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 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 \(\left( [0,1]^N\right) ^{\mathbb {Z}}\), 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.
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Notes
A dynamical system (X, T) is called an infinite minimal system if X is an infinite set and (X, T) is a minimal dynamical system. This is equivalent to the condition that (X, T) is a minimal system having no periodic points.
It becomes a bit simpler because we do not need to use harmonic analysis. Indeed the authors first found the proof of this discrete signal version and then turned their attention to band-limited signals.
When the authors were trying to prove Theorem 1.4, we were stuck here for a while. After some struggle, we realized that we can overcome the difficulty by using band-limited signals.
Here we have used real algebraic geometry in an essential way. We cannot hope that the topological dimension of \(\bigcup _{a\ge 0} a \mathcal {B}\) will behave nicely if \(\mathcal {B}\) is a fractal.
Another reason why short intervals cause a difficulty comes from the nature of mean dimension theory. Mean dimension provides a good control of the behavior of sufficiently long orbit segments of a dynamical system. But it tells us nothing about short ones. See Sect. 7.1 for more precise explanation.
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Acknowledgements
The authors would like to deeply thank Professor Elon Lindenstrauss. His influence is prevailing in the paper. The authors learned from him the most important ideas such as signal processing and Voronoi diagram in one-dimension higher space. The authors also would like to thank the referee for many helpful comments.
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Y.G. was partially supported by the Marie Curie Grant PCIG12-GA-2012-334564 and the National Science Center (Poland) Grants 2013/08/A/ST1/00275 and 2016/22/E/ST1/00448. M.T. was supported by Grant-in-Aid for Young Scientists (B) 25870334 from JSPS.
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Gutman, Y., Tsukamoto, M. Embedding minimal dynamical systems into Hilbert cubes. Invent. math. 221, 113–166 (2020). https://doi.org/10.1007/s00222-019-00942-w
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DOI: https://doi.org/10.1007/s00222-019-00942-w