Low Complexity of Optimizing Measures Over an Expanding Circle Map

In this paper, we prove that for a real analytic expanding circle map, all optimizing measures of a real analytic potential function have zero entropy, unless the potential is cohomologous to a constant. We use the group structure of the symbolic space to solve a transversality problem involved. We also discuss applications to optimizing measures for generic smooth potentials and to Lyapunov optimizing measures.