arXiv subjects
Publications and source records attributed to Hanfeng Li.
Answering a question of Glasner, we show that any finitely generated nonabelian free group has a minimal null action which is a RIM non-open extension of an effective strongly proximal action.
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.
Let $Γ$ be a countable Abelian group and $f \in \Z[Γ]$, where $\Z[Γ]$ denotes the integral group ring of $Γ$. Consider the Pontryagin dual $X_f$ of the cyclic $\Z[Γ]$-module $\Z[Γ]/\Z[Γ] f$ and suppose that $f$ is weakly expansive (e.g., $f$ is invertible in $\ell^1(Γ)$, or, when $Γ$ is not virtually $\Z$ or $\Z^2$, $f$ is well-balanced) and that $X_f$ is connected. We prove that if $τ\colon X_f \to X_f$ is a $Γ$-equivariant continuous map, then $τ$ is surjective if and only if the restriction of $τ$ to each $Γ$-homoclinicity class is injective. We also show that this equivalence remains valid in the case when $Γ= \Z^d$ and $f \in \Z[Γ] = \Z[u_1,u_1^{-1}, \ldots, u_d, u_d^{-1}]$ is an irreducible atoral polynomial such that its zero-set $Z(f)$ is contained in the image of the intersection of $[0,1]^d$ and a finite union of hyperplanes in $\R^d$ under the quotient map $\R^d \to \T^d$ (e.g., when $d \geq 2$ such that $Z(f)$ is finite). These two results are analogues of the classical Garden of Eden theorem of Moore and Myhill for cellular automata with finite alphabet over $Γ$.
We establish a Garden of Eden theorem for expansive algebraic actions of amenable groups with the weak specification property, i.e. for any continuous equivariant map T from the underlying space to itself, T is pre-injective if and only if it is surjective. In particular, this applies to all expansive principal algebraic actions of amenable groups and expansive algebraic actions of Z^d with CPE.
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.
We give algebraic characterizations for expansiveness of algebraic actions of countable groups. The notion of p-expansiveness is introduced for algebraic actions, and we show that for countable amenable groups, a finitely presented algebraic action is 1-expansive exactly when it has finite entropy. We also study the local entropy theory for actions of countable amenable groups on compact groups by automorphisms, and show that the IE group determines the Pinsker factor for such actions. For an expansive algebraic action of a polycyclic-by-finite group on X, it is shown that the entropy of the action is equal to the entropy of the induced action on the Pontryagin dual of the homoclinic group, the homoclinic group is a dense subgroup of the IE group, the homoclinic group is nontrivial exactly when the action has positive entropy, and the homoclinic group is dense in X exactly when the action has completely positive entropy.
An \textit{algebraic} action of a discrete group $Γ$ is a homomorphism from $Γ$ to the group of continuous automorphisms of a compact abelian group $X$. By duality, such an action of $Γ$ is determined by a module $M=\widehat{X}$ over the integer group ring $\mathbb{Z}Γ$ of $Γ$. The simplest examples of such modules are of the form $M=\mathbb{Z}Γ/\mathbb{Z}Γf$ with $f\in \mathbb{Z}Γ$; the corresponding algebraic action is the \textit{principal algebraic $Γ$-action} $α_f$ defined by $f$. In this note we prove the following extensions of results by Hayes \cite{Hayes} on ergodicity of principal algebraic actions: If $Γ$ is a countably infinite discrete group which is not virtually cyclic, and if $f\in\mathbb{Z}Γ$ satisfies that right multiplication by $f$ on $\ell ^2(Γ,\mathbb{R})$ is injective, then the principal $Γ$-action $α_f$ is ergodic (Theorem \ref{t:ergodic2}). If $Γ$ contains a finitely generated subgroup with a single end (e.g. a finitely generated amenable subgroup which is not virtually cyclic), or an infinite nonamenable subgroup with vanishing first $\ell ^2$-Betti number (e.g., an infinite property $T$ subgroup), the injectivity condition on $f$ can be replaced by the weaker hypothesis that $f$ is not a right zero-divisor in $\mathbb{Z}Γ$ (Theorem \ref{t:ergodic1}). Finally, if $Γ$ is torsion-free, not virtually cyclic, and satisfies Linnell's \textit{analytic zero-divisor conjecture}, then $α_f$ is ergodic for every $f\in \mathbb{Z}Γ$ (Remark \ref{r:analytic zero divisor}).
