The Hilbert cube contains a minimal subshift of full mean dimension

Abstract

We construct a minimal dynamical system of mean dimension equal to $1$, which can be embedded in the shift action on the Hilbert cube $[0,1{]}^{\mathbb{Z}}$. This clarifies a seemingly plausible impression about embedding possibility in relation to mean dimension. Our result finally leads to a full understanding of a pair of exact ranges of all the possible values of mean dimension, within which there will always be a minimal dynamical system that can/cannot be embedded in the shift action on the Hilbert cube.