Embedding minimal dynamical systems into Hilbert cubes

摘要

We study the problem of embedding minimal dynamical systems into the shift action on the Hilbert cube mml:mfenced close=")" open="("[0,1]NZ 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 can be embedded in mml:mfenced close=")" open="("[0,1]NZ and asked what is the optimal value for c. We solve this problem by showing embedding is possible when c=1/2 The value c=1/2 is optimal. The proof exhibits a new interaction between harmonic analysis and dynamical coding techniques.