Dimensions of Points in Self-Similar Fractals

Abstract

We use nontrivial connections between the theory of computing and the finescale geometry of Euclidean space to give a complete analysis of the dimensions of individual points in fractals that are computably self-similar.

A Divergence Formula for Randomness and Dimension

“…It is known that cdim β (X) = sup S∈X dim β (S) and that cDim β (X) = sup S∈X Dim β (S) [14]. Constructive dimensions are thus investigated in terms of the dimensions of individual sequences.…”

“…We thus understand (1.1) and (1.2) to say that dim(S) and Dim(S) are the lower and upper information densities of the sequence S. These constructive dimensions and their analogs at other levels of effectivity have been investigated extensively in recent years [10]. The constructive dimensions dim(S) and Dim(S) have recently been generalized to incorporate a probability measure ν on the sequence space Σ ∞ as a parameter [14]. Specifically, for each such ν and each sequence S ∈ Σ ∞ , we now have the constructive dimension dim ν (S) and the constructive strong dimension Dim ν (S) of S with respect to ν.…”

Lines missing every random point1

“…Recent work along these lines has included algorithmic classifications of points lying on computable curves and arcs [14,26,6,32,23] and in more exotic sets [22,19,8,15]. This paper concerns a simple, fundamental question: Can the direction of a line in Euclidean space force the line to meet at least one random point?…”

Connectivity properties of dimension level sets

“…The properties of constructive dimension and its relationships with algorithmic randomness, Kolmogorov complexity, and other topics in the theory of computing have been extensively investigated over the past few years [6]. Intuitively, the dimension of a sequence S is the asymptotic density of information in S [13,12].…”

“…Gu, Lutz, and Mayordomo [4] have noted that (1.1), in combination with classical results in geometric measure theory, implies that every point x ∈ R n that lies on a computable curve of finite length has dimension dim(x) ≤ 1. For another example, Lutz and Mayordomo [12] have recently carried out a pointwise analysis of the dimensions of self-similar fractals, using information-theoretic methods to show that, for every computably self-similar fractal F ⊆ R n , every point x ∈ F , and every symbolic sequence T that naturally encodes x in the construction of F , the dimension identity…”