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We construct a minimal dynamical system of mean dimension equal to $1$, which can be embedded in the shift action on the Hilbert cube $[0,1]^mathbb{Z}$. Our result clarifies a seemingly plausible impression and finally enables us to have a full understanding of (a pair of) the exact ranges of all possible values of mean dimension, within which there will always be a minimal dynamical system that can be (resp. cannot be) embedded in the shift action on the Hilbert cube. The key ingredient of our idea is to produce a dense subset of the alphabet $[0,1]$ more gently.
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We introduce mean dimensions for continuous actions of countable sofic groups on compact metrizable spaces. These generalize the Gromov-Lindenstrauss-Weiss mean dimensions for actions of countable amenable groups, and are useful for distinguishing continuous actions of countable sofic groups with infinite entropy.
We introduce some notions of conditional mean dimension for a factor map between two topological dynamical systems and discuss their properties. With the help of these notions, we obtain an inequality to estimate the mean dimension of an extension system. The conditional mean dimension for $G$-extensions are computed. We also exhibit some applications in the dynamical embedding problems.
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 $Xtimes Y$. The product inequality for mean dimension is well known: $mathrm{mdim}(Xtimes Y,G)lemathrm{mdim}(X,G)+mathrm{mdim}(Y,G)$, 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 $mathrm{mdim}(Xtimes X,G)<2cdotmathrm{mdim}(X,G)$. 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 $mathrm{mdim}(X^n,G)=ncdotmathrm{mdim}(X,G)$, 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.
We investigate the mean dimension of a cellular automaton (CA for short) with a compact non-discrete space of states. A formula for the mean dimension is established for (near) strongly permutative, permutative algebraic and unit one-dimensional automata. In higher dimensions, a CA permutative algebraic or having a spaceship has infinite mean dimension. However, building on Meyerovitchs example, we give an example of algebraic surjective cellular automaton with positive finite mean dimension.
We introduce an invariant, called mean rank, for any module M of the integral group ring of a discrete amenable group $Gamma$, as an analogue of the rank of an abelian group. It is shown that the mean dimension of the induced $Gamma$-action on the Pontryagin dual of M, the mean rank of M, and the von Neumann-Luck rank of M all coincide. As applications, we establish an addition formula for mean dimension of algebraic actions, prove the analogue of the Pontryagin-Schnirelmnn theorem for algebraic actions, and show that for elementary amenable groups with an upper bound on the orders of finite subgroups, algebraic actions with zero mean dimension are inverse limits of finite entropy actions.
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