TY - JOUR

T1 - Lyapunov exponents of cocycles over non-uniformly hyperbolic systems

AU - Kalinin, Boris

AU - Sadovskaya, Victoria

N1 - Funding Information:
2010 Mathematics Subject Classification. Primary: 37H15, 37D25. Key words and phrases. Cocycles, Lyapunov exponents, non-uniformly hyperbolic systems, hyperbolic measures, periodic orbits. The first author was supported in part by Simons Foundation grant 426243, the second author was supported in part by NSF grant DMS-1301693.

PY - 2018/10

Y1 - 2018/10

N2 - 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 A over f is Hölder continuous and takes values in GL(d, ℝ) or, more generally, in the group of invertible bounded linear operators on a Banach space. For a GL(d, ℝ)-valued cocycle A we prove that the Lyapunov exponents of A with respect to µ can be approximated by the Lyapunov exponents of 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 A with respect to µ, but they cannot always be approximated by the exponents of A on periodic orbits. We prove that they can be approximated in terms of the norms of the return values of A on hyperbolic periodic orbits of f.

AB - 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 A over f is Hölder continuous and takes values in GL(d, ℝ) or, more generally, in the group of invertible bounded linear operators on a Banach space. For a GL(d, ℝ)-valued cocycle A we prove that the Lyapunov exponents of A with respect to µ can be approximated by the Lyapunov exponents of 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 A with respect to µ, but they cannot always be approximated by the exponents of A on periodic orbits. We prove that they can be approximated in terms of the norms of the return values of A on hyperbolic periodic orbits of f.

UR - http://www.scopus.com/inward/record.url?scp=85052021400&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85052021400&partnerID=8YFLogxK

U2 - 10.3934/dcds.2018224

DO - 10.3934/dcds.2018224

M3 - Article

AN - SCOPUS:85052021400

VL - 38

SP - 5105

EP - 5118

JO - Discrete and Continuous Dynamical Systems

JF - Discrete and Continuous Dynamical Systems

SN - 1078-0947

IS - 10

ER -