Embedding minimal dynamical systems into Hilbert cubes

TL;DR

This paper proves that minimal dynamical systems with mean dimension less than half of N can be embedded into the Hilbert cube shift space, establishing an optimal bound and employing Fourier and complex analysis techniques.

Contribution

It establishes the optimal mean dimension bound of N/2 for embedding minimal systems into Hilbert cube shifts, improving previous results.

Findings

Minimal systems with mean dimension less than N/2 can be embedded.

The bound N/2 is proven to be optimal.

Fourier and complex analysis are used in the proof.

Abstract

We study the problem of embedding minimal dynamical systems into the shift action on the Hilbert cube $\left([0,1]^N\right)^{\mathbb{Z}}$. This problem is intimately related to the theory of mean dimension, which counts the averaged number of parameters of dynamical systems. Lindenstrauss proved that minimal systems of mean dimension less than $N/36$ can be embedded into $\left([0,1]^N\right)^{\mathbb{Z}}$, and he proposed the problem of finding the optimal value of the mean dimension for the embedding. We solve this problem by proving that minimal systems of mean dimension less than $N/2$ can be embedded into $\left([0,1]^N\right)^{\mathbb{Z}}$. The value $N/2$ is optimal. The proof uses Fourier and complex analysis.