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Dimension function

In mathematics, the notion of an (exact) dimension function (also known as a gauge 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

Consider a metric space (X,&nbsp;d) and a subset E of X. Given a number s&nbsp;≥&nbsp;0, the s-dimensional Hausdorff measure of E, denoted μ<sup>s</sup>(E), is defined by

where

μ<sub>δ</sub><sup>s</sup>(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<sup>s</sup>(E) is non-increasing. In fact, for all values of s, except possibly one, H<sup>s</sup>(E) is either 0 or +∞; this exceptional value is called the Hausdorff dimension of E, here denoted dim<sub>H</sub>(E). Intuitively speaking, μ<sup>s</sup>(E)&nbsp;=&nbsp;+∞ for s&nbsp;&lt;&nbsp;dim<sub>H</sub>(E) for the same reason as the 1-dimensional linear length of a 2-dimensional disc in the Euclidean plane is +∞; likewise, μ<sup>s</sup>(E)&nbsp;=&nbsp;0 for s&nbsp;&gt;&nbsp;dim<sub>H</sub>(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)<sup>s</sup> for some s, and to look for the same property of the Hausdorff measure being finite and non-zero.

Definition

Let (X,&nbsp;d) be a metric space and E&nbsp;⊆&nbsp;X. Let h&nbsp;:&nbsp;[0,&nbsp;+∞)&nbsp;→&nbsp;[0,&nbsp;+∞] be a function. Define μ<sup>h</sup>(E) by

where

Then h is called an (exact) dimension function (or gauge function) for E if μ<sup>h</sup>(E) is finite and strictly positive. There are many conventions as to the properties that h should have: Rogers (1998), for example, requires that h should be monotonically increasing for t&nbsp;≥&nbsp;0, strictly positive for t&nbsp;&gt;&nbsp;0, and continuous on the right for all t&nbsp;≥&nbsp;0.

Packing dimension

Packing dimension is constructed in a very similar way to Hausdorff dimension, except that one "packs" E from inside with pairwise disjoint balls of diameter at most δ. Just as before, one can consider functions h&nbsp;:&nbsp;[0,&nbsp;+∞)&nbsp;→&nbsp;[0,&nbsp;+∞] more general than h(δ)&nbsp;=&nbsp;δ<sup>s</sup> and call h an exact dimension function for E if the h-packing measure of E is finite and strictly positive.

Example

Almost surely, a sample path X of Brownian motion in the Euclidean plane has Hausdorff dimension equal to 2, but the 2-dimensional Hausdorff measure μ<sup>2</sup>(X) is zero. The exact dimension function h is given by the logarithmic correction

I.e., with probability one, 0&nbsp;&lt;&nbsp;μ<sup>h</sup>(X)&nbsp;&lt;&nbsp;+∞ for a Brownian path X in R<sup>2</sup>. For Brownian motion in Euclidean n-space R<sup>n</sup> with n&nbsp;&ge;&nbsp;3, the exact dimension function is

References