Dimensions of surface repellers and attractors of non-linear planar IFSs

Abstract

We establish that for a $C^r$-generic surface repeller ($1 < r \leq \infty$), its Hausdorff and box dimensions are exactly the unique zero of the sub-additive topological pressure function. As an application of our framework, we show that the attractor of a uniformly non-conformal and weakly irreducible planar non-linear iterated function system (IFS) satisfying the strong separation condition (SSC) attains its expected Hausdorff and box dimensions. Furthermore, as a direct consequence of our generic repeller theorem, we deduce that this dimension formula also holds for a generic $C^r$ planar IFS satisfying the SSC. Using this approach, we also extend the dichotomy result for graphs of Weierstrass-type functions of Ren and Shen (2021) by weakening their real-analytic requirement to arbitrary $C^r$ regularity for $r > 1$.