Abstract
The purpose of this survey is to explain basic ideas of mean dimension theory. Mean dimension represents a number of parameters per unit time for describing a dynamical system. It was introduced by Gromov in 1999. His motivation is to propose a new approach to geometric analysis over noncompact manifolds. Rather independently of this original motivation, Lindenstrauss and Weiss found applications of mean dimension to classical problems in topological dynamics. In this survey we start from the viewpoint of Lindenstrauss–Weiss and first review the applications to topological dynamics. Next we explain the relation to geometric analysis, in particular holomorphic curve theory. Mean dimension is connected to rate distortion theory, which is a branch of information theory studying the trade-off between distortion and compression rate in lossy data compression. In the last three sections we review a variational principle between mean dimension and rate distortion function and then discuss its possible application to holomorphic curve theory.
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