Finite flowers and maximizing measures for generic Lipschitz functions on the circle

Given a circle endowed with a doubling map, we consider the problem of maximizing measures for Lipschitz functions. We provide a reduction result for the conjecture stating that in a dense open set in that space the maximizing measure is unique and supported by a periodic orbit. More precisely we show that it follows from a generic finite form in the Conze–Guivarc'h–Mane lemma. This unsolved point is discussed in the last section as well as other open questions.