  In this paper we consider the space of smooth conjugacy classes of an Anosov
diffeomorphism of the two-torus. The only 2-manifold that supports an Anosov
diffeomorphism is the 2-torus, and Franks and Manning showed that every such
diffeomorphism is topologically conjugate to a linear example, and furthermore,
the eigenvalues at periodic points are a complete smooth invariant. The
question arises: what sets of eigenvalues occur as the Anosov diffeomorphism
ranges over a topological conjugacy class? This question can be reformulated:
what pairs of cohomology classes (one determined by the expanding eigenvalues,
and one by the contracting eigenvalues) occur as the diffeomorphism ranges over
a topological conjugacy class? The purpose of this paper is to answer this
question: all pairs of H\"{o}lder reduced cohomology classes occur.
