Mean Dimension
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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the mean (topological) dimension of a
topological dynamical system In mathematics, topological dynamics is a branch of the theory of dynamical systems in which qualitative, asymptotic properties of dynamical systems are studied from the viewpoint of general topology. Scope The central object of study in topolog ...
is a non-negative extended real number that is a measure of the complexity of the system. Mean dimension was first introduced in 1999 by
Gromov Gromov (russian: Громов) is a Russian male surname, its feminine counterpart is Gromova (Громова). Gromov may refer to: * Alexander Georgiyevich Gromov (born 1947), Russian politician and KGB officer * Alexander Gromov (born 1959), R ...
. Shortly after it was developed and studied systematically by Lindenstrauss and Weiss. In particular they proved the following key fact: a system with finite
topological entropy In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing h ...
has zero mean dimension. For various topological dynamical systems with infinite topological entropy, the mean dimension can be calculated or at least bounded from below and above. This allows mean dimension to be used to distinguish between systems with infinite topological entropy. Mean dimension is also related to the problem of embedding topological  dynamical systems in shift spaces (over Euclidean cubes).


General definition

A topological dynamical system consists of a compact Hausdorff topological space \textstyle X and a continuous self-map \textstyle T:X\rightarrow X. Let \textstyle \mathcal denote the collection of open finite covers of \textstyle X. For \textstyle \alpha\in\mathcal define its order by : \operatorname(\alpha)=\max_\sum_1_U(x)-1 An open finite cover \textstyle \beta refines \textstyle \alpha, denoted \textstyle \beta\succ\alpha, if for every \textstyle V\in\beta, there is \textstyle U\in\alpha so that \textstyle V\subset U. Let : D(\alpha)=\min_ \operatorname(\beta) Note that in terms of this definition the
Lebesgue covering 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 ...
is defined by \dim_\mathrm(X)=\sup_D(\alpha). Let \textstyle \alpha,\beta be open finite covers of \textstyle X. The join of \textstyle \alpha and \textstyle \beta is the open finite cover by all sets of the form \textstyle A\cap B where \textstyle A\in\alpha, \textstyle B\in\beta. Similarly one can define the join \textstyle \bigvee_^n\alpha_i of any finite collection of open covers of \textstyle X. The mean dimension is the non-negative extended real number: : \operatorname(X,T)=\sup_\lim_\frac where \textstyle \alpha^n=\bigvee_^T^\alpha.


Definition in the metric case

If the compact Hausdorff topological space \textstyle X is
metrizable In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty) ...
and \textstyle d is a compatible metric, an equivalent definition can be given. For \textstyle \varepsilon>0, let \textstyle\operatorname_\varepsilon(X,d) be the minimal non-negative integer \textstyle n, such that there exists an open finite cover of \textstyle X by sets of diameter less than \textstyle \varepsilon such that any \textstyle n+2 distinct sets from this cover have empty intersection. Note that in terms of this definition the
Lebesgue covering 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 ...
is defined by \textstyle \dim_\mathrm(X)=\sup_ \operatorname_\varepsilon(X,d). Let : d_n(x,y)=\max_d(T^i x,T^i y) The mean dimension is the non-negative extended real number: : \operatorname(X,d)=\sup_\lim_\frac


Properties

* Mean dimension is an invariant of topological dynamical systems taking values in \textstyle ,\infty/math>. * If the Lebesgue covering dimension of the system is finite then its mean dimension vanishes, i.e. \textstyle \dim_\mathrm(X)<\infty\Rightarrow \operatorname(X,T)=0. * If the topological entropy of the system is finite then its mean dimension vanishes, i.e. \textstyle \dim_\mathrm(X,T)<\infty\Rightarrow \operatorname(X,T)=0.


Example

Let \textstyle d\in\mathbb. Let \textstyle X=( ,1d)^ and \textstyle T:X\rightarrow X be the shift homeomorphism \textstyle (\ldots,x_,x_,\mathbf,x_1,x_2,\ldots)\rightarrow(\ldots,x_,x_0,\mathbf,x_2,x_3,\ldots), then \textstyle \operatorname(X,T)=d.


See also

*
Dimension theory 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 coord ...
*
Topological entropy In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing h ...
* Universal spaces (in topology and topological dynamics)


References

* {{refend


External links

What is Mean Dimension?
Entropy and information Topological dynamics