Dimension function

In mathematics, the notion of an (exact) dimension function is a tool in the study of fractals and other subsets of metric spaces. Dimension functions are a generalisation of the simple "diameter to the dimension" power law used in the construction of s-dimensional Hausdorff measure.

Motivation: s-dimensional Hausdorff measure

Main article: Hausdorff dimension

Consider a metric space (X, d) and a subset E of X. Given a number s ≥ 0, the s-dimensional Hausdorff measure of E, denoted μ^s(E), is defined by

$\mu ^{s}(E)=\lim _{\delta \to 0}\mu _{\delta }^{s}(E),$

where

$\mu _{\delta }^{s}(E)=\inf \left\{\left.\sum _{i=1}^{\infty }\mathrm {diam} (C_{i})^{s}\right|\mathrm {diam} (C_{i})\leq \delta ,\bigcup _{i=1}^{\infty }C_{i}\supseteq E\right\}.$

μ_δ^s(E) can be thought of as an approximation to the "true" s-dimensional area/volume of E given by calculating the minimal s-dimensional area/volume of a covering of E by sets of diameter at most δ.

As a function of increasing s, μ^s(E) is non-increasing. In fact, for all values of s, except possibly one, H^s(E) is either 0 or +∞; this exceptional value is called the Hausdorff dimension of E, here denoted dim_H(E). Intuitively speaking, μ^s(E) = +∞ for s < dim_H(E) for the same reason as the 1-dimensional linear length of a 2-dimensional disc in the Euclidean plane is +∞; likewise, μ^s(E) = 0 for s > dim_H(E) for the same reason as the 3-dimensional volume of a disc in the Euclidean plane is zero.

The idea of a dimension function is to use different functions of diameter than just diam(C)^s for some s, and to look for the same property of the Hausdorff measure being finite and non-zero.

Definition

Let (X, d) be a metric space and E ⊆ X. Let h : [0, +∞) → [0, +∞] be a function. Define μ^h(E) by

$\mu ^{h}(E)=\lim _{\delta \to 0}\mu _{\delta }^{h}(E),$

where

$\mu _{\delta }^{h}(E)=\inf \left\{\left.\sum _{i=1}^{\infty }h\left(\mathrm {diam} (C_{i})\right)\right|\mathrm {diam} (C_{i})\leq \delta ,\bigcup _{i=1}^{\infty }C_{i}\supseteq E\right\}.$

Then h is called an (exact) dimension function for E if μ^h(E) is finite and strictly positive.

Reference

• Rogers, C. A. (1998). Hausdorff measures. Cambridge Mathematical Library (Third edition ed.). Cambridge: Cambridge University Press. pp. xxx+195. ISBN 0-521-62491-6. {{cite book}}: |edition= has extra text (help)