In mathematics, the conformal dimension of a
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 settin ...
''X'' is the infimum of the
Hausdorff dimension
In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of ...
over the
conformal gauge
Conformal may refer to:
* Conformal (software), in ASIC Software
* Conformal coating in electronics
* Conformal cooling channel, in injection or blow moulding
* Conformal field theory in physics, such as:
** Boundary conformal field theory
** ...
of ''X'', that is, the class of all metric spaces
quasisymmetric to ''X''.
[John M. Mackay, Jeremy T. Tyson, ''Conformal Dimension : Theory and Application'', University Lecture Series, Vol. 54, 2010, Rhodes Island]
Formal definition
Let ''X'' be a metric space and
be the collection of all metric spaces that are quasisymmetric to ''X''. The conformal dimension of ''X'' is defined as such
:
Properties
We have the following
inequalities
Inequality may refer to:
Economics
* Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy
* Economic inequality, difference in economic well-being between population groups
* ...
, for a metric space ''X'':
:
The second inequality is true by definition. The first one is deduced from the fact that the
topological dimension
In mathematics, the Lebesgue covering dimension or topological dimension of a topological space is one of several different ways of defining the dimension of the space in a
topologically invariant way.
Informal discussion
For ordinary Euclidean ...
T is invariant by
homeomorphism
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
, and thus can be defined as the
infimum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest low ...
of the
Hausdorff dimension
In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of ...
over all spaces homeomorphic to ''X''.
Examples
* The conformal dimension of
is ''N'', since the topological and Hausdorff dimensions of
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
s agree.
* The
Cantor set
In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883.
Thr ...
''K'' is of null conformal dimension. However, there is no metric space quasisymmetric to ''K'' with a 0 Hausdorff dimension.
See also
*
Anomalous scaling dimension
In theoretical physics, the scaling dimension, or simply dimension, of a local operator in a quantum field theory characterizes the rescaling properties of the operator under spacetime dilations x\to \lambda x. If the quantum field theory is sca ...
References
Fractals
Metric geometry
Dimension theory
{{geometry-stub