TY - JOUR

T1 - Double variational principle for mean dimension

AU - Lindenstrauss, Elon

AU - Tsukamoto, Masaki

N1 - Publisher Copyright:
© 2019, Springer Nature Switzerland AG.

PY - 2019/8/1

Y1 - 2019/8/1

N2 - We develop a variational principle between mean dimension theory and rate distortion theory. We consider a minimax problem about the rate distortion dimension with respect to two variables (metrics and measures). We prove that the minimax value is equal to the mean dimension for a dynamical system with the marker property. The proof exhibits a new combination of ergodic theory, rate distortion theory and geometric measure theory. Along the way of the proof, we also show that if a dynamical system has the marker property then it has a metric for which the upper metric mean dimension is equal to the mean dimension.

AB - We develop a variational principle between mean dimension theory and rate distortion theory. We consider a minimax problem about the rate distortion dimension with respect to two variables (metrics and measures). We prove that the minimax value is equal to the mean dimension for a dynamical system with the marker property. The proof exhibits a new combination of ergodic theory, rate distortion theory and geometric measure theory. Along the way of the proof, we also show that if a dynamical system has the marker property then it has a metric for which the upper metric mean dimension is equal to the mean dimension.

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U2 - 10.1007/s00039-019-00501-8

DO - 10.1007/s00039-019-00501-8

M3 - Article

AN - SCOPUS:85066493351

SN - 1016-443X

VL - 29

SP - 1048

EP - 1109

JO - Geometric and Functional Analysis

JF - Geometric and Functional Analysis

IS - 4

ER -