TY - JOUR
T1 - Embedding minimal dynamical systems into Hilbert cubes
AU - Gutman, Yonatan
AU - Tsukamoto, Masaki
N1 - Publisher Copyright:
© 2020, Springer-Verlag GmbH Germany, part of Springer Nature.
PY - 2020/7/1
Y1 - 2020/7/1
N2 - We study the problem of embedding minimal dynamical systems into the shift action on the Hilbert cube ([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 can be embedded in ([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. The value c= 1 / 2 is optimal. The proof exhibits a new interaction between harmonic analysis and dynamical coding techniques.
AB - We study the problem of embedding minimal dynamical systems into the shift action on the Hilbert cube ([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 can be embedded in ([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. The value c= 1 / 2 is optimal. The proof exhibits a new interaction between harmonic analysis and dynamical coding techniques.
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U2 - 10.1007/s00222-019-00942-w
DO - 10.1007/s00222-019-00942-w
M3 - Article
AN - SCOPUS:85077600476
VL - 221
SP - 113
EP - 166
JO - Inventiones Mathematicae
JF - Inventiones Mathematicae
SN - 0020-9910
IS - 1
ER -