Estimates of Kolmogorov Complexity in Approximating Cantor Sets - Archive ouverte HAL
Article Dans Une Revue Nonlinearity Année : 2011

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.
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Dates et versions

hal-02899843 , version 1 (15-07-2020)

Identifiants

  • HAL Id : hal-02899843 , version 1

Citer

C Bonanno, J.-R Chazottes, P. Collet. Estimates of Kolmogorov Complexity in Approximating Cantor Sets. Nonlinearity, 2011. ⟨hal-02899843⟩
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