A categorical theory for the discretization of a large class of dynamical systems with variable coefficients is proposed. It is based on the existence of covariant functors between the Rota category of Galois differential algebras and suitable catego
ries of abstract dynamical systems. The integrable maps obtained share with their continuous counterparts a large class of solutions and, in the linear case, the Picard-Vessiot group.
The notion of entropy is ubiquitous both in natural and social sciences. In the last two decades, a considerable effort has been devoted to the study of new entropic forms, which generalize the standard Boltzmann-Gibbs (BG) entropy and are widely app
licable in thermodynamics, quantum mechanics and information theory. In [23], by extending previous ideas of Shannon [38], [39], Khinchin proposed an axiomatic definition of the BG entropy, based on four requirements, nowadays known as the Shannon-Khinchin (SK) axioms. The purpose of this paper is twofold. First, we show that there exists an intrinsic group-theoretical structure behind the notion of entropy. It comes from the requirement of composability of an entropy with respect to the union of two statistically independent subsystems, that we propose in an axiomatic formulation. Second, we show that there exists a simple universal class of admissible entropies. This class contains many well known examples of entropies and infinitely many new ones, a priori multi-parametric. Due to its specific relation with the universal formal group, the new family of entropies introduced in this work will be called the universal-group entropy. A new example of multi-parametric entropy is explicitly constructed.
