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
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
![{\displaystyle \operatorname {ord} (\alpha )=\max _{x\in X}\sum _{U\in \alpha }1_{U}(x)-1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1780e89ebd108a6241f5e7a5fc61a6694101c333)
An open finite cover
refines
, denoted
, if for every
, there is
so that
. Let
![{\displaystyle D(\alpha )=\min _{\beta \succ \alpha }\operatorname {ord} (\beta )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/049f5d18d268e26dd6b3602eea6dce266ca4b3ab)
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:
![{\displaystyle \operatorname {mdim} (X,T)=\sup _{\alpha \in {\mathcal {\mathcal {O}}}}\lim _{n\rightarrow \infty }{\frac {D(\alpha ^{n})}{n}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/960957da66d65ac0655427d06b7a593e73a4a323)
where
Definition in the metric case
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
![{\displaystyle d_{n}(x,y)=\max _{0\leq i\leq n-1}d(T^{i}x,T^{i}y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e34e90a1d9dca9d8e5c6437cab658af8bf04f15c)
The mean dimension is the non-negative extended real number:
![{\displaystyle \operatorname {mdim} (X,d)=\sup _{\varepsilon >0}\lim _{n\rightarrow \infty }{\frac {\operatorname {Widim} _{\varepsilon }(X,d_{n})}{n}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f6aa72aac6623fdf8ea8ed99f3ad31b2ec76c00)
Properties
- 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
Let
. Let
and
be the shift homeomorphism
, then
.
See also
- Dimension theory
- Topological entropy
- Universal spaces (in topology and topological dynamics)
References
- ^ 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.
- ^ 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.
- Adler, R.; Downarowicz, T.; Misiurewicz, M. (2008). "Topological entropy". Scholarpedia. 3 (2): 2200. Bibcode:2008SchpJ...3.2200A. doi:10.4249/scholarpedia.2200.
External links
What is Mean Dimension?