Variational description of uniform Lyapunov exponents via adapted metrics on exterior products

Abstract: In this work, we provide a comprehensive study on the relation between uniform Lyapunov exponents, Liouville trace formula and adapted metrics for cocycles in Hilbert spaces. Firstly, we prove that uniform Lyapunov exponents can be approximated by constructing adapted metrics on exterior products. Secondly, we develop a general theory of computation in an abstract context; establish a generalized Liouville trace formula; pose and discuss the symmetrization problem related to the computation. Thirdly, we discuss ergodic properties and upper semicontinuity in the context of general subadditive families over a noncompact base. Moreover, we use adapted metrics and the trace formula to obtain, for the first time, effective dimension estimates for a general class of delay equations. In particular, we illustrate the approach by deriving such estimates for chaotic attractors arising in the Mackey-Glass equations and periodically forced Suarez-Schopf delayed oscillator. As the delay value tends to infinity, the estimates seem to be asymptotically sharp.