Embedding minimal dynamical systems into Hilbert cubes

By Yonatan Gutman, Masaki Tsukamoto. Inventiones Mathematicae, 2015

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We study the problem of embedding minimal dynamical systems into the shift action on the Hilbert cube $$\left( [0,1]^N\right) ^{\mathbb {Z}}$$ [ 0 , 1 ] N Z . This problem is intimately related to the theory of mean dimension, which counts the average number of parameters for describing a dynamical system. Lindenstrauss proved that minimal systems of mean dimension less than cN for $$c=1/36$$ c = 1 / 36 can be embedded in $$\left( [0,1]^N\right) ^{\mathbb {Z}}$$ [ 0 , 1 ] N Z , and asked what is the optimal value for c . We solve this problem by showing embedding is possible when $$c=1/2$$ c = 1 / 2 . The value $$c=1/2$$ c = 1 / 2 is optimal. The proof exhibits a new interaction between harmonic analysis and dynamical coding techniques.

Subfields: Computational Theory and Mathematics, Mathematical Physics

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