Siddhartha Bhattacharya, Tullio Ceccherini-Silberstein, Michel Coornaert
Let $X$ be a compact metrizable group and $Γ$ a countable group acting on $X$ by continuous group automorphisms. We give sufficient conditions under which the dynamical system $(X,Γ)$ is surjunctive, i.e., every injective continuous map $τ\colon X \to X$ commuting with the action of $Γ$ is surjective.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Let $X$ be a compact metrizable group and $Γ$ a countable group acting on $X$ by continuous group automorphisms. We give sufficient conditions under which the dynamical system $(X,Γ)$ is surjunctive, i.e., every injective continuous map $τ\colon X \to X$ commuting with the action of $Γ$ is surjective.
Let $G$ be a group. Let $X$ be an algebraic group over an algebraically closed field $K$. Denote by $A=X(K)$ the set of rational points of $X$. We study algebraic group cellular automata $τ\colon A^G \to A^G$ whose local defining map is induced by a homomorphism of algebraic groups $X^M \to X$ where $M$ is a finite memory. When $G$ is sofic and $K$ is uncountable, we show that if $τ$ is post-surjective then it is weakly pre-injective. Our result extends the dual version of Gottschalk's Conjecture for finite alphabets proposed by Capobianco, Kari, and Taati. When $G$ is amenable, we prove that …
We explore the dual version of Gottschalk's conjecture recently introduced by Capobianco, Kari, and Taati, and the notion of dual surjunctivity in general. We show that dual surjunctive groups satisfy Kaplansky's direct finiteness conjecture for all fields of positive characteristic. By quantifying the notions of injectivity and post-surjectivity for cellular automata, we show that the image of the full topological shift under an injective cellular automaton is a subshift of finite type in a quantitative way. Moreover we show that dual surjunctive groups are closed under ultraproducts, under e…
Generalized Gottschalk's conjecture for sofic groups and applications
该研究将元胞自动机的满射性与对偶满射性定理推广到sofic群上元胞自动机的局部及全局一致有界奇异扰动情形，得到代数非均匀元胞自动机的满射性，并证明sofic群上扭群环的稳定有限性，推广了Kaplansky相关猜想的已知结果。
An algebraic $Z^{d}$-action is an action of $Z^{d}$ on a compact abelian group $X$ by automorphisms of $X$. We prove that for $d \ge 8$, there exist mixing zero entropy algebraic $Z^{d}$-actions which do not exhibit isomorphism rigidity property.
We investigate rigidity of measurable structure for higher rank abelian algebraic actions. In particular, we show that ergodic measures for these actions fiber over a 0 entropy measure with Haar measures along the leaves. We deduce various rigidity theorems for isomorphisms and joinings as corollaries.
Local rigidity of weak or no hyperbolicity algebraic actions
针对弱双曲或无双曲的阿贝尔代数作用，本文提出证明其 $C^\infty$ 局部刚性的通用策略，并首次建立半单李群齐性空间上一大类抛物作用的强局部刚性。
We show that all $C^\infty$ Anosov $Z^r$-actions on tori and nilmanifolds without rank-one factor actions are, up to $C^\infty$ conjugacy, actions by automorphisms.
We show that the space of expanding maps contains an open and dense set where smooth conjugacy classes of expanding maps are determined by the values of the Jacobians of return maps at periodic points.
Local rigidity of higher rank partially hyperbolic algebraic actions
本研究结合表示论、调和分析与KAM迭代，完整解决高阶部分双曲代数作用局部分类问题，证明多类阿贝尔遍历代数作用的$C^\infty$局部刚性，首次得到非可达部分双曲作用及带非阿贝尔幂零根基扭空间的相关新结果。
In the moduli space of polynomials of degree 3 with marked critical points c_1 and c_2, let C_{1,n} be the locus of maps for which c_1 has period n and let C_{2,m} be the locus of maps for which c_2 has period m. A consequence of Thurston's rigidity theorem is that the curves C_{1,n} and C_{2,m} intersect transversally. We give a purely algebraic proof that the intersection points are 3-adically integral and use this to prove transversality. We also prove an analogous result when c_1 or c_2 or both are taken to be preperiodic with tail length exactly 1.
Bi-Lipschitz rigidity of discrete subgroups
本文证明连通单实代数群中 Zariski 稠密离散子群的双利普希茨刚性，并据此排除高阶半单代数群中此类子群在非极大抛物商空间上具有 $C^1$ 光滑薄极限集的可能。
Consider homogeneous G/H and G/F, for an S-algebraic group G. A lattice Γ acts on the left strictly conservatively. The following rigidity results are obtained: morphisms, factors and joinings defined apriori only in the measurable category are in fact algebraically constrained. Arguing in an elementary fashion we manage to classify all the measurable Φ commuting with the Γ-action: assuming ergodicity, we find they are algebraically defined.
