Free energies as invariants of Teichmüller like structures

A Teichmüller like structure on the space of d-degree holomorphic maps on the circle S1, marked by conjugations to the map z $mapsto$ zd, can be defined. Here we introduce a definition of free energy associated to this kind of dynamics as an invariant of equivalence classes in the Teichmüller space. This quantity encodes a length spectrum of rotation cycles in S1.