  We construct the complete set of orders of growth and we define on it the
generalized entropy of a dynamical systems. With this object we provide a
framework where we can study the separation of orbits of a map beyond the scope
of exponential growth. We are going to show that this construction is
particularly useful to study families of dynamical systems with vanishing
entropy. Moreover, we are going to see that the space of orders of growth in
which orbits are separated is wilder than expected. This is going to be
achieved with different types of examples.
