Two-scale branching functions and inhomogeneous attractors
Vilma Orgoványi, Alex Rutar
TL;DR
This work introduces and develops the two-scale branching function $\beta_E$ for metric spaces, linking lower/upper box dimensions with the Assouad spectrum and enabling a unified approach to multi- scale fractal dimensions. It proves that, for quasi-doubling spaces, $\beta_E$ can be approximated by Lipschitz functions and provides a complete classification of attainable Lipschitz branching functions, which in turn yields a new proof of Assouad spectrum classification. The paper then applies this framework to inhomogeneous self-conformal sets, showing that the attractor's two-scale branching function can be explicitly computed from the condensation set and the homogeneous attractor, and deriving exact formulas for the lower box dimension and Assouad spectrum. A key mechanism is the projection to monotone subspaces via $\Phi_h$ and the associated operators $\Gamma$, $\Psi$, and $\Omega_h$, which produce sharp dimension relations such as $\operatorname{dim}^{\theta}_{A} \Lambda = \max\{h, \overline{\operatorname{dim}}^{\theta}_{A} F\}$. The results culminate in a precise dimension formula for inhomogeneous attractors under minimal distortion, with conformal IFSs shown to be minimally distorting, thereby broadening the class of systems for which the theory applies.
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.