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.

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

μ s ( E ) = lim δ → 0 μ δ s ( E ) , {\displaystyle \mu ^{s}(E)=\lim _{\delta \to 0}\mu _{\delta }^{s}(E),}

where

μ δ s ( E ) = inf { ∑ i = 1 ∞ d i a m ( C i ) s | d i a m ( C i ) ≤ δ , ⋃ i = 1 ∞ C i ⊇ E } . {\displaystyle \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.

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

μ h ( E ) = lim δ → 0 μ δ h ( E ) , {\displaystyle \mu ^{h}(E)=\lim _{\delta \to 0}\mu _{\delta }^{h}(E),}

where

μ δ h ( E ) = inf { ∑ i = 1 ∞ h ( d i a m ( C i ) ) | d i a m ( C i ) ≤ δ , ⋃ i = 1 ∞ C i ⊇ E } . {\displaystyle \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 (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 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, +∞) → [0, +∞] 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.

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 . {\displaystyle h(r)=r^{2}\cdot \log {\frac {1}{r}}\cdot \log \log \log {\frac {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 . {\displaystyle h(r)=r^{2}\cdot \log \log {\frac {1}{r}}.}

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