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. 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 formula_1 and a continuous self-map formula_2. Let formula_3 denote the collection of open finite covers of formula_1. For formula_5 define its order by
An open finite cover formula_7 refines formula_8, denoted formula_9, if for every formula_10, there is formula_11 so that formula_12. Let
Note that in terms of this definition the Lebesgue covering dimension is defined by formula_14.
Let formula_15 be open finite covers of formula_1. The join of formula_8 and formula_7 is the open finite cover by all sets of the form formula_19 where formula_20, formula_21. Similarly one can define the join formula_22 of any finite collection of open covers of formula_1.
The mean dimension is the non-negative extended real number:
where formula_25
Definition in the metric case.
If the compact Hausdorff topological space formula_1 is metrizable and formula_27 is a compatible metric, an equivalent definition can be given. For formula_28, let formula_29 be the minimal non-negative integer formula_30, such that there exists an open finite cover of formula_1 by sets of diameter less than formula_32 such that any formula_33 distinct sets from this cover have empty intersection. Note that in terms of this definition the Lebesgue covering dimension is defined by formula_34. Let
The mean dimension is the non-negative extended real number:
Example.
Let formula_40. Let formula_41 and formula_2 be the shift homeomorphism formula_43, then formula_44.
External links.
What is Mean Dimension? 