Authors’ introduction: We connect the notion of a weakly Gibbsian state, as has recently emerged from the statistical mechanical study of certain lattice spin systems, with the concept of intermittency, as modelled by Manneville-Pomeau maps.
Weakly Gibbsian states were introduced by R. L. Dobrushin in his last conference talk in Renkum [A Gibbsian representation for non-Gibbsian fields, Workshop on Probability and Physics, Renkum (1995)]. What was sought was a Gibbsian restoration of certain physically relevant examples of non-Gibbsian states? A first part of the Dobrushin programme has been recently completed in [C. Maes, F. Redig, S. Shlosman and A. Van Moffaert, Commun. Math. Phys. 209, 517-545 (2000; Zbl 0945.60098)] where it is shown that essentially all restrictions to a sublattice of the low-temperature phases in the realm of the Pirogov-Sinai theory for lattice spin systems are weakly Gibbsian. The typical scenario is the occurrence of a ‘configuration-dependent range of the interaction’. This implies that the relative energies are no longer uniformly bounded (as is the case for the usual Gibbsian set-up) but can be unbounded as dictated by configuration-dependent length scales. This divides the set of lattice spin configurations into two disjoint sets: the ‘good’ ones for which the effective interaction is short range, and the ‘bad’ ones, for which the total interaction is diverging. Instead of introducing the somewhat abstract formalism defining weakly Gibbsian states, we refer to C. Maes, F. Redig, A. Van Moffaert and K. U. Leuven [Stochastic Processes Appl. 79, 1-15 (1999; Zbl 0963.60094)] for general definitions and properties and we only underline the above via a concrete and, for our purposes, illustrative example.