  A fundamental problem in the dimension theory of self-affine sets is the
construction of high-dimensional measures which yield sharp lower bounds for
the Hausdorff dimension of the set. A natural strategy for the construction of
such high-dimensional measures is to investigate measures of maximal Lyapunov
dimension; these measures can be alternatively interpreted as equilibrium
states of the singular value function introduced by Falconer. Whilst the
existence of these equilibrium states has been well-known for some years their
structure has remained elusive, particularly in dimensions higher than two. In
this article we give a complete description of the equilibrium states of the
singular value function in the three-dimensional case, showing in particular
that all such equilibrium states must be fully supported. In higher dimensions
we also give a new sufficient condition for the uniqueness of these equilibrium
states. As a corollary, giving a solution to a folklore open question in
dimension three, we prove that for a typical self-affine set in $\mathbb{R}^3$,
removing one of the affine maps which defines the set results in a strict
reduction of the Hausdorff dimension.
