In [6], Shub and Sullivan posed the problem of finding complete measure theoretic invariants for analytic Lebesgue measure preserving expanding endomorphisms of $S^{1}$. In [2], the author gave necessary and sufficient conditions for two such endomorphisms to be isomorphic. These complete invariants were a
mixture of a topological and measure-theoretic nature. Establishing them required finding a coboundary equation with no obvious method of construction. In this note we use a result by Arteaga to furnish a different set of complete isomorphism invariants, still of a mixed topological and measure-theoretic nature but far more easily checked than the ones established in [2].