On the exponential stability and hyperbolicity of linear cocycles

Abstract

We prove that any uniformly exponentially stable linear cocycle of matrices defined over a topological dynamical system can be reduced via suitable change of variables to a linear cocycle whose generator has a spectral norm less than 1 at each point. Moreover, we establish an analogous result for nonuniformly exponentially stable linear cocycles, i.e. cocycles with negative Lyapunov exponents. In addition, by using tools from ergodic theory, we obtain converse results that give conditions for (non)uniform exponential stability of linear cocyles under the assumption that they admit a suitable reduction to a linear cocycle whose generator has a spectral norm less than 1 at points that form a ‘large’ subset of our topological dynamical system in the appropriate sense. Finally, we discuss the versions of our results in the case of (non)uniformly hyperbolic linear cocyles.

COMMUNICATED BY:

• B. Kuzma

Keywords:

• Linear cocycle
• exponential stability
• hyperbolicity

2010 Mathematics Subject Classifications:

• Primary: 37D20
• 37D25

Acknowledgements

I would like to thank anonymous referees of my paper [4] since their comments and suggestions heavily influenced results and the structure of the present paper. I am also indebted to the anonymous referees and the handling editor of the present paper whose suggestions helped me to improve the presentation of the results.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The author was supported by the Croatian Science Foundation (Hrvatska Zaklada za Znanost) under the project IP-2014-09-2285 and by the University of Rijeka under the project number 17.15.2.2.01