What is Mean Dimension?

Yonatan Gutman

Before we discuss mean dimension, we need to discuss other concepts. If you know this material you can always skip to the discussion of mean dimension by scrolling down or by clicking here. I will try to keep the technical parts at bay. Topological dynamics and ergodic theory are subfields of the more general field of dynamical systems. So our first question is:

What is a Dynamical System?

The best non-technical explanation available on the web, I know of, is the one written by Mike Hochman here. Let us summarize some of the ideas: examples of dynamical systems are the solar system, the weather, the production of white cells in the blood, the motion of billiard balls on a billiard table, the movement of gas molecules in a container, sugar dissolving in a cup of coffee, the stock market, the formation of traffic jams etc. etc. Although these examples and numerous others in the fields of physics, biology, chemistry, engineering, mathematics, ecology... may differ substantially from each other, they can all be modeled by a phase space and an evolution rule. The phase space consists of all possible world-states (e.g. the locations and velocities of the planets of the solar system) whereas the evolution rule is the transformation which sends every world-state, representing the state of the system "now'', to the world-state representing the state of the system "one unit time later''.

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:

What is Dimension?

When we talk about dimension we mean the so called Lebesgue covering dimension. This should not be confused with Hausdorff dimension or other fractal dimensions. The Lebesgue covering dimension is defined for every topological space. Its possible values are $0,1,2,\ldots$ and it can also be $\infty$. Two of its most important properties are:

• 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!).

Combining these two facts you get Brouwer's invariance of domain theorem which states that $\mathbb{{R}}^{n}$ is not homeomorphic to $\mathbb{{R}}^{m}$, if $m\neq n$. You may consider this fact obvious - but try to prove it! So how is Lebesgue covering dimension of a topological space $X$ defined? To simplify matters we will assume that $X$ is compact and metric and let us call the metric $d$. We now say that $dim(X)\leq n$ $(n\in\mathbb{{Z}}_{+})$, if for every $\epsilon>0$ we can find a finite cover by of $X$ by open sets of diameter (measured with $d$) less than $\epsilon$, such that any $n+2$ distinct sets from this cover have empty intersection, or in other words any element $x\in X$ is contained in at most $n+1$ sets of the cover. To motivate this definition take a rectangular A4 paper and try to cover it by small discs. You will find out, that because the dimension of the plane is $2$, you can find a cover where any $4$ distinct discs do not intersect, i.e. any point is contained in at most $3$ discs. You will not be able to find a cover by small discs where any $3$ distinct discs do not intersect. Finally we say that $dim(X)=n$ if $dim(X)\leq n$ but it does not hold $dim(X)\leq n-1$, and $dim(X)=\infty$ if it does not hold $dim(X)\leq n$ for any $n$. It is convenient to introduce the notation $Widim_{\epsilon}(X,d)$ to be the minimal $n$, such that there exists a finite cover of $X$ by open sets of diameter less than $\epsilon$ such that any $n+2$ distinct sets from this cover have empty intersection. A moment of thought will show that in terms of $Widim_{\epsilon}(X,d)$, $dim(X,d)$ may be defined by the following formula: \[ dim(X,d)=\sup_{\epsilon>0}Widim_{\epsilon}(X,d) \] We will call $Widim_{\epsilon}(X,d)$ the $\epsilon$-(coarse) dimension of $(X,d)$ - Heuristically it is the dimension of $X$ when we neglect the fine structure smaller than $\epsilon$. It is possible to prove that $dim(X,d)$ only depends on the topology of $X$, so any compatible metric will yield the same number, however this will not be important for us. We are now ready to introduce mean dimension.

What is Mean Dimension?