Let $(\az,F)$ be a bipermutative algebraic cellular automaton. We present conditions which force a probability measure which is invariant for the $\N\times\Z$-action of $F$ and the shift map $\s$ to be the Haar measure on $\gs$, a closed shift-invariant subgroup of the Abelian compact group $\az$. This generalizes simultaneously results of B. Host, A. Maass and S. Mart\'ınez \cite{Host-Maass-Martinez-2003} and M. Pivato \cite{Pivato-2003}. This result is applied to give conditions which also force a $(F,\s)$-invariant probability measure to be the uniform Bernoulli measure when $F$ is a partic…
We study rational self-maps of $\mathbb{P}^{1}$ whose critical points all have finite forward orbit. Thurston's rigidity theorem states that outside a single well-understood family, there are finitely many such maps over $\mathbb{C}$ of fixed degree and critical orbit length. We provide an algebraic proof of this fact for tamely ramified maps for which at least one of the critical points is periodic. We also produce wildly ramified counterexamples.
Smooth rigidity for very non-algebraic Anosov diffeomorphisms of codimension one
该研究提出基于匹配函数的Anosov微分同胚光滑刚性新方法，证明三维环面线性双曲自同构小邻域内极非代数余维一Anosov微分同胚光滑共轭等价于雅可比周期数据匹配，并推广至高维情形。
Rigidity of the dynamics of ${\rm Aut}({\mathsf{F}}_n)$ on representations into a compact group
研究${\rm Aut}({\mathsf{F}}_n)$在紧凑群表示上的动力学稳定性，揭示轨道闭包和不变测度的代数性质。
Algebraic actions II. Groupoid rigidity
研究代数作用的局部变换群oid的刚性，证明在特定条件下，群oid的全局化作用可相互嵌入，尤其在代数数论中，群oid能保持初始作用的同构性，解决了一个关于Cartan对同构的问题。
Baumslag-Solitar groups, relative profinite completions and measure equivalence rigidity
本文为保测度等价关系的非周期包含引入代数不变量，借此证明了非顺从Baumslag-Solitar群直积自由pmp作用的稳定轨道等价刚性，并完成了其相对profinite完备化直积的完全分类。
We prove an operator algebraic superrigidity statement for homomorphisms of irreducible lattices, and also their commensurators, in certain higher-rank groups into unitary groups of finite factors. This extends the authors' previous work regarding non-free measure-preserving actions, and also answers a question of Connes for such groups.
In several contexts the defining invariant structures of a hyperbolic dynamical system are smooth only in systems of algebraic origin (smooth rigidity), and we prove new results of this type for a class of flows. For a compact Riemannian manifold and a uniformly quasiconformal transversely symplectic Anosov flow we define the longitudinal KAM-cocycle and use it to prove a rigidity result: The joint stable/unstable subbundle is Zygmund-regular, and higher regularity implies vanishing of the longitudinal KAM-cocycle, which in turn implies that the subbundle is Lipschitz-continuous and indeed tha…
We show that for certain classes of actions of Z^d, d >= 2, by automorphisms of the torus any measurable conjugacy has to be affine, hence measurable conjugacy implies algebraic conjugacy; similarly any measurable factor is algebraic, and algebraic and affine centralizers provide invariants of measurable conjugacy. Using the algebraic machinery of dual modules and information about class numbers of algebraic number fields we consruct various examples of Z^d-actions by Bernoulli automorphisms whose measurable orbit structure is rigid, including actions which are weakly isomorphic but not isomor…
We survey some recent developments and applications of the study of the rigidity properties of natural algebraic actions of multidimensional abelian groups.
Weyl groups and rigidity of von Neumann algebras
本文研究了非紧致半单代数群G的Weyl群与von Neumann代数的刚性关系，通过分析群测度空间von Neumann代数M和群von Neumann代数M之间的关系，证明了Aut_M(M)与G的Weyl群同构，并为Connes的刚性猜想提供了新见解。
We consider partially hyperbolic abelian algebraic high-rank actions on compact homogeneous spaces obtained from simple indefinite orthogonal and unitary groups. In the first part of the paper, we show local differentiable rigidity for such actions. The conclusions are based on progress towards computations of the Schur multipliers of these non-split groups, which is the main aim of the second part.
We prove that topologically isomorphic linear cellular automaton shifts are algebraically isomorphic. Using this, we show that two distinct such shifts cannot be isomorphic. We conclude that the automorphism group of a linear cellular automaton shift is a finitely generated abelian group.
Local rigidity of Julia sets
本文给出判据，说明两个有理函数Julia集之间的局部同胚若满足一定正则性条件，则来自代数对应，并指出该结果将用于曲线动力学André-Oort猜想的证明。
Let X_1 and X_2 be mixing connected algebraic dynamical systems with the Descending Chain Condition. We show that every equivariant continuous map X_1 to X_2 is affine (that is, X_2 is topologically rigid) if and only if the system X_2 has finite topological entropy.
