Dimension function explained

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

See 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} \mu ^{s} _{\delta} (E),

where

\mu ^{s} _{\delta} (E) = inf \left\{ \left. \sum ^{inf ty} _{i =1} diam (C _{i} ) ^{s} \right| diam (C _{i} ) \leq \delta, c up ^{inf ty} _{i =1} 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,&nbsp;+∞)&nbsp;→&nbsp;[0,&nbsp;+∞] be a function. Define μ^h(E) by

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

where

\mu ^{h} _{\delta} (E) = inf \left\{ \left. \sum ^{inf ty} _{i =1} h \left( diam (C _{i} ) \right) \right| diam (C _{i} ) \leq \delta, c up ^{inf ty} _{i =1} C _{i} \supseteq E \right\}.

Then h is called an (exact) dimension function (or gauge function) for E if μ^h(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 ≥ 0, strictly positive for t > 0, and continuous on the right for all t ≥ 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 : [0,&nbsp;+∞)&nbsp;→&nbsp;[0,&nbsp;+∞] more general than h(δ) = δ^s 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 μ²(X) is zero. The exact dimension function h is given by the logarithmic correction

h(r) = r ^{2} ⋅ log (1 {r})/(⋅) log log log (1 {r}.)/()

I.e., with probability one, 0 < μ^h(X) < +∞ for a Brownian path X in R². For Brownian motion in Euclidean n-space Rⁿ with n ≥ 3, the exact dimension function is

h(r) = r ^{2} ⋅ log log (1 r.)/()

References

• Olsen, L.. The exact Hausdorff dimension functions of some Cantor sets. Nonlinearity. 16. 2003. 3. 963 - 970. 10.1088/0951-7715/16/3/309.
• Book: Rogers, C. A.. Hausdorff measures. Third. Cambridge Mathematical Library. Cambridge University Press. Cambridge. 1998. xxx+195. 0-521-62491-6.