 10.3934/dcds.2017275. |
[2] |
L. Barreira and Ya. Pesin,
Nonuniform Hyperbolicity: Dynamics Of Systems with Nonzero Lyapunov Exponents, Encyclopedia of Mathematics and Its Applications 115 Cambridge University Press, 2007.
doi: 10.1017/CBO9781107326026. |
[3] |
S. Gouëzel and A. Karlsson, Subadditive and multiplicative ergodic theorems, Preprint. |
[4] |
G. Grabarnik and M. Guysinsky,
Livšic theorem for Banach rings, Discrete and Continuous Dynamical Systems, 37 (2017), 4379-4390.
doi: 10.3934/dcds.2017187. |
[5] |
B. Kalinin,
Livšic theorem for matrix cocycles, Annals of Math., 173 (2011), 1025-1042.
doi: 10.4007/annals.2011.173.2.11. |
[6] |
B. Kalinin and V. Sadovskaya,
Cocycles with one exponent over partially hyperbolic systems, Geometriae Dedicata, 167 (2013), 167-188.
doi: 10.1007/s10711-012-9808-z. |
[7] |
B. Kalinin and V. Sadovskaya, Periodic approximation of Lyapunov exponents for Banach cocycles,
Ergodic Theory and Dynamical Systems, 2017, arXiv: 1608.05757
doi: 10.1017/etds.2017.43. |
[8] |
A. Karlsson and G. Margulis,
A multiplicative ergodic theorem and nonpositively curved spaces, Communications in Mathematical Physics, 208 (1999), 107-123.
doi: 10.1007/s002200050750. |
[9] |
A. Katok,
Lyapunov exponents, entropy and periodic points of diffeomorphisms, Publ. Math. IHES, 51 (1980), 137-173.
|
[10] |
V. Oseledets,
A multiplicative ergodic theorem. Liapunov characteristic numbers for dynamical systems, Trans. Mosc. Math. Soc., 19 (1968), 179-210.
|
[11] |
Ya. Pesin,
Families of invariant manifolds corresponding to nonzero characteristic exponents, Math. USSR-Izv., 40 (1976), 1261-1305.
|
[12] |
Ya. Pesin,
Characteristic Ljapunov exponents and smooth ergodic theory, Russ. Math. Surv., 32 (1977), 55-112,187.
|
[13] |
M. Pollicott,
Lectures on Ergodic Theory and Pesin Theory on Compact Manifolds, Cambridge University Press, 1993.
doi: 10.1017/CBO9780511752537. |
[14] |
Z. Wang and W. Sun,
Lyapunov exponents of hyperbolic measures and hyperbolic periodic orbits, Trans. Amer. Math. Soc., 362 (2010), 4267-4282.
doi: 10.1090/S0002-9947-10-04947-0. |
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