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 X {\displaystyle \textstyle X} and a continuous self-map T : X → X {\displaystyle \textstyle T:X\rightarrow X} . Let O {\displaystyle \textstyle {\mathcal {O}}} denote the collection of open finite covers of X {\displaystyle \textstyle X} . For α ∈ O {\displaystyle \textstyle \alpha \in {\mathcal {O}}} define its order by
ord ( α ) = max x ∈ X ∑ U ∈ α 1 U ( x ) − 1 {\displaystyle \operatorname {ord} (\alpha )=\max _{x\in X}\sum _{U\in \alpha }1_{U}(x)-1}
An open finite cover β {\displaystyle \textstyle \beta } refines α {\displaystyle \textstyle \alpha } , denoted β ≻ α {\displaystyle \textstyle \beta \succ \alpha } , if for every V ∈ β {\displaystyle \textstyle V\in \beta } , there is U ∈ α {\displaystyle \textstyle U\in \alpha } so that V ⊂ U {\displaystyle \textstyle V\subset U} . Let
D ( α ) = min β ≻ α ord ( β ) {\displaystyle D(\alpha )=\min _{\beta \succ \alpha }\operatorname {ord} (\beta )}
Note that in terms of this definition the Lebesgue covering dimension is defined by dim L e b ( X ) = sup α ∈ O D ( α ) {\displaystyle \dim _{\mathrm {Leb} }(X)=\sup _{\alpha \in {\mathcal {O}}}D(\alpha )} . Let α , β {\displaystyle \textstyle \alpha ,\beta } be open finite covers of X {\displaystyle \textstyle X} . The join of α {\displaystyle \textstyle \alpha } and β {\displaystyle \textstyle \beta } is the open finite cover by all sets of the form A ∩ B {\displaystyle \textstyle A\cap B} where A ∈ α {\displaystyle \textstyle A\in \alpha } , B ∈ β {\displaystyle \textstyle B\in \beta } . Similarly one can define the join ⋁ i = 1 n α i {\displaystyle \textstyle \bigvee _{i=1}^{n}\alpha _{i}} of any finite collection of open covers of X {\displaystyle \textstyle X} . The mean dimension is the non-negative extended real number:
mdim ( X , T ) = sup α ∈ O lim n → ∞ D ( α n ) n {\displaystyle \operatorname {mdim} (X,T)=\sup _{\alpha \in {\mathcal {\mathcal {O}}}}\lim _{n\rightarrow \infty }{\frac {D(\alpha ^{n})}{n}}}
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