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

Let (X-k)(k >= 0) be a stationary and ergodic process with joint distribution mu, where the random variables X-k take values in a finite set A. Let R-n be the first time this process repeats its first n symbols of output. It is well known that (1/n) log R-n converges almost surely to the entropy of the process. Refined properties of R-n (large deviations, multifractality, etc) are encoded in the return-time L-q-spectrum defined as R(q) = lim(n) 1/n log integral R(n)(q )d mu (q is an element of R) provided the limit exists. We consider the case where (X-k)(k >= 0) is distributed according to the equilibrium state of a potential phi : A(N) -> R with summable variation, and we prove that R(q) = {P((1 - q)phi) for q >= q(phi)*, sup(eta) integral phi d eta for q < q(phi)*, where P((1 - q)phi) is the topological pressure of (1 - q)phi, the supremum is taken over all shift-invariant measures, and q(phi)* is the unique solution of P((1 - q)phi) = sup(eta) integral phi d eta. Unexpectedly, this spectrum does not coincide with the L-q-spectrum of mu(phi), which is P((1 - q)phi), and it does not coincide with the waiting-time L-q-spectrum in general. In fact, the return-time L-q-spectrum coincides with the waiting-time L-q-spectrum if and only if the equilibrium state of phi is the measure of maximal entropy. As a by-product, we also improve the large deviation asymptotics of (1/n) log R-n.