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For linear cocycles in Banach spaces, the evolution of length elements in agiven metric can be characterized by what we call infinitesimal growthexponents. Then the maximization procedure of such quantities results in themaximized exponent which provides an upper bound for the largest uniformLyapunov exponent in the metric. Under certain assumptions covering practicalpurposes, we prove that the maximized exponents computed in adapted Lyapunovmetrics, which are not necessarily coercive for continuous-time cocycles, canbe arbitrarily close to the standard largest Lyapunov exponent. For cocycles inHilbert spaces, this gives a variational description of uniform Lyapunovexponents via metrics on exterior products. From this, we give a comprehensivelook at the Liouville trace formula, which is widely used to obtain effectivedimension estimates for global attractors. We also prove that the sum ofuniform Lyapunov exponents and the Lyapunov dimension depend uppersemicontinuously on perturbations of the cocycle. Our results explain knownexact computations of the Lyapunov dimension and the robustness of upperdimension estimates observed in many works.  We use adapted metrics and the trace formula to obtain (to the best of ourknowledge, for the first time) effective dimension estimates for a generalclass of delay equations. In particular, we illustrate the approach by derivingsuch estimates for chaotic attractors arising in the Mackey-Glass equations andperiodically forced Suarez-Schopf delayed oscillator. As the delay value tendsto infinity, the estimates seem to be asymptotically sharp.

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