Two-scale branching functions and inhomogeneous attractors

Abstract

We introduce the notion of a two-scale branching function associated with an arbitrary metric space, which encodes the lower and upper box dimensions as well as the Assouad spectrum. If the metric space is quasi-doubling, this function is approximately Lipschitz. We fully classify the attainable Lipschitz two-scale branching functions, which gives a new proof of the classification of Assouad spectra due to the second author. We then study inhomogeneous self-conformal sets satisfying standard separation conditions. We show that the two-scale branching function of the attractor is given explicitly in terms of the two-scale branching function of the condensation set and the Hausdorff dimension of the homogeneous attractor. In particular, this gives formulas for the lower box dimension and the Assouad spectrum of the attractor.