  In this article, we continue the structural study of factor maps betweeen
symbolic dynamical systems and the relative thermodynamic formalism. Here, one
is studying a factor map from a shift of finite type $X$ (equipped with a
potential function) to a sofic shift $Z$, equipped with a shift-invariant
measure $\nu$. We study relative equilibrium states, that is shift-invariant
measures on $X$ that push forward under the factor map to $\nu$ which maximize
the relative pressure: the relative entropy plus the integral of $\phi$.
  In the non-relative case (where $Z$ is the one point shift and the factor map
is trivial), these measures have a very broad range of application: in
hyperbolic dynamics, information theory, geometry, Teichm\"uller theory and
elsewhere). Relative equilibrium states have also been shown to arise naturally
in some contexts in geometric measure theory as a description of measures
achieving the Hausdorff dimension in ambient spaces. Previous articles have
identified relative versions of well-known notions of degree appearing in
one-dimensional symbolic settings, and established bounds in terms of these on
the number of ergodic relative equilibrium states.
  In this paper, we establish a new connection to multiplicative ergodic theory
by relating these factor triples to a cocycle of Ruelle Perron-Frobenius
operators, and showing that the principal Lyapunov exponent of this cocycle is
the relative pressure; and the dimension of the leading Oseledets space is
equal to the number of measures of relative maximal entropy, counted with a
previously-identified concept of multiplicity.
