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Bingbing Liang

Publications and source records attributed to Bingbing Liang.

13 recordsLinked to original sources
arXiv2026

Mean weak length

We introduce a weak version of the classical length function, termed the weak length function, defined on subsets of $R$-modules relative to the ambient $R$-modules over a unital ring $R$. We further consider the concept of mean weak length for each $R{\Gamma}$-module relative to a bigger $R{\Gamma}$-module associated with an amenable group ${\Gamma}$. Under an appropriate upgrading condition together with certain mild assumptions, we establish that the mean weak length satisfies an addition formula with respect to short exact sequences. This result has three applications. First, we provide a purely algebraic proof of the additivity of algebraic entropy, which is a property originally established via topological entropy methods. Second, within our unified framework, we give an alternative and conceptual proof of the additivity of mean length, previously obtained by Li-Liang and Virili using different approaches. Third, we recover a particular case of amenable extension for Sylvester rank functions in the spirit of Jiang-Li.

math.RA↗
arXiv2024

Naive mean dimension

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.

math.DS↗
arXiv2024

Conditional sofic mean dimension

We undertake a study of the conditional mean dimensions for a factor map between continuous actions of a sofic group on two compact metrizable spaces. When the group is infinitely amenable, all these concepts recover as the conditional mean dimensions introduced in \cite{L22}. A range of results established for actions of amenable groups are extended to the sofic framework. Additionally, our exploration encompasses the study of the relative mean dimension introduced by Tsukamoto, shedding light on its inherent correlation with the conditional metric mean dimension within the sofic context. A lower bound on the conditional metric mean dimension, originally proposed by Shi-Tsukamoto, is extended to the sofic case.

math.DS↗
arXiv2023

Mean dimension of natural extension of algebraic systems

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.

math.DS↗
arXiv2021

Conditional mean dimension

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.

math.DS↗
arXiv2021

The non-coexistence of distality and expansivity for group actions on infinite compacta

Let $X$ be a compact metric space and $G$ a finitely generated group. Suppose $ϕ:G\rightarrow {\rm Homeo}(X)$ is a continuous action. We show that if $ϕ$ 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.

math.DS↗
arXiv2021

On the structure theory of cubespace fibrations

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.

math.DS↗
arXiv2019

Sofic Mean Length

Given a length function L on the R-modules of a unital ring R, for each sofic group $Γ$ we define a mean length for every locally L-finite $RΓ$-module relative to a bigger $RΓ$-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, $RΓ$ is stably direct finite. The second one shows that for any $ZΓ$-module M, the mean topological dimension of the induced $Γ$-action on the Pontryagin dual of M coincides with the von Neumann-Lück rank of M.

math.GR↗
arXiv2017

Dynamical correspondences of $L^2$-Betti numbers

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.

math.DS↗
arXiv2017

Entropy on modules over the group ring of a sofic group

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

math.DS↗
arXiv2015

Mean Dimension, Mean Rank, and von Neumann-Lück Rank

We introduce an invariant, called mean rank, for any module M of the integral group ring of a discrete amenable group $Γ$, as an analogue of the rank of an abelian group. It is shown that the mean dimension of the induced $Γ$-action on the Pontryagin dual of M, the mean rank of M, and the von Neumann-Lück 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.

math.DS↗