Mean dimension

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.

[2] 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).

A topological dynamical system consists of a compact Hausdorff topological space

and a continuous self-map

denote the collection of open finite covers of

define its order by An open finite cover

β ≻ α

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

α ∈

α , β

be open finite covers of

is the open finite cover by all sets of the form

Similarly one can define the join

of any finite collection of open covers of

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

If the compact Hausdorff topological space

is a compatible metric, an equivalent definition can be given.

be the minimal non-negative integer

, such that there exists an open finite cover of

by sets of diameter less than

distinct sets from this cover have empty intersection.

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

{\displaystyle \textstyle \dim _{\mathrm {Leb} }(X)=\sup _{\varepsilon >0}\operatorname {Widim} _{\varepsilon }(X,d)}

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

be the shift homeomorphism