Reverse Iterated Function Systems: Density, Dimensions, and $p$-adic Extension

In 1996, Strichartz introduced reverse iterated function systems (RIFS) $\mathcal{F}=\{f_i(x)=r_i x+b_i\}_{i=1}^m$ of expanding mappings on $\mathbb{Z}$ and left the determination of the general dimension formulas of invariant sets as an open problem. In this paper we study the topological and geometric properties as well as the dimensions of the forward orbits generated by such systems, thereby providing a complete solution. We first work in a general locally compact complete metric space to show that the non-empty invariant sets of $\mathcal{F}$ are unions of forward orbits, along with giving necessary and sufficient conditions for their existence. Specialising to the RIFS $\mathcal{F}$ on $\mathbb{R}$, we determine the upper and lower mass dimensions, the Beurling dimension, and the discrete Hausdorff dimension of its forward orbits and invariant sets. Moreover, we establish a fundamental connection with the box-counting and similarity dimensions of the attractor generated by the dual contractive IFS $\mathcal{F}^{-1}=\{f_i^{-1}(x)=r_i^{-1}(x-b_i)\}_{i=1}^m$. Under the assumptions that the orbit is non-overlapping and uniformly discrete, renewal theory yields the precise asymptotic central density: in the non-arithmetic case it converges to an explicitly computable constant, while in the arithmetic case it approaches a multiplicatively periodic function. Finally, an analogous treatment is given for $p$-adic systems, where the mass dimension of a forward orbit equals the $p$-adic box dimension of the corresponding $p$-adic attractor.