Gliding flight

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{{Expert-subject|Mathematics|date=November 2008}}

In [[mathematics]], the notion of an ('''exact''') '''dimension function''' (also known as a '''gauge function''') is a tool in the study of [[fractal]]s and other subsets of [[metric space]]s. 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|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

:<math>\mu^{s} (E) = \lim_{\delta \to 0} \mu_{\delta}^{s} (E),</math>

where

:<math>\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\}.</math>

''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 dimH(''E''). Intuitively speaking, ''μ''''s''(''E'') = +∞ for ''s'' < dimH(''E'') for the same reason as the 1-dimensional linear [[length]] of a 2-dimensional [[Disk (mathematics)|disc]] in the [[Euclidean plane]] is +∞; likewise, ''μ''''s''(''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 ''μ''''h''(''E'') by

:<math>\mu^{h} (E) = \lim_{\delta \to 0} \mu_{\delta}^{h} (E),</math>

where

:<math>\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\}.</math>

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 [[monotone function|monotonically increasing]] for ''t'' ≥ 0, strictly positive for ''t'' > 0, and [[continuous function|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 [[disjoint sets|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.

==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''2(''X'') is zero. The exact dimension function ''h'' is given by the [[logarithm]]ic correction

:<math>h(r) = r^{2} \cdot \log \frac1{r} \cdot \log \log \log \frac1{r}.</math>

I.e., with probability one, 0 < ''μ''''h''(''X'') < +∞ for a Brownian path ''X'' in '''R'''2. For Brownian motion in Euclidean ''n''-space '''R'''''n'' with ''n'' ≥ 3, the exact dimension function is

:<math>h(r) = r^{2} \cdot \log \log \frac1r.</math>

==References==

* {{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]]

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