We discuss the growth of the singular values of symplectic transfer matrices
associated with ergodic discrete Schr\"odinger operators in one dimension, with
scalar and matrix-valued potentials. While for an individual value of the
spectral parameter the rate of exponential growth is almost surely governed by
the Lyapunov exponents, this is not, in general, true simultaneously for all
the values of the parameter. The structure of the exceptional sets is
interesting in its own right, and is also of importance in the spectral
analysis of the operators. We present new results along with amplifications and
generalisations of several older ones, and also list a few open questions.
Here are two sample results. On the negative side, for any square-summable
sequence pnβ there is a residual set of energies in the spectrum on which the
middle singular value (the W-th out of 2W) grows no faster than pnβ1β.
On the positive side, for a large class of cocycles including the i.i.d.\ ones,
the set of energies at which the growth of the singular values is not as given
by the Lyapunov exponents has zero Hausdorff measure with respect to any gauge
function Ο(t) such that Ο(t)/t is integrable at zero.
The employed arguments from the theory of subharmonic functions also yield a
generalisation of the Thouless formula, possibly of independent interest: for
each k, the average of the first k Lyapunov exponents is the logarithmic
potential of a probability measure