Thermodynamic formalism for invariant measures in iterated function systems with overlaps

Abstract

We study images of equilibrium (Gibbs) states for a class of non-invertible transformations associated to conformal iterated function systems (IFSs) with overlaps [Formula: see text]. We prove exact dimensionality for these image measures, and find a dimension formula using their overlap numbers. In particular, we obtain a geometric formula for the dimension of self-conformal measures for IFSs with overlaps, in terms of the overlap numbers. This implies a necessary and sufficient condition for dimension drop. If [Formula: see text] is a self-conformal measure, then [Formula: see text] if and only if the overlap number [Formula: see text]. Examples are also discussed.