Fractal information density

Fractal sets are generated by simple generating formulas (iterated functions) and therefore have an almost zero algorithmic (Kolmogorov) complexity. Yet when observed as data with no knowledge of the iterated function, for instance, when observing pixel values of any region of a fractal image, the fractal set is very complex. It has rich and complicated patterns that appear at any arbitrary level of magnification. This suggests that fractal sets have a rich information content despite their essentially zero algorithmic complexity. This highlights a significant gap between algorithmic complexity of sets and their information richness. To explain this, we propose an information-based complexity measure of fractal sets. We extend a well-known notion of compression ratio of general binary sequences to two-dimensional sets and apply it to fractal sets. We introduce a notion of set information density and boundary information density, and as an application, we estimate them for two well-known fractal sets.