Estimates of Kolmogorov complexity in approximating cantor sets

• Published in: Nonlinearity Vol. 24; no. 2; pp. 459 - 479
• Main Authors: Bonanno, C, Chazottes, J-R, Collet, P
• Format: Journal Article
• Language: English
• Published: Bristol IOP Publishing 01.02.2011; Institute of Physics
• Subjects: Approximation; Complexity; Counting; Dynamical Systems; Estimates; Exact sciences and technology; Global analysis, analysis on manifolds; Mathematical methods in physics; Mathematics; Metric Geometry; Nonlinearity; Other topics in mathematical methods in physics; Physics; Probability; Running; Sciences and techniques of general use; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds; Turing machines; Cantor set; On line; Nonlinear analysis; Turing machine; Kolmogorov complexity; Counting; Mathematical physics; box counting dimension; C^k Cantor sets; iterated function system; scaling function; random Cantor sets
• ISSN: 0951-7715; 1361-6544
• DOI: 10.1088/0951-7715/24/2/005

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.