Tilings from tops of overlapping iterated function systems

The top of the attractor of a hyperbolic iterated function system \bigl\{f_{i}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}| i=1,2,\ldots,M \bigr\} is defined and used to extend self-similar tilings to overlapping systems. The theory provides sequences of approximate supertiles that converge to tilings. Individual tiles in a tiling are limits of nested decreasing sequences of approximate tiles. Examples include systems of finite type, tilings related to aperiodic monotiles, and ones where there are infinitely many distinct but related prototiles.