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October  2018, 38(10): 5105-5118. doi: 10.3934/dcds.2018224

Lyapunov exponents of cocycles over non-uniformly hyperbolic systems

Boris Kalinin and  Victoria Sadovskaya

Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA

Received  December 2017 Published  July 2018

Fund Project: The first author was supported in part by Simons Foundation grant 426243, the second author was supported in part by NSF grant DMS-1301693.

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 $f$.

Keywords: Cocycles, Lyapunov exponents, non-uniformly hyperbolic systems, hyperbolic measures, periodic orbits.
Mathematics Subject Classification: Primary: 37H15, 37D25.
Citation: Boris Kalinin, Victoria Sadovskaya. Lyapunov exponents of cocycles over non-uniformly hyperbolic systems. Discrete & Continuous Dynamical Systems, 2018, 38 (10) : 5105-5118. doi: 10.3934/dcds.2018224
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.  Google Scholar

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S. Gouëzel and A. Karlsson, Subadditive and multiplicative ergodic theorems, Preprint. Google Scholar

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G. Grabarnik and M. Guysinsky, Livšic theorem for Banach rings, Discrete and Continuous Dynamical Systems, 37 (2017), 4379-4390.  doi: 10.3934/dcds.2017187.  Google Scholar

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

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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.  Google Scholar

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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 doi: 10.1017/etds.2017.43.  Google Scholar

[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.  Google Scholar

[9]

A. Katok, Lyapunov exponents, entropy and periodic points of diffeomorphisms, Publ. Math. IHES, 51 (1980), 137-173.   Google Scholar

[10]

V. Oseledets, A multiplicative ergodic theorem. Liapunov characteristic numbers for dynamical systems, Trans. Mosc. Math. Soc., 19 (1968), 179-210.   Google Scholar

[11]

Ya. Pesin, Families of invariant manifolds corresponding to nonzero characteristic exponents, Math. USSR-Izv., 40 (1976), 1261-1305.   Google Scholar

[12]

Ya. Pesin, Characteristic Ljapunov exponents and smooth ergodic theory, Russ. Math. Surv., 32 (1977), 55-112,187.   Google Scholar

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M. Pollicott, Lectures on Ergodic Theory and Pesin Theory on Compact Manifolds, Cambridge University Press, 1993. doi: 10.1017/CBO9780511752537.  Google Scholar

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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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

S. Gouëzel and A. Karlsson, Subadditive and multiplicative ergodic theorems, Preprint. Google Scholar

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

A. Katok, Lyapunov exponents, entropy and periodic points of diffeomorphisms, Publ. Math. IHES, 51 (1980), 137-173.   Google Scholar

[10]

V. Oseledets, A multiplicative ergodic theorem. Liapunov characteristic numbers for dynamical systems, Trans. Mosc. Math. Soc., 19 (1968), 179-210.   Google Scholar

[11]

Ya. Pesin, Families of invariant manifolds corresponding to nonzero characteristic exponents, Math. USSR-Izv., 40 (1976), 1261-1305.   Google Scholar

[12]

Ya. Pesin, Characteristic Ljapunov exponents and smooth ergodic theory, Russ. Math. Surv., 32 (1977), 55-112,187.   Google Scholar

[13]

M. Pollicott, Lectures on Ergodic Theory and Pesin Theory on Compact Manifolds, Cambridge University Press, 1993. doi: 10.1017/CBO9780511752537.  Google Scholar

[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.  Google Scholar

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