We show that any Lipschitz projection-valued function p on a connected closed Riemannian manifold can be approximated uniformly by smooth projection-valued functions q with Lipschitz constant close to that of p. This answers a question of Rieffel.
We show that for any amenable group Γand any ZΓ-module M of type FL with vanishing Euler characteristic, the entropy of the natural Γ-action on the Pontryagin dual of M is equal to the L2-torsion of M. As a particular case, the entropy of the principal algebraic action associated with the module ZΓ/ZΓf is equal to the logarithm of the Fuglede-Kadison determinant of f whenever f is a non-zero-divisor in ZΓ. This confirms a conjecture of Deninger. As a key step in the proof we provide a general Szegő-type approximation theorem for the Fuglede-Kadison determinant on the group von Neumann algebra of an amenable group. As a consequence of the equality between L2-torsion and entropy, we show that the L2-torsion of a non-trivial amenable group with finite classifying space vanishes. This was conjectured by Lück. Finally, we establish a Milnor-Turaev formula for the L2-torsion of a finite Δ-acyclic chain complex.
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 undertake a local analysis of combinatorial independence as it connects to topological entropy within the framework of actions of sofic groups.
We give a new short proof for the uniqueness of the universal minimal space. The proof holds for the uniqueness of the universal object in every collection of topological dynamical systems closed under taking projective limits and possessing universal objects.
We prove a conjecture of Rosenberg about the minimal value for energies of untaries in the two-dimensional noncommutative tori and answer a question of his about lower bounds for energies of unital *-endomorphisms of the two-dimensional noncommutative tori.
For a countable amenable group Γand an element f in the integral group ring ZΓbeing invertible in the group von Neumann algebra of Γ, we show that the entropy of the shift action of Γon the Pontryagin dual of the quotient of ZΓby its left ideal generated by f is the logarithm of the Fuglede-Kadison determinant of f. For the proof, we establish an \ell^p-version of Rufus Bowen's definition of topological entropy, addition formulas for group extensions of countable amenable group actions, and an approximation formula for the Fuglede-Kadison determinant of f in terms of the determinants of perturbations of the compressions of f.
We prove a number of identities relating the sofic entropy of a certain class of non-expansive algebraic dynamical systems, the sofic entropy of the Wired Spanning Forest and the tree entropy of Cayley graphs of residually finite groups. We also show that homoclinic points and periodic points in harmonic models are dense under general conditions.
Recently Lewis Bowen introduced a notion of entropy for measure-preserving actions of a countable sofic group on a standard probability space admitting a generating partition with finite entropy. By applying an operator algebra perspective we develop a more general approach to sofic entropy which produces both measure and topological dynamical invariants, and we establish the variational principle in this context. In the case of residually finite groups we use the variational principle to compute the topological entropy of principal algebraic actions whose defining group ring element is invertible in the full group C*-algebra.
In a previous paper the authors developed an operator-algebraic approach to Lewis Bowen's sofic measure entropy that yields invariants for actions of countable sofic groups by homeomorphisms on a compact metrizable space and by measure-preserving transformations on a standard probability space. We show here that these measure and topological entropy invariants both coincide with their classical counterparts when the acting group is amenable.
For a family F (a collection of subsets of Z_+), the notion of F-independence is defined both for topological dynamics (t.d.s.) and measurable dynamics (m.d.s.). It is shown that there is no non-trivial {syndetic}-independent m.d.s.; a m.d.s. is {positive-density}-independent if and only if it has completely positive entropy; and a m.d.s. is weakly mixing if and only if it is {IP}-independent. For a t.d.s. it is proved that there is no non-trivial minimal {syndetic}-independent system; a t.d.s. is weakly mixing if and only if it is {IP}-independent. Moreover, a non-trivial proximal topological K system is constructed, and a topological proof of the fact that minimal topological K implies strong mixing is presented.