Mean dimension and a sharp embedding theorem: extensions of aperiodic subshifts

Abstract

We show that if $(X,T)$ is an extension of an aperiodic subshift (a subsystem of $({1,2,...,l}^{\mathbb{Z}},\mathrm{shift})$ for some $l\in \mathbb{N}$) and has mean dimension $mdim(X,T)<\frac{D}{2}$ $(D\in \mathbb{N}$), then it embeds equivariantly in (([0,1]^{D})^{\mathbb{Z}},\mathrm{shift})$.Theresultissharp.If$(X,T)$isanextensionofanaperiodiczero-dimensionalsystemthenitembedsequivariantlyin$(([0,1]^{D+1})^{\mathbb{Z}},\mathrm{shift})$.

References in corpus (2)

• Mean Dimension & Jaworski-type Theorems
• Dynamical Embedding in Cubical Shifts & the Topological Rokhlin and Small Boundary Properties

Cited by in corpus (9)

• Mean Dimension & Jaworski-type Theorems
• The embedding problem in topological dynamics and Takens' theorem
• Minimal subshifts of arbitrary mean topological dimension
• Dynamical Embedding in Cubical Shifts & the Topological Rokhlin and Small Boundary Properties
• Mean dimension and a non-embeddable example for amenable group actions
• Takens-type reconstruction theorems of one-sided dynamical systems on compact metric spaces
• Dynamical correspondences of $L^2$-Betti numbers
• Mean dimension and an embedding theorem for real flows
• Conditional mean dimension