Epsilon-Distortion Complexity for Cantor Sets

The way we measure the complexity of a (compact) set can be colloquially described as follows. We define the ε-distortion complexity of a set C as the minimal length of the programs producing a finite set ε-close to C , in the sense of Hausdorff distance. As in the classical notion of Kolmogorov complexity of sequences, by programs we mean programs running on a universal Turing machine [11]. We are interested in the behavior of the ε-distortion complexity when ε is getting small, and in eventual relations of this behavior with other characteristics of the set (e.g., fractal dimension).

In the present article we consider various classes of Cantor sets on the real line generated by iterated function systems (IFS’s) [2] and compute bounds from above and below of their ε-distortion complexity as a function of ε. We first consider IFS’s with polynomial contractions and obtain the upper bound const×log(ε -1 ) for the ε-distortion complexity of the generated Cantor set, where the (finite) constant may depend on the polynomials. We can produce “many” polynomial IFS’s with a lower bound of the same order using a probabilistic construction. It turns out that some particular Cantor sets like the usual middle third Cantor set are of much lower complexity. For analytic IFS’s, we obtain the upper bound const × (log(ε -1 )) 2 . Next we consider random central Cantor sets produced by affine IFS’s, for which the contraction rate is chosen at random at each step of the construction. In this case, we get the upper bound const × (log(ε -1 )) 2 and the lower bound const × (log(ε -1 )) 2-δ , for any δ > 0, for almost all such Cantor sets (where the constant in our bound depends on δ and tends to 0 when δ → 0). Finally, we consider C k IFS’s. Contrarily to the previous cases, the leading, asymptotic behavior of the ε-distortion complexity depends on the box counting dimension D of the generated Cantor set. Indeed, we obtain the upper bound const × ε -D k -δ , for any δ > 0 (where the constant in our bound depends on δ and blows up when δ → 0). We then construct “many” C k (random) Cantor sets with a lower bound const × ε -D k +δ (for any δ > 0), by constructing their scaling function [14].

The case of sets reduced to one point on the line was investigated in [7] where in particular the Hausdorff dimension of the set of reals with given asymptotic complexity is computed. For graphs of functions, from the point of view of determining the values of a function at given precision, relations with ε-entropy are obtained in [1]. Another notion of complexity is to ask about the smallest execution time of the programs generating a given set with ε-precision, in the sense of Hausdorff distance [15]. This was used in [6] to show that a class of Julia sets was polynomial time computable.

This article is organized as follows. In Section 2 we define the ε-distortion complexity of a compact set and state our results. Section 3 is devoted to the proofs.

For a compact set C ⊂ R d , we define its ε-distortion complexity as follows.

Definition 2.1. The ε-distortion complexity of a compact set C ⊂ R d at precision ε > 0 is defined by

where the minimum is taken over all binary programs P ∈ {0, 1} * running on a universal Turing machine U , which produce a finite subset C(P) ∈ R d ; ℓ(P) is the program length; d H denotes the Hausdorff distance.

Notice that, because of the compactness of C , we can use a minimum in the above definition, which always leads to a finite number.

For the reader’s convenience, we recall that the Hausdorff distance d H between two closed subsets F 1 , F 2 of a metric space with metric d is given by (see, e.g., [12,2])

We now recall the definition of Cantor sets generated by iterated function systems [2]. For the sake of simplicity, we restrict ourselves to Cantor sets in the unit interval [0, 1] ⊂ R, although several results can be easily generalised to arbitrary finite dimension.

Let A = [0, 1] and let I be a finite set of indices with at least two elements. An Iterated Function System (IFS for short) is a collection

of injective contractions on A with uniform contraction rate ρ ∈ (0, 1), and such that φ i (A) ∩ φ j (A) = ∅ for i = j.

For any infinite word ω ∈ I ∞ and for any n ∈ N, let ω n 1 ∈ I n denote the prefix of length n given by the first n symbols of ω, and let

The map π :

is a Cantor set and satisfies

(

We are interested in the behaviour of

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