Embedding minimal dynamical systems into Hilbert cubes

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.

38 pages

References in corpus (1)

• Mean Dimension & Jaworski-type Theorems

Cited by in corpus (12)

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• Mean dimension and metric mean dimension for non-autonomous dynamical systems
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• Double variational principle for mean dimension
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• From rate distortion theory to metric mean dimension: variational principle
• Embedding asymptotically expansive system
• Properties of mean dimension and metric mean dimension coming from the topological entropy
• Double variational principle for mean dimension with potential
• Dynamical correspondences of $L^2$-Betti numbers
• Mean dimension of full shifts