dimension function
   HOME

TheInfoList



OR:

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 space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...
s. Dimension functions are a generalisation of the simple "
diameter In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid fo ...
to the
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
" power law used in the construction of ''s''-dimensional Hausdorff measure.


Motivation: ''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 :\mu^ (E) = \lim_ \mu_^ (E), where :\mu_^ (E) = \inf \left\. ''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 disc in the Euclidean plane is +∞; likewise, ''μ''''s''(''E'') = 0 for ''s'' > dimH(''E'') for the same reason as the 3-dimensional
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). Th ...
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, +∞be_a_function._Define_''μ''''h''(''E'')_by :\mu^_(E)_=_\lim__\mu_^_(E), where :\mu_^_(E)_=_\inf_\left\. 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_ , +∞) → [0, +∞be_a_function._Define_''μ''''h''(''E'')_by :\mu^_(E)_=_\lim__\mu_^_(E), where :\mu_^_(E)_=_\inf_\left\. 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_ In__mathematics,_a_monotonic_function_(or_monotone_function)_is_a_function_between__ordered_sets_that_preserves_or_reverses_the_given__order._This_concept_first_arose_in_calculus,_and_was_later_generalized_to_the_more_abstract_setting_of_orde_...
_for_''t'' ≥ 0,_strictly_positive_for_''t'' > 0,_and_continuous_function.html" ;"title="monotone_function.html" "title=", +∞.html" ;"title=", +∞) → [0, +∞">, +∞) → [0, +∞be a function. Define ''μ''''h''(''E'') by :\mu^ (E) = \lim_ \mu_^ (E), where :\mu_^ (E) = \inf \left\. 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 In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of orde ...
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, +∞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 In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 of ...
ic correction :h(r) = r^ \cdot \log \frac1 \cdot \log \log \log \frac1. I.e., with probability one, 0 < ''μ''''h''(''X'') < +∞ for a Brownian path ''X'' in R2. For Brownian motion in Euclidean ''n''-space R''n'' with ''n'' ≥ 3, the exact dimension function is :h(r) = r^ \cdot \log \log \frac1r.


References

* * {{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 Dimension theory Fractals Metric geometry