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