Lyapunov minimizing measures for expanding maps of the circle

We consider the set of maps f is an element of Falpha+ = boolean OR C-beta > alpha(1+beta) of the circle which are covering maps of degree D, expandin g, min(x is an element of S1) f'(x) > 1 and orientation preserving. We are interested in characterizing the set of such maps f which admit a unique f- invariant probability measure A minimizing f In f'd mu over all f-invariant probability measures. We show there exists a set G(+) subset of Falpha+, o pen and dense in the C1+alpha-topology, admitting a unique minimizing measu re supported on a periodic orbit. We also show that, if f admits a minimizi ng measure not supported on a finite set of periodic points, then f is a li mit in the C1+alpha-topology of maps admitting a unique minimizing measure supported on a strictly ergodic set of positive topological entropy. We use in an essential way a sub-cohomological equation to produce the pert urbation. In the context of Lagrangian systems, the analogous equation was introduced by R. Mane and A. Fathi extended it to the all configuration spa ce in [8]. We will also present some results on the set of f-invariant measures A maxi mizing integral A d mu for a fixed C-1-expanding map f and a general potent ial A, not necessarily equal to -ln f'.