Estimates of Kolmogorov complexity in approximating cantor sets - Bonanno, C; Chazottes, J-R; Collet, P (Nonlinearity, 2011)
Our aim is to quantify how complex a Cantor set is as we approximate it better and better. We formalize this by asking what is the shortest program, running on a universal Turing machine, which produces this set at the precision [varepsilon] in the sense of Hausdorff distance. This is the Kolmogorov complexity of the approximated Cantor set, which we call the '[varepsilon]-distortion complexity'. How does this quantity behave as [varepsilon] tends to 0? And, moreover, how this behaviour relates to other characteristics of the Cantor set? This is the subject of this work: we estimate this quantity for several types of Cantor sets on the line generated by iterated function systems and exhibit very different behaviours. For instance, the [varepsilon]-distortion complexity of most C super(k) Cantor sets is proven to behave as [varepsilon] super(-)Dkwhere D is its box counting dimension.