Infinite Ergodic Theory and Non-extensive Entropies

We recapitulate results from the infinite ergodic theory that are relevant to the theory of non-extensive entropies. In particular, we recall that the Lyapunov exponent of the corresponding systems is zero and that the deviation between neighboring trajectories does not necessarily grow polynomially. Nonetheless, as we show, no single quantity can describe this subexponential growth, the generalized q-exponential exp _q being, in particular, ruled out. We also revisit a number of dynamical systems preserving nonfinite ergodic measure.