dimension function

{{main|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

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

where

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

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

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

where

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

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

:$h(r)\; =\; r^\{2\}\; \backslash cdot\; \backslash log\; \backslash frac1\{r\}\; \backslash cdot\; \backslash log\; \backslash log\; \backslash log\; \backslash frac1\{r\}.$

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

:$h(r)\; =\; r^\{2\}\; \backslash cdot\; \backslash log\; \backslash log\; \backslash frac1r.$

{{reflist}}

• {{cite journal

| author = Olsen, L.

| title = The exact Hausdorff dimension functions of some Cantor sets

| journal = Nonlinearity

| volume = 16

| year = 2003

| issue = 3

| pages = 963–970

| doi = 10.1088/0951-7715/16/3/309

}}

• {{cite book

| author = Rogers, C. A.

| title = Hausdorff measures

| edition = Third

| series = Cambridge Mathematical Library

| publisher = Cambridge University Press

| location = Cambridge

| year = 1998

| pages = xxx+195

| isbn = 0-521-62491-6

}}

Category:Dimension theory

Category:Fractals

Category:Metric geometry