Mean dimension of product spaces: a fundamental formula

Abstract

Mean dimension is a topological invariant of dynamical systems, which originates with Mikhail Gromov in 1999 and which was studied with deep applications around 2000 by Elon Lindenstrauss and Benjamin Weiss within the framework of amenable group actions. Let a countable discrete amenable group G act continuously on compact metrizable spaces X and Y. Consider the product action of G on the product space $$X\times Y$$ . The product inequality for mean dimension is well known: $${{\,\mathrm{\textrm{mdim}}\,}}(X\times Y,G)\le {{\,\mathrm{\textrm{mdim}}\,}}(X,G)+{{\,\mathrm{\textrm{mdim}}\,}}(Y,G)$$ , while it was unknown for a long time if the product inequality could be an equality. In 2019, Masaki Tsukamoto constructed the first example of two different continuous actions of G on compact metrizable spaces X and Y, respectively, such that the product inequality becomes strict. However, there is still one longstanding problem which remains open in this direction, asking if there exists a continuous action of G on some compact metrizable space X such that $${{\,\mathrm{\textrm{mdim}}\,}}(X\times X,G)<2\cdot {{\,\mathrm{\textrm{mdim}}\,}}(X,G)$$ . We solve this problem. Somewhat surprisingly, we prove, in contrast to (topological) dimension theory, a rather satisfactory theorem: If an infinite (countable discrete) amenable group G acts continuously on a compact metrizable space X, then we have $${{\,\mathrm{\textrm{mdim}}\,}}(X^n,G)=n\cdot {{\,\mathrm{\textrm{mdim}}\,}}(X,G)$$ , for any positive integer n. Our product formula for mean dimension, together with the example and inequality (stated previously), eventually allows mean dimension of product actions to be fully understood.