Upper box dimension of inhomogeneous attractors of $C^1$ iterated function systems

We prove that the upper box dimension of an inhomogeneous attractor of a $C^1$ iterated function system in ${\mathbb{R}}^d$ is bounded above by the maximum of the singularity dimension of the system and the upper box dimension of the condensation set. This extends the upper bound of Burrell and Fraser (2020) for inhomogeneous self-affine sets to the $C^1$ setting. The covering argument of Burrell and Fraser does not apply directly to nonlinear compositions. We overcome this difficulty by combining their approach with an extension of the recursive covering estimate of Feng and Simon (2023). We also obtain the corresponding bound for orbital sets driven by arbitrary compact subshifts.