Tsukamoto - Introduction to mean dimension

Abstract

Mean dimension is a topological invariant of dynamical systems introduced by Gromov in 1999. It is a dynamical version of topological dimension. It evaluates the number of parameters per unit time for describing a given dynamical system. Gromov introduced this notion for the purpose of exploring a new direction of geometric analysis. Independently of this original motivation, Elon Lindenstrauss and Benjamin Weiss found deep applications of mean dimension in topological dynamics. I plan to survey some highlights of the mean dimension theory.

Gromov

Gromov tried to dynamicalize concepts from geometry. i.e. find analogs

• Number of holes ⇒ topological entropy
• Pidgeon hole principle ⇒ Symbolic coding
• Topological dimension ⇒ Mean dimension

Mean dimension is the number of parameter per unit times for describing a dynamical system.

Topological entropy and symbolic dynamics

Definition of Dynamisches System. (on compact space) In other words a compact metrizible space equipped with Z action ⇒ We could generalize from Z to other groups.

Topological entropy. For $ϵ>0$ and a compact metric space $(X,d)$ we define $\#(X,d,ϵ)$ as the minimim number $n$ for which there is an open covering $X=U_1\cup U_2\cup ...\cup U_n$ Satisfying $Diam(U_{i)}\le ϵ$.

Let $T$ be a homeo. For $N\ge 1$ define metric bby $d_N(x,y)={\max }_{0\le x<n}d(T^{n}x,T^{n}y)$ The entropy is defined by $h_{top}(X,T)={\lim }_{ϵ\to 0}({\lim }_{N\to \infty }\frac{\log \#(X,d_N,ϵ)}{n?})$

$\#(X,d_N,ϵ)$ is the covering number. Now to symbolic dynamics.

Recipe for dynamcalization Given somt geoemtric/topological invariant $Q(K)$ of $Spaces$K try to define its dynamicsl version $Q_{dynamics}(X,T)$ so that $Q_{dynamicalization}=...$

Definition of an embedding: For $n\ge 1$ we can define $P_n(X,T)$ as the set of $n$-periodic points. If $(X,T)$ embeds ind $(Y,S)$ then

• $h_{top}(X,T)\le h_{top}(Y,S)$
• …

Krieger embedding theorem Let $A,B$ finite sets. $X$ a subshift of $A^Z$ this staisfies:

• $h_{top}(X,\sigma )\le \log |B|$
• $|P_n(X,\sigma |\le |B{|}^n$ for all $n\ge 1$.

Then $X$ embeds in $B^Z$.

Mean dimension and topological dynamics

A conti map $f:X\to Y$ is called $\text{epsilo}$-embedding if $Diam{f}^{-1}(y)<ϵ$ for all $y\in Y$.

Defint $Widi{m}_{ϵ}(X,d)$ as the minimom interger $n$ for whicth there is an $n$-deimension simplicial complex $P$ and en $ϵ$-embedding $f:X\to P$.

We define topological dimension by $dimX={\lim }_{\to 0}Widi{m}_{eo}(X,d)$ This is a topological invariant.

Mnger Nöbeling theorem If $dimX<\frac{N}{2}$then X topologiccaly embedding in $[0,1{]}^N$.

Dynamical embedding problem $[0,1{]}^Z=...\times [0,1]\times ...$ Consider shift maps. Then the pair is called the shift of the Hilbert cube.

Given a dynamical system $(X,T)$ decide wheter we can embed it in $([0,1{]}^N,\sigma )$.

Jaworski If a finite dimensional dynamicsl system $(X,T)$ has no periodic point then it embeds in $([0,1{]}^Z,\sigma )$. (very surpising but I dont understand why)

Question of Auslander Can we embed every minimal dynamicsl system in $([0,1{]}^Z,\sigma )$. It is minimal, if every orbit is dense in $X$.

Mean dimension and geometric analysis

His paper title: Topological invariant of dynamical systems and spaces of holomorphic maps: 1

uppose a grop $\Gamma$ cocompactly acts on an open manifold $M$. We consider a geometric PDE on $M$ and lot $X$ be the space of its soolutions. If the qeuation is invarint under $\Gamma$ then $\Gamma$ also acts on $X$.

Mean dimension of ${\mathbb{R}}^k$-actions. Let $T:{\mathbb{R}}^k\times X\to X$ bie a cntinuous action. For a R^n subset $\Omega$ we define $d_{\Omega }(x,y)={\sup }_{uin\Omega }d(T^{u}x,T^{u}y)$ We define mean dimension by $mdim(X,T)={\lim }_{{\lim }_{L\to \infty }}\frac{Widim...}{}$