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
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 μs(E), is defined by
\( {\displaystyle \mu ^{s}(E)=\lim _{\delta \to 0}\mu _{\delta }^{s}(E),} \)
where
\( {\displaystyle \mu _{\delta }^{s}(E)=\inf \left\{\left.\sum _{i=1}^{\infty }\mathrm {diam} (C_{i})^{s}\right|\mathrm {diam} (C_{i})\leq \delta ,\bigcup _{i=1}^{\infty }C_{i}\supseteq E\right\}.}\)
μδs(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, μs(E) is non-increasing. In fact, for all values of s, except possibly one, Hs(E) is either 0 or +∞; this exceptional value is called the Hausdorff dimension of E, here denoted dimH(E). Intuitively speaking, μs(E) = +∞ for s < dimH(E) for the same reason as the 1-dimensional linear length of a 2-dimensional disc in the Euclidean plane is +∞; likewise, μs(E) = 0 for s > dimH(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 μh(E) by
\( {\displaystyle \mu ^{h}(E)=\lim _{\delta \to 0}\mu _{\delta }^{h}(E),}\)
where
\( {\displaystyle \mu _{\delta }^{h}(E)=\inf \left\{\left.\sum _{i=1}^{\infty }h\left(\mathrm {diam} (C_{i})\right)\right|\mathrm {diam} (C_{i})\leq \delta ,\bigcup _{i=1}^{\infty }C_{i}\supseteq E\right\}.}\)
Then h is called an (exact) dimension function (or gauge function) for E if μh(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, +∞) → [0, +∞] more general than h(δ) = δs 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
\( {\displaystyle h(r)=r^{2}\cdot \log {\frac {1}{r}}\cdot \log \log \log {\frac {1}{r}}.} \)
I.e., with probability one, 0 < μh(X) < +∞ for a Brownian path X in R2. For Brownian motion in Euclidean n-space Rn with n ≥ 3, the exact dimension function is
\( {\displaystyle h(r)=r^{2}\cdot \log \log {\frac {1}{r}}.} \)
References
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.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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