Expansive subdynamics for algebraic $Z^d$-actions

Entropy geometry and disjointness for zero-dimensional algebraic actions

We show that many algebraic actions of higher-rank abelian groups on zero-dimensional groups are mutually disjoint. The proofs exploit differences in the entropy geometry arising from subdynamics and a form of Abramov--Rokhlin formula for half-space entropies.

Homoclinic groups, IE groups, and expansive algebraic 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 a…

Pseudo-Orbit Tracing and Algebraic actions of countable amenable groups

Consider a countable amenable group acting by homeomorphisms on a compact metrizable space. Chung and Li asked if expansiveness and positive entropy of the action imply existence of an off-diagonal asymptotic pair. For algebraic actions of polycyclic-by-finite groups, Chung and Li proved it does. We provide examples showing that Chung and Li's result is near-optimal in the sense that the conclusion fails for some non-algebraic action generated by a single homeomorphism, and for some algebraic actions of non-finitely generated abelian groups. On the other hand, we prove that every expansive act…

Garden of Eden and Specification

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.

Homoclinic points, atoral polynomials, and periodic points of algebraic Z^d-actions

Cyclic algebraic Z^d-actions are defined by ideals of Laurent polynomials in d commuting variables. Such an action is expansive precisely when the complex variety of the ideal is disjoint from the multiplicative d-torus. For such expansive actions it is known that the limit for the growth rate of periodic points exists and is equal to the entropy of the action. In an earlier paper the authors extended this result to ideals whose variety intersects the d-torus in a finite set. Here we further extend it to the case when the dimension of intersection of the variety with the d-torus is at most d-2…

Expansive algebraic actions of discrete residually finite amenable groups and their entropy

We prove an entropy formula for certain expansive actions of a countable discrete residually finite group $Γ$ by automorphisms of compact abelian groups in terms of Fuglede-Kadison determinants. This extends an earlier result proved by the first author under somewhat more restrictive conditions. The main tools for this generalization are a representation of the $Γ$-action by means of a `fundamental homoclinic point', and the description of entropy in terms of the renormalized logarithmic growth-rate of the set of $Γ_n$-fixed points, where $(Γ_n, n\ge1)$ is a decreasing sequence of finite index…

Self Duality and Codings for Expansive Group Automorphisms

Lind and Schmidt have shown that the homoclinic group of a cyclic $\Z^k$ algebraic dynamical system is isomorphic to the dual of the phase group. We show that this duality result is part of an exact sequence if $k=1$. The exact sequence is a well known algebraic object, which has been applied by Schmidt in his work on rigidity. We show that it can be derived from dynamical considerations only. The constructions naturally lead to an almost 1-1-coding of certain Pisot automorphisms by their associated $β$-shift, generalizing similar results for Pisot automorphisms of the torus.

Entropy and growth rate of periodic points of algebraic Z^d-actions

Expansive algebraic Z^d-actions corresponding to ideals are characterized by the property that the complex variety of the ideal is disjoint from the multiplicative unit torus. For such actions it is known that the limit for the growth rate of periodic points exists and equals the entropy of the action. We extend this result to actions for which the complex variety intersects the multiplicative torus in a finite set. The main technical tool is the use of homoclinic points which decay rapidly enough to be summable.

A generalization of the simulation theorem for semidirect products

We generalize a result of Hochman in two simultaneous directions: Instead of realizing an effectively closed $\mathbb{Z}^d$ action as a factor of a subaction of a $\mathbb{Z}^{d+2}$-SFT we realize an action of a finitely generated group analogously in any semidirect product of the group with $\mathbb{Z}^2$. Let $H$ be a finitely generated group and $G = \mathbb{Z}^2 \rtimes H$ a semidirect product. We show that for any effectively closed $H$-dynamical system $(Y,f)$ where $Y$ is a Cantor set, there exists a $G$-subshift of finite type $(X,σ)$ such that the $H$-subaction of $(X,σ)$ is an extens…

代数群作用的内禀遍历性、生成元与符号表示

Intrinsic Ergodicity, Generators and Symbolic Representations of Algebraic Group Actions

本文为可数无限顺从群的内禀遍历主代数作用构造自然符号表示，并据此找到显式生成划分。

Ergodicity of algebraic actions of nilpotent groups

An algebraic $Γ$-action is an action of a countable group $Γ$ on a compact abelian group $X$ by continuous automorphisms of $X$. We prove that any expansive algebraic action of a finitely generated nilpotent group $Γ$ on a connected group $X$ is ergodic. We also show that this result does not hold for actions of polycyclic groups.

Representations of toral automorphisms

This survey gives an account of an algebraic construction of symbolic covers and representations of ergodic automorphisms of compact abelian groups, initiated by A.M. Vershik around 1992 for hyperbolic automorphisms of finite-dimensional tori. The key ingredient in this approach, which was subsequently extended to arbitrary expansive automorphisms of compact abelian groups, is the use of homoclinic points of the automorphism. Although existence and abundance of homoclinic points is intimately connected to expansiveness of the automorphism, it is nevertheless possible to extend certain aspects …

阿贝尔群上的代数扩张性

Algebraic expansivity on abelian groups

本文研究了阿贝尔群上端态射的代数扩张性，探讨了其基本性质、与Weiss代数熵的关系，并证明正向扩张满射仅限于有限系统，同时通过庞特里亚金对偶性展示了其与拓扑动力学的联系。

A Garden of Eden theorem for principal algebraic actions

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 the natural action of $Γ$ on $X_f$ is expansive 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. This is an analogue of the classical Garden of Eden theorem of Moore and Myhill for cellular automata with finite alphabet over $Γ$.

连续群作用的马尔可夫性质：代数作用、熵与同宿群

Markovian properties of continuous group actions: algebraic actions, entropy and the homoclinic group

本研究提出统一框架关联代数作用与拓扑熵相关结论，引入马尔可夫性质证明熵与渐近对的关联，并借助独立熵对刻画两类顺从群有限表现膨胀代数作用的同宿群。