Estimates of Kolmogorov Complexity in Approximating Cantor Sets
Résumé
Our aim is to quantify how complex is a Cantor set 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 ε in the sense of Hausdorff distance. This is the Kolmogorov complexity of the approximated Cantor set, that we call the "ε-distortion complexity". How does this quantity behave as ε tends to 0? And, moreover, how does this behaviour relates to other characteristics of the Cantor set? This is the subject of the present work: we estimate this quantity for several types of Cantor sets on the line generated by iterated function systems (IFS's) and exhibit very different behaviours. For instance, the ε-distortion complexity of a C^k Cantor set is proven to behave like ε^{−D/k}, where D is its box counting dimension.
Origine | Fichiers produits par l'(les) auteur(s) |
---|
Loading...