Dimension and embeddability for Dynamical systems

It is well known that a compact metric space embeds in a finite-dimensional Euclidean space if and only if it has finite Lebesgue covering dimension. In 1999, Gromov introduced the notion of mean dimension, a fundamental invariant for dynamical systems that could be considered as a dynamical analogue of Lebesgue covering dimension.

A (discrete time) dynamical system is a pair (X,\Phi), where $\Phi:X\to X$ be a homeomorphism of a compact metric space $X$. The \emph{shift embeddability} or \emph{sampling-rate problem} asks: Under what conditions do there exists a finite number of continuous real-valued functions $f_1,\ldots,f_d:X \to \mathbb{R}$ so that a point $x \in X$ can be uniquely recovered by sampling the values of the $f_1,\ldots,f_d$ along the orbit of $x$?

In this talk I will describe historical developments around the shift embeddability problem and some exciting recent developments.

We will not assume any specialized background (in particular, we will recall the definition of Lebesgue covering dimension).