We study the rigidity properties of a class of algebraic Z^3-actions with entropy rank two. For this class, conditions are found which force an invariant measure to be the Haar measure on an affine subset. This is applied to show isomorphism rigidity for such actions, and to provide examples of non-isomorphic Z^3-actions with all their Z^2-sub-actions isomorphic. The proofs use lexicographic half-space entropies and total ergodicity along critical directions.
We study equivalence relations that arise from translation actions $Γ\curvearrowright G$ which are associated to dense embeddings $Γ<G$ of countable groups into second countable locally compact groups. Assuming that $G$ is simply connected and the action $Γ\curvearrowright G$ is strongly ergodic, we prove that $Γ\curvearrowright G$ is orbit equivalent to another such translation action $Λ\curvearrowright H$ if and only if there exists an isomorphism $δ:G\rightarrow H$ such that $δ(Γ)=Λ$. If $G$ is moreover a real algebraic group, then we establish analogous rigidity results for the translation…
In this paper we study characters on special linear groups SL_n(R), where R is either an infinite field or the localization of an order in a number field. We give several applications to the theory of measure preserving actions, operator-algebraic superrigidity, and almost homomorphisms.
Let $d > 1$, and let $(X,α)$ and $(Y,β)$ be two zero-entropy ${\mathbb{Z}}^d$-actions on compact abelian groups by $d$ commuting automorphisms. We show that if all lower rank subactions of $α$ and $β$ have completely positive entropy, then any measurable equivariant map from $X$ to $Y$ is an affine map. In particular, two such actions are measurably conjugate if and only if they are algebraically conjugate.
We prove the locally differentiable rigidity of generic partially hyperbolic abelian algebraic high-rank actions on compact homogeneous spaces obtained from split symplectic Lie groups. We also gave a non-generic action rigidity example on compact homogeneous spaces obtained from SL(2n,R) or SL(2n,C). The conclusions are based on geometric Katok-Damjanovic way and progress towards computations of the generating relations in these groups.
Rigidity in Complex Dynamics: Multiplier Spectrum and Dynamical André-Oort Conjecture
本研究介绍一维复动力学刚性问题的近期进展，证明了曲线的动力André-Oort猜想及乘子谱的一般单射性，其证明结合代数几何、Arakelov几何与复动力学思想。
On rigidity properties of time-changes of unipotent flows
本文研究半单线性群有限体积商上单幂流时间变换的可测同构，在谱间隙条件下证明其要么产生平凡联结极限，要么为仿射同构，推广了Ratner的horocycle流结果。
The holomorphic endomorphism f of projective space is called post-critically finite (PCF) if the forward image of the critical locus, under iteration of f, has algebraic support (i.e., is a finite union of hypersurfaces). In the case of dimension 1, a deep result of Thurston implies that there are no algebraic families of PCF morphisms, other than a well-understood exceptional class known as the flexible Lattes maps. This note proves a corresponding result in arbitrary dimension, for a certain subclass of morphism. Specifically, we restrict attention to morphisms f of degree at least two, with…
We develop a model-theoretic framework for the study of distal factors of strongly ergodic, measure-preserving dynamical systems of countable groups. Our main result is that all such factors are contained in the (existential) algebraic closure of the empty set. This allows us to recover some rigidity results of Ioana and Tucker-Drob as well as prove some new ones: for example, that strongly ergodic, distal systems are coalescent and that every two such systems that are weakly equivalent are isomorphic. We also prove the existence of a universal distal, ergodic system that contains any other di…
Joining rigidity for rational maps
本文建立有理映射的联结刚性理论：若最大熵测度的联结支持局部双全纯图，则其整体化为代数曲线，否则产生非离散局部全纯关系族。
Let $k$ be a number field, $\mathbf{G}$ an algebraic group defined over $k$, and $\mathbf{G}(k)$ the group of $k$-rational points in $\mathbf{G}.$ We determine the set of functions on $\mathbf{G}(k)$ which are of positive type and conjugation invariant, under the assumption that $\mathbf{G}(k)$ is generated by its unipotent elements. An essential step in the proof is the classification of the $\mathbf{G}(k)$-invariant ergodic probability measures on an adelic solenoid naturally associated to $\mathbf{G}(k);$ this last result is deduced from Ratner's measure rigidity theorem for homogeneous spa…
In this paper we study perturbations of constant cocycles for actions of higher rank semi-simple algebraic groups and their lattices. Roughly speaking, for ergodic actions, Zimmer's cocycle superrigidity theorems implies that the perturbed cocycle is measurably conjugate to a constant cocycle modulo a compact valued cocycle. The main point of this article is to see that a cocycle which is a continuous perturbation of a constant cocycle is actually continuously conjugate back to the original constant cocycle modulo a cocycle that is continuous and "small". We give some applications to perturbat…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。