Mean topological dimension
Given a continuous map \(f\) on a compact metric space \(X\), an invariant called mean topological dimension has been recently introduced by M. Gromov. The paper being reviewed develops some basic properties of this invariant. In some cases, it gives a way of distinguishing dynamical systems with infinite topological entropy. Given a finite open cover \(\alpha\) of \(X\), its order is one less than the cardinality of any collection of subsets of \(\alpha\) with nonempty intersection, and \({\mathcal D}(\alpha)\) is defined to be the minimum order of any finite cover of \(X\) that refines \(\alpha\). Under fairly general conditions on \(\alpha\), it is shown that if \(X\) has topological dimension at most \(k\), then \({\mathcal D}(\alpha)\leq k\). If \(T:X\rightarrow X\) is continuous, then the mean dimension of the system \((X,T)\) is the supremum over all finite open covers \(\alpha\) of \(\lim_{n\rightarrow \infty}({\mathcal D}(\alpha_n)/n)\), where \(\alpha_n\) is the common join of the covers \(T^{-i}(\alpha)\), \(0\leq i < n\). It follows that if \(X\) is finite dimensional, then this mean dimension is 0. Moreover, the mean dimension of the shift dynamical system generated by a \(k\)-dimensional, compact space of symbols is at most \(k\). This is used to show that if a minimal system is embeddable in the shift space based on the symbols \(0\leq s\leq 1\), then the mean dimension of the system is at most 1. Examples of minimal systems of mean dimension greater than one are given. Finally, an alternative approach, reminiscent of one definition of topological entropy, is described; here there is explicit dependence upon the metric. It is shown that the new metric version of mean dimension is an upper bound for the original mean dimension. It follows from this that if the topological entropy is finite, then the mean dimension is 0.
scientific article; zbMATH DE number 1436095