  For statistical systems that violate one of the four Shannon-Khinchin axioms,
entropy takes a more general form than the Boltzmann-Gibbs entropy. The
framework of superstatistics allows one to formulate a maximum entropy
principle with these generalized entropies, making them useful for
understanding distribution functions of non-Markovian or non-ergodic complex
systems. For such systems where the composability axiom is violated there exist
only two ways to implement the maximum entropy principle, one using escort
probabilities, the other not. The two ways are connected through a duality.
Here we show that this duality fixes a unique escort probability, which allows
us to derive a complete theory of the generalized logarithms that naturally
arise from the violation of this axiom. We then show how the functional forms
of these generalized logarithms are related to the asymptotic scaling behavior
of the entropy.
