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).

General definition

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A topological dynamical system consists of a compact Hausdorff topological space   and a continuous self-map  . Let   denote the collection of open finite covers of  . For   define its order by

 

An open finite cover   refines  , denoted  , if for every  , there is   so that  . Let

 

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

Let   be open finite covers of  . The join of   and   is the open finite cover by all sets of the form   where  ,  . 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  

Definition in the metric case

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If the compact Hausdorff topological space   is metrizable and   is a compatible metric, an equivalent definition can be given. For  , let   be the minimal non-negative integer  , such that there exists an open finite cover of   by sets of diameter less than   such that any   distinct sets from this cover have empty intersection. Note that in terms of this definition the Lebesgue covering dimension is defined by  . Let

 

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

 

Properties

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  • Mean dimension is an invariant of topological dynamical systems taking values in  .
  • If the Lebesgue covering dimension of the system is finite then its mean dimension vanishes, i.e.  .
  • If the topological entropy of the system is finite then its mean dimension vanishes, i.e.  .[2]

Example

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Let  . Let   and   be the shift homeomorphism  , then  .

See also

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References

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  1. ^ Gromov, Misha (1999). "Topological invariants of dynamical systems and spaces of holomorphic maps I". Mathematical Physics, Analysis and Geometry. 2 (4): 323–415. doi:10.1023/A:1009841100168. S2CID 117100302.
  2. ^ a b Lindenstrauss, Elon; Weiss, Benjamin (2000-12-01). "Mean topological dimension". Israel Journal of Mathematics. 115 (1). p. 14: 1–24. CiteSeerX 10.1.1.30.3552. doi:10.1007/BF02810577. ISSN 0021-2172.
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What is Mean Dimension?