  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 computed by constructing adapted metrics on exterior products. This
explains known exact computations of the (global) Lyapunov dimension. Secondly,
we pose and discuss the symmetrization problem related to the computation of
uniform Lyapunov exponents via the Liouville formula. Thirdly, we discuss
ergodic properties and upper semicontinuity of the Lyapunov dimension.
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.
