Möbius Transforms, Cycles and q-triplets in Statistical Mechanics

In the realm of Boltzmann-Gibbs (BG) statistical mechanics and its q-generalisation for complex systems, we analysed sequences of q-triplets, or q-doublets if one of them was the unity, in terms of cycles of successive Möbius transforms of the line preserving unity ([Formula: see text] corresponds to the BG theory). Such transforms have the form [Formula: see text] , where a is a real number; the particular cases [Formula: see text] and [Formula: see text] yield, respectively, [Formula: see text] and [Formula: see text] , currently known as additive and multiplicative dualities. This approach seemingly enables the organisation of various complex phenomena into different classes, named N-complete or incomplete. The classification that we propose here hopefully constitutes a useful guideline in the search, for non-BG systems whenever well described through q-indices, of new possibly observable physical properties.