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We investigate the dynamical property of the naive mean dimension for continuous actions of any countable group on compact metrizable spaces. It is shown that naive mean dimension serves as an upper bound of sofic mean dimension for actions of nonamenable groups. For algebraic actions we obtain more satisfactory results by looking at the naive version of mean rank for modules over integral group rings. We also consider the naive metric mean dimension and investigate its relations with sofic metric mean dimension., Comment: 25 pages. Comments are welcome!

Mean dimension may decrease after taking the natural extension. In this paper we show that mean dimension is preserved by natural extension for an endomorphism on a compact metrizable abelian group. As an application, we obtain that the mean dimension of an algebraic cellular automaton coincides withthe mean dimension of its natural extension, which strengthens a result of Burguet and Shi \cite{BS21} with a different proof., Comment: 11pages. Some typos are fixed. to be published

Let $X$ be a compact metric space and $G$ a finitely generated group. Suppose $\phi:G\rightarrow {\rm Homeo}(X)$ is a continuous action. We show that if $\phi$ is both distal and expansive, then $X$ must be finite. A counterexample is constructed to show the necessity of finite generation condition on $G$. This is also a supplement to a result due to Auslander-Glasner-Weiss which says that every distal action by a finitely generated group on a zero-dimensional compactum is equicontinuous.

We show that if $G$ is a finitely generated group of subexponential growth and $X$ is a Suslinian continuum, then any action of $G$ on $X$ cannot be expansive., Comment: 7 pages

We study fibrations in the category of cubespaces/nilspaces. We show that a fibration of finite degree $f \colon X\rightarrow Y$ between compact ergodic gluing cubespaces (in particular nilspaces) factors as a (possibly countable) tower of compact abelian Lie group principal fiber bundles over $Y$. If the structure groups of $f$ are connected then the fibers are (uniformly) isomorphic (in a strong sense) to an inverse limit of nilmanifolds. In addition we give conditions under which the fibers of $f$ are isomorphic as subcubespaces. We introduce regionally proximal equivalence relations relative to factor maps between minimal topological dynamical systems for an arbitrary acting group. We prove that any factor map between minimal distal systems is a fibration and conclude that if such a map is of finite degree then it factors as a (possibly countable) tower of principal abelian Lie compact group extensions, thus achieving a refinement of both the Furstenberg's and the Bronstein-Ellis structure theorems in this setting., Comment: Some typos are fixed. The proof of Theorem 1.29 is modified due to a gap in Lemma 5.2

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., Comment: Some typos are corrected. To appear in Journal of Ergodic Theory and Dynamical Systems

We partially generalize Peters' formula on modules over the group ring ${\mathbb F} \Gamma$ for a given finite field ${\mathbb F}$ and a sofic group $\Gamma$. It is also discussed that how the values of entropy are related to the zero divisor conjecture.

We investigate dynamical analogues of the $L^2$-Betti numbers for modules over integral group ring of a discrete sofic group. In particular, we show that the $L^2$-Betti numbers exactly measure the failure of addition formula for dynamical invariants., Comment: The main result is reformulated a little bit and the proof is more self-contained

Given a length function L on the R-modules of a unital ring R, for each sofic group $\Gamma$ we define a mean length for every locally L-finite $R\Gamma$-module relative to a bigger $R\Gamma$-module. We establish an addition formula for the mean length. We give two applications. The first one shows that for any unital left Noetherian ring R, $R\Gamma$ is stably direct finite. The second one shows that for any $Z\Gamma$-module M, the mean topological dimension of the induced $\Gamma$-action on the Pontryagin dual of M coincides with the von Neumann-L\"{u}ck rank of M., Comment: 54 pages. Minor changes. To appear in Adv. Math

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-L\"uck 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., Comment: 35 pages, minor changes. To appear in J. Reine Angew. Math

We define the topological pressure for any sub-additive potentials of the countable discrete amenable group action and any given open cover. A local variational principle for the topological pressure is established., Comment: 16 pages