Any modeling of a dynamical system consists of keeping some information and discarding some other information, for example, is a gas molecule modeled as a point or a ball or a more complicated structure? In topological dynamics the information retained is the "closeness'' of world-states, specified for instance, by a distance function $d(x,y)$ between all pairs $x,y$ of worlds-states. In ergodic theory the information retained is the relative probability of different world-states. More technically, for us a topological dynamical system (t.d.s), will be given by a compact metric space $(X,d)$ and a homeomorphism (continuous invertible map) $T:X\rightarrow T$. $X$ represents the phase space and $T$ represents the evaluation rule. The metric $d:X\times X\rightarrow\mathbb{{R}}$ is the distance function.

A fundamental problem in the theory of dynamical systems is the problem of deciding if two given systems are isomorphic ("the same''). A very powerful tool is given by invariants. For example, an invariant can be given by a series of steps which you apply to a dynamical system, and which result with some quantity (e.g. a real number). This procedure must have the property, that if you apply the same steps to another system, which is actually isomorphic to the first system, but its real-world manifestation is say very different, then you get exactly the same quantity. This property is referred to as invariance and that is why invariants are called invariants. A major invariant for topological dynamical systems is the invariant of topological entropy. You can read about it here. We would like to introduce the invariant of mean dimension so we first need to answer the question:

- It is invariant under homeomorphisms. So if $X$ and $Y$ are homeomorphic then $dim(X)=dim(Y)$.
- $dim(\mathbb{{R}}^{n})=n$, in words, the dimension of the "$n$-dimensional'' Euclidean space is $n$ (dah!).

- Mean dimension is an invariant of topological dynamical systems.
- Mean dimension takes values in $[0,\infty]$.
- If the topological entropy of the system is finite then its mean dimension vanishes, i.e $h_{top}(X,T)<\infty\Rightarrow mdim(X,T)=0$ (Lindenstrauss-Weiss).
- Let $d\in\mathbb{N}$. Denote $X=([0,1]^{d})^{\mathbb{Z}}$. Let $T:X\rightarrow$ be the shift homeomorphism: \[ (\ldots,x_{-2},x_{-1},\mathbf{x_{0}},x_{1},x_{2},\ldots) \] \[ \downarrow \] \[ (\ldots,x_{-1},x_{0},\mathbf{x_{1}},x_{2},x_{3},\ldots) \]

An excellent treatment of mean dimension in the perspective of Dimension Theory is given by the monograph [Coo]. Mean dimension has also been developed in the context of amenable and sofic actions ([CK,Kri,Li,LW]).

[CK] Michel Coornaert and Fabrice Krieger. Mean topological dimension for actions of discrete amenable groups. Discrete Contin. Dyn. Syst., 13(3):779-793, 2005.

[Coo] Michel Coornaert. Dimension topologique et systèmes dynamiques, vol- ume 14 of Cours Spécialisés [Specialized Courses]. Société Mathématique de France, Paris, 2005.

[CSC] Tullio Ceccherini-Silberstein and Michel Coornaert. Cellular automata and groups. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2010.

[Gou1] A. Gournay. Dimension moyenne et espaces d'applications pseudo-holomorphes. PhD thesis, 2008.

[Gou2] Antoine Gournay. A dynamical approach to von Neumann dimension. Discrete Contin. Dyn. Syst., 26(3):967-987, 2010.

[Gro] Misha Gromov. Topological invariants of dynamical systems and spaces of holomorphic maps. I. Math. Phys. Anal. Geom., 2(4):323-415, 1999.

[GT] Yonatan Gutman and Masaki Tsukamoto. Mean dimension and a sharp embedding theorem: extensions of aperiodic subshifts. Ergodic Theory Dynam. Systems 34, 1888-1896, 2014.

[Gut1] Yonatan Gutman. Embedding $\mathbb{Z}^k$-actions in cubical shifts and $\mathbb{Z}^k$-symbolic extensions. Ergodic Theory Dynam. Systems, 31(2):383-403, 2011.

[Gut2] Yonatan Gutman. Mean dimension and Jaworski-type theorems. Proceedings of the London Mathematical Society, 111(4):831-850, 2015.

[Gut3] Yonatan Gutman. Embedding topological dynamical systems with periodic points in cubical shifts. Ergodic Theory Dynam. Systems 37, 512-538, 2017.

[Kri] Fabrice Krieger. Minimal systems of arbitrary large mean topological dimension. Israel J. Math., 172:425-444, 2009.

[Li] Hanfeng Li. Sofic mean dimension. Adv. Math., 244:570-604, 2013.

[Lin] Elon Lindenstrauss. Mean dimension, small entropy factors and an em- bedding theorem. Inst. Hautes Études Sci. Publ. Math., 89(1):227-262, 1999.

[LL] Hanfeng Li and Bingbing Liang. Mean dimension, mean rank, and von Neumann-Lück rank. Journal für die reine und angewandte Mathematik (Crelles Journal), 2013.

[LT12] Elon Lindenstrauss and Masaki Tsukamoto. Mean dimension and an embedding problem: an example. Israel J. Math, 199:573-584, 2014.

[LW] Elon Lindenstrauss and Benjamin Weiss. Mean topological dimension. Israel J. Math., 115:1-24, 2000.

[MT] Shinichiroh Matsuo and Masaki Tsukamoto. Instanton approximation, periodic ASD connections, and mean dimension. J. Funct. Anal., 260(5):1369-1427, 2011.

[Tsu1] Masaki Tsukamoto. Moduli space of Brody curves, energy and mean dimension. Nagoya Math. J., 192:27-58, 2008.

[Tsu2] Masaki Tsukamoto. Deformation of Brody curves and mean dimension. Ergodic Theory Dynam. Systems, 29(5):1641-1657, 2009.