Generalized entropies: Rényi and correlation integral approach

We study the generalized Rényi entropies which were introduced in the physics literature. The proper defintion of these entropies needs, in our opinion, further clarification. We show that the Rényi entropies do not contain any new information. On the other hand, we introduce the notion of the correlation entropies, which are not invariants of the dynamical systems in the measure-theoretic sense, but they are invariant under transformations of the state space with bounded distortion. With these correlation entropies we provide a formal definition which can serve as a basis for the results reported in the physics literature.