  In topics such as the thermodynamic formalism of linear cocycles, the
dimension theory of self-affine sets, and the theory of random matrix products,
it has often been found useful to assume positivity of the matrix entries in
order to simplify or make feasible certain types of calculation. It is natural
to ask how positivity may be relaxed or generalised in a way which enables
similar calculations to be made in more general contexts. On the one hand one
may generalise by considering almost additive or asymptotically additive
potentials which mimic the properties enjoyed by the logarithm of the norm of a
positive matrix cocycle; on the other hand one may consider matrix cocycles
which are dominated, a condition which includes positive matrix cocycles but is
more general. In this article we explore the relationship between almost
additivity and domination for planar cocycles. We show in particular that a
locally constant linear cocycle in the plane is almost additive if and only if
it is either conjugate to a cocycle of isometries, or satisfies a property
slightly weaker than domination which is introduced in this paper. Applications
to matrix thermodynamic formalism are presented.
