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

Abstract

Strichartz initiated the study of reverse iterated function systems (RIFSs) on integer lattices and the dimensions of their invariant sets. We develop a dimension theory for RIFSs through forward orbits, first on locally compact complete metric spaces and then for one-dimensional affine systems. For an affine RIFS $\mathcal F=\{f_i(x)=r_i x+b_i\}_{i=1}^m$ on $\mathbb R$, we introduce the semigroup-growth exponent\[ d_{\mathcal F}=\lim_{R\to\infty}\frac{\log\#\{g\in G^*(\mathcal F):|g'|\le R\}}{\log R}, \] where $G^*(\mathcal F)$ is the semigroup of distinct maps generated by $\mathcal F$. We prove that this limit exists and that $\dim_{\mathrm B}K(\mathcal F^{-1})\le d_{\mathcal F}\le s$. Every forward orbit has mass dimension $d_{\mathcal F}$ if it is locally finite and infinity otherwise; its Beurling dimension is $d_{\mathcal F}$ if it is uniformly locally finite and infinity otherwise. These formulas extend to suitably discrete invariant sets. Moreover, $d_{\mathcal F}=s$ if and only if the dual IFS has no exact overlaps, while $d_{\mathcal F}=0$ if and only if $\mathcal F$ is degenerate. Under finite overlaps, we obtain two-sided polynomial orbit-counting estimates. For non-overlapping uniformly locally finite orbits, renewal theory gives a limiting central density in the non-arithmetic case and a multiplicatively periodic asymptotic profile in the arithmetic case. We also determine the lower and discrete Hausdorff dimensions of rational lattice orbits with finite overlaps. Finally, for expansion ratios that are signed powers of a fixed prime $p$, we identify the Euclidean mass and Beurling dimensions of every rational orbit with the $p$-adic box-counting dimensions of the orbit and its attractor, without any separation assumption.