Return-time L^q-spectrum for equilibrium states with potentials of summable variation

Let be a stationary and ergodic process with joint distribution, where the random variables take values in a finite set. Let be the first time this process repeats its first n symbols of output. It is well known that converges almost surely to the entropy of the process. Refined properties of (large deviations, multifractality, etc) are encoded in the return-time -spectrum defined as provided the limit exists. We consider the case where is distributed according to the equilibrium state of a potential with summable variation, and we prove that where is the topological pressure of, the supremum is taken over all shift-invariant measures, and is the unique solution of. Unexpectedly, this spectrum does not coincide with the -spectrum of, which is, and it does not coincide with the waiting-time -spectrum in general. In fact, the return-time -spectrum coincides with the waiting-time -spectrum if and only if the equilibrium state of is the measure of maximal entropy. As a by-product, we also improve the large deviation asymptotics of.