Let $(X,T)$ be a t.d.s, that means $X$ is a compact metric space (with metric we call $d$) and $T$ is a homeomorphism. The fact the space is equipped with an evolution law given by $T$, enables us to "refine our metric'' in a manner yielding information related to the dynamics of the system. Let us define $d_{n}(x,y)=\max{\{d(x,y),d(Tx,Ty),\ldots d(T^{n-1}x,T^{n-1}y)\}}$ or in other words \[ d_{n}(x,y)=\max_{0\leq i\leq n-1}d(T^{i}x,T^{i}y) \] This means that $x$ and $y$ are $\epsilon-d_{n}$ close ($d_{n}(x,y)<\epsilon$) if $x$ and $y$ stay $\epsilon$-close for the first $n$ iterates of the map $T$ (where we start our count from iterate zero $T^{0}=Id$). If we think of $T$ as giving the state of the system after one second, then $d_{n}(x,y)<\epsilon$ means $x$ and $y$ are $\epsilon$-close for the first $n-1$ seconds. For fixed $\epsilon>0$ consider the quantities $Widim_{\epsilon}(X,d_{1}),Widim_{\epsilon}(X,d_{2}),\ldots$ Obviously this is an increasing sequence which indicates how much $\epsilon$-coarse dimension one can see using an increasing number of iterations. To quantify the increase we resort to the simplest approximation: linear approximation, i.e. we look for a constant $g_{\epsilon}$ such that $Widim_{\epsilon}(X,d_{n})=g_{\epsilon}n+o(n)$. This translates to the following definition: \[ g_{\epsilon}=\lim_{n\rightarrow\infty}\frac{Widim_{\epsilon}(X,d_{n})}{n} \] You may be worried the limit does not exist but actually it does (why? Hint: the sequence is a sub-additive). We may call this constant $g_{\epsilon}$ the growth rate (per unit time) of $\epsilon$-coarse dimension or in short the $\epsilon$-dimension growth rate. Similarly to the definition of dimension we define mean dimension to be the supremum over all $\epsilon$-dimension growth rates, i.e. $mdim(X,d,T)=\sup_{\epsilon>0}g_{_{\epsilon}}$, or in other words: \[ mdim(X,d,T)=\sup_{\epsilon>0}\lim_{n\rightarrow\infty}\frac{Widim_{\epsilon}(X,d_{n})}{n} \] One can prove $mdim(X,d,T)$ only depends on the topology of $X$ and the map $T:X\rightarrow X$ (i.e. every compatible metric will yield the same quantity) and is an invariant of topological dynamical systems. We thus use the notation $mdim(X,T)$. A natural question is:

Which Properties does Mean Dimension have?

• 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: \[ (\ldots,x_{-2},x_{-1},\mathbf{x_{0}},x_{1},x_{2},\ldots) \] \[ \downarrow \] \[ (\ldots,x_{-1},x_{0},\mathbf{x_{1}},x_{2},x_{3},\ldots) \]

$(([0,1]^{d})^{\mathbb{Z}},shift)$ is referred to as the full topological shift on the alphabet $[0,1]^{d}$ or simply the $d$-cubical shift. For its mean dimension it holds: \[ mdim(([0,1]^{d})^{\mathbb{Z}},shift)=d \]

History and a Partial Bibliography for Applications of Mean Dimension

Mean dimension was introduced by Gromov in [Gro] and developed and studied systematically by Lindenstrauss and Weiss in [LW]. Since its introduction mean dimension has been used in the several fields of mathematics. Follows a partial list of relevant sources:

Symbolic Dynamics ([BD,Gut1]).

Holomorphic Functions ([Gou1,Gro,Tsu1,Tsu2])

Mathematical Physics ([Gro,MT]),

Cellular Automata ([CSC]) (Note: Strictly speaking a closely related concept is used),

Topological Dynamics ([Gut1,Gut2,Gut3,GT,Lin,LT,LW]),

Von Neumann Dimension & related ([Gou2,LL]).

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]).

Universal Embedding for Topological Dynamical Systems

Personally I have been interested in the application of mean dimension in the context of embedding topological dynamical systems in certain natural systems. The motivating phenomenon in dimension theory is the so-called Menger-Nöbeling theorem (1931) that (in particular) says that any compact metric space of dimension $d$ (or less) can be embedded in the $(2d+1)$dimensional Euclidean cube $I_{2d+1}=[0,1]^{2d+1}$. We thus say that $I_{2d+1}$ is universal for compact metric space of dimension $\leq d$. One may wonder if a similar phenomenon holds in the context of topological dynamical systems. First one must however decide what is the appropriate analogue of the Euclidean cube in the dynamical context. Well, you can find such a candidate in the end of this section - The full topological shift over the $d$-dimensional cube $(([0,1]^{d})^{\mathbb{Z}},shift)$, also known as the $d$-cubical shift. So what is the analogue of the Menger-Nöbeling theorem in the dynamical context? This is actually an active area of research so if you are interested I would recommend starting with my work Mean dimension & Jaworski-type theorems.

References

[BD] Mike Boyle and Tomasz Downarowicz. The entropy theory of symbolic extensions. Invent. Math., 156(1):119-161, 2004.
[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.