Mean Dimension & Jaworski-type Theorems

Abstract

According to the celebrated Jaworski Theorem, a finite dimensional aperiodic dynamical system $(X,T)$ embeds in the $1$-dimensional cubical shift $([0,1{]}^{\mathbb{Z}},shift)$. If $X$ admits periodic points (still assuming $\dim (X)<\infty$) then we show in this paper that periodic dimension $perdim(X,T)<\frac{d}{2}$ implies that $(X,T)$ embeds in the $d$-dimensional cubical shift $(([0,1{]}^d{)}^{\mathbb{Z}},shift)$. This verifies a conjecture by Lindenstrauss and Tsukamoto for finite dimensional systems. Moreover for an infinite dimensional dynamical system, with the same periodic dimension assumption, the set of periodic points can be equivariantly immersed in $(([0,1{]}^d{)}^{\mathbb{Z}},shift)$. Furthermore we introduce a notion of markers for general topological dynamical systems, and use a generalized version of the Bonatti-Crovisier tower theorem, to show that an extension $(X,T)$ of an aperiodic finite-dimensional system whose mean dimension obeys $mdim(X,T)<\frac{d}{16}$ embeds in the $(d+1)$-cubical shift $(([0,1{]}^{d+1}{)}^{\mathbb{Z}},shift)$.

To appear in Proceedings of the London Mathematical Society

References in corpus (2)

• The Rokhlin dimension of topological Z^m-actions
• Mean dimension and a sharp embedding theorem: extensions of aperiodic subshifts

Cited by in corpus (28)

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• Rokhlin dimension for actions of residually finite groups
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• The nuclear dimension of C*-algebras associated to homeomorphisms
• Symbolic dynamics in mean dimension theory
• The C*-algebra of a minimal homeomorphism with finite mean dimension has finite radius of comparison
• Dynamic asymptotic dimension for actions of virtually cyclic groups
• Dynamical Embedding in Cubical Shifts & the Topological Rokhlin and Small Boundary Properties
• Double variational principle for mean dimension
• An explicit compact universal space for real flows
• Large dynamics of Yang--Mills theory: mean dimension formula
• Application of signal analysis to the embedding problem of ${\mathbb{Z}}^k$-actions
• From rate distortion theory to metric mean dimension: variational principle
• Mean dimension and a non-embeddable example for amenable group actions
• $G$-index, topological dynamics and marker property
• Mean dimension of ${\mathbb{Z}}^k$-actions
• Takens-type reconstruction theorems of one-sided dynamical systems on compact metric spaces
• Double variational principle for mean dimension with potential
• Mean dimension and an embedding theorem for real flows
• Dynamical correspondences of $L^2$-Betti numbers
• The nuclear dimension of $C^{\ast }$-algebras associated to topological flows and orientable line foliations
• Conditional mean dimension
• Mean dimension of Bernstein spaces and universal real flows
• Finite mean dimesnion and marker property