Mean dimension explained

In mathematics, the mean (topological) dimension of a topological dynamical system 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.^[1] 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 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

style X and a continuous self-map style T:X → X . Let style l{O} denote the collection of open finite covers of style X . For style \alpha\in l{O} define its order by\operatorname{ord}(\alpha) =max _{x\in} \sum _{U\in\alpha} 1 _{U(x)-1}

An open finite cover

style \beta refines style \alpha , denoted style \beta\succ\alpha , if for every style V\in\beta , there is style U\in\alpha so that style V\subset U . LetD(\alpha) =min _{\beta\succ\alpha} \operatorname{ord}(\beta)

Note that in terms of this definition the Lebesgue covering dimension is defined by

\dim _{Leb(X)=\sup} _{\alpha\inl{O} }D(\alpha).

Let

style \alpha,\beta be open finite covers of style X . The join of style \alpha and style \beta is the open finite cover by all sets of the form style A\cap B where style A\in\alpha , style B\in\beta . Similarly one can define the join style v ee ^{n\alpha} _{i} of any finite collection of open covers of style X .

The mean dimension is the non-negative extended real number:

\operatorname{mdim}(X,T) = \sup _{\alpha\inl{l{O} }}\lim_\frac

where

style \alpha ^{n -1} _{i =0} T ^{-i} \alpha.

Definition in the metric case

If the compact Hausdorff topological space

style X is metrizable and style d is a compatible metric, an equivalent definition can be given. For style \varepsilon> 0 , let style\operatorname{Widim} _{\varepsilon(X,d)} be the minimal non-negative integer style n , such that there exists an open finite cover of style X by sets of diameter less than style \varepsilon such that any style n +2 distinct sets from this cover have empty intersection. Note that in terms of this definition the Lebesgue covering dimension is defined by style \dim _{Leb(X)=\sup} _{\varepsilon>0} \operatorname{Widim} _{\varepsilon(X,d)} . Letd _{n(x,y)=max} _{0\leq} d(T ^{i} x,T ^{i} y)

The mean dimension is the non-negative extended real number:

\operatorname{mdim}(X,d) = \sup _{\varepsilon>0} \lim _{n → infty} (\operatorname{Widim)/(_{\varepsilon(X,d}) _{n)}{n}}

Properties

• Mean dimension is an invariant of topological dynamical systems taking values in

style [ 0 , inf ty] .

• If the Lebesgue covering dimension of the system is finite then its mean dimension vanishes, i.e.

style \dim _{Leb(X)<infty ⇒} \operatorname{mdim}(X,T) =0 .

• If the topological entropy of the system is finite then its mean dimension vanishes, i.e.

style \dim _{top(X,T)<infty ⇒} \operatorname{mdim}(X,T) =0 .^[2]

Example

Let

style d\in {N }. Let style X = ([ 0 , 1 ] ^{d)} ^{Z} and style T:X → X be the shift homeomorphism style (\ldots,x _{-2} ,x _{-1} , x _{0} ,x _{1 ,x} _{2,\ldots) → (\ldots,x} _{-1} ,x _{0 , x _{1} ,x} _{2,x} _{3,\ldots)} , then style \operatorname{mdim}(X,T) = d .

See also

• • • Dimension theory Topological entropy Universal spaces (in topology and topological dynamics)

References

• R. . Adler . T. . Downarowicz . M. . Misiurewicz . Topological entropy . Scholarpedia . 3 . 2 . 2200 . 2008 . 10.4249/scholarpedia.2200. 2008SchpJ...3.2200A . free .

External links

What is Mean Dimension?

Notes and References

1. Misha . Gromov . Topological invariants of dynamical systems and spaces of holomorphic maps I . Mathematical Physics, Analysis and Geometry . 2 . 4 . 323–415 . 1999 . 10.1023/A:1009841100168 . free . 117100302 .
2. Mean topological dimension. Israel Journal of Mathematics. 2000-12-01. 0021-2172. 1–24. 115. 1. 10.1007/BF02810577. free. Elon. Lindenstrauss. Benjamin. Weiss. p. 14. 10.1.1.30.3552.