紧阿贝尔群上仿射作用的轨道等价与拓扑共轭

诱导类、斜率类和共轭类

Inducing, slopes, and conjugacy classes

证明了最终扩张连续分段仿射区间映射的共轭类包含在参数空间的光滑余维1子流形中。特别是共轭类有空的内部。这是基于对诱导马尔可夫映射和各态历经理论行为之间关系的研究。

Topological conjugacy for unimodal nonautonomous discrete dynamical systems

The goal of this article is to study how combinatorial equivalence implies topological conjugacy. For that, we introduce the concept of kneading sequences for nonautonomous discrete dynamical systems and show that these sequences are a complete invariant for topological conjugacy classes.

Cantor极小系统的完全群、翻转共轭和轨道等价

Full groups, flip conjugacy, and orbit equivalence of Cantor minimal systems

本文考虑Cantor极小系统$(X,\f)$的全群$[ϕ]$和拓扑全群$[[ϕ]]$.证明了换位子群$D([\f])$和$D([[\f]])$是单的，并证明了交换子子群$D([\f])$和$D([[\f]])$分别完全决定了$\f$的轨道等价类和翻转共轭类.这些结果改进了在\cite{gps:1999}中发现的分类。作为所使用的技术的必然结果，我们建立了一个事实，即$\f$可以被写为从$[\f]$的三个对合的产品。

可数sofic群作用的测度共轭不变量

Measure conjugacy invariants for actions of countable sofic groups

Sofic群是Gromov在[Gr 99]中隐式定义的，Weiss在[We 00]中显式定义的。所有剩余有限群（因此每个线性群）都是sofic的。本文的目的是在概率空间上，对每一个可数sofic群$G$，引入一类保测度$G$ -作用的测度共轭不变量.这些不变量将Kolmogorov-Sinai熵推广到顺从群的作用。精确计算了$G$上的Bernoulli移位，从而对Bernoulli系统进行了完整的分类，直到许多群的测度共轭，包括所有可数的线性群. Y. Kida和S.利用Popa对映射类群和性质T群上的Bernoulli移位分别进行了轨道等价和von Neumann等价的分类。

Balanced strong shift equivalence, balanced in-splits, and eventual conjugacy

We introduce the notion of balanced strong shift equivalence between square nonnegative integer matrices, and show that two finite graphs with no sinks are one-sided eventually conjugate if and only if their adjacency matrices are conjugate to balanced strong shift equivalent matrices. Moreover, we show that such graphs are eventually conjugate if and only if one can be reached by the other via a sequence of out-splits and balanced in-splits; the latter move being a variation of the classical in-split move introduced by Williams in his study of shifts of finite type. We also relate one-sided e…

Affine map equivalence versus critical set equivalence for quadratic maps of the plane

In recent work [Nien et al. 2016], the authors enumerated a classification of quadratic maps of the plane according to their critical sets and images. It is straightforward to show that quadratic maps which are affinely map equivalent are also equivalent in the critical set classification. The question remained whether maps that are equivalent in the critical set classification are also affinely map equivalent. This paper establishes a complete enumeration of the affine map equivalence classes. As a consequence, the relationship between affine map equivalence and critical set equivalence is es…

极小同胚与测度上的近似共轭

Minimal Homeomorphisms and Approximate Conjugacy in Measure

设X是具有有限覆盖维的无限紧度量空间。设$\afhpa,\bt: X\to X$是两个极小同胚.设两个交叉积C* 代数s的$K_0$ -群的值域在实仿射连续函数空间中稠密.证明了$\af$和$\bt$在测度上一致近似共轭当且仅当它们具有仿射同胚不变概率测度空间.

The complexity of conjugacy, orbit equivalence, and von Neumann equivalence of actions of nonamenable groups

Building on work of Popa, Ioana, and Epstein--Törnquist, we show that, for every nonamenable countable discrete group $Γ$, the relations of conjugacy, orbit equivalence, and von Neumann equivalence of free ergodic (or weak mixing) measure preserving actions of $Γ$ on the standard atomless probability space are not Borel, thus answering questions of Kechris. This is an optimal and definitive result, which establishes a neat dichotomy with the amenable case, since any two free ergodic actions of an amenable group on the standard atomless probability space are orbit equivalent by classical result…

Generalized Bowen-Franks Groups and Profinite Conjugacy for Hyperbolic Toral Automorphisms

We show that a collection of generalized Bowen-Franks group, what we call the principal Bowen-Franks $R$-modules, form a complete set of $R$-module invariants for the equivalence relation of profinite conjugacy for similar hyperbolic toral automorphisms. We also show that these principal generalized Bowen-Franks $R$-modules are the principal invariants in a large class of generalized Bowen-Franks $R$-module invariants for similar hyperbolic automorphisms. We further show how the conjugacy invariant of cyclicity, when applied to hyperbolic toral automorphisms, is realized for profinite conjugac…

Local Conjugacy of Irreducible Hyperbolic Toral Automorphisms

This paper is dedicated to the conjugacy problem in $GL(n,\Z)$ and its connection with algebraic number theory. This connection may be summed up in the Latimer-MacDuffee-Taussky Theorem, which, in a very broad sense, identifies the conjugacy relation in $GL(n,\Z)$ with the relation of arithmetic equivalence between certain ideals of algebraic numbers. The main purpouse is to clarify the link between a weaker relation between ideals, weak equivalence, and local conjugacy in $GL(n,\Z)$, i.e., the conjugacy of $GL(n,\Z)$ matrices in the groups $GL(n,\Z_{p})$ of invertible matrices over the $p$-ad…

Slices of Parameter Space of Cubic Polynomials

In this paper, we study slices of the parameter space of cubic polynomials, up to affine conjugacy, given by a fixed value of the multiplier at a non-repelling fixed point. In particular, we study the location of the $main\, cubioid$ in this parameter space. The $main\, cubioid$ is the set of affine conjugacy classes of complex cubic polynomials that have certain dynamical properties generalizing those of polynomials $z^2+c$ for $c$ in the filled main cardioid.

Equivalence of Collet-Eckmann conditions for slowly recurrent rational maps

In this short note we observe that within the family of slowly recurrent rational maps on the Riemann sphere, the Collet-Eckmann, second Collet-Eckmann, and topological Collet-Eckmann conditions are equivalent and also invariant under topological conjugacy.

On the complexity of topological conjugacy of Toeplitz subshifts

In this paper, we study the descriptive set theoretic complexity of the equivalence relation of conjugacy of Toeplitz subshifts of a residually finite group $G$. On the one hand, we show that if $G = \mathbb{Z}$, then topological conjugacy on Toeplitz subshifts with separated holes is amenable. In contrast, if $G$ is non-amenable, then conjugacy of Toeplitz $G$-subshifts is a non-amenable equivalence relation. The results were motivated by a general question, asked by Gao, Jackson and Seward, about the complexity of conjugacy for minimal, free subshifts of countable groups.

符号矩阵的拓扑共轭及其相关关系

Topological conjugacy and its relations for symbolic matrices

本文研究了符号矩阵中整数子矩阵与表示矩阵的关系，用子矩阵表述了新版本的分解定理，并证明通过固定对角线的整数子矩阵可实现表示矩阵的强移位等价（共轭）。

Rigidity of some abelian-by-cyclic solvable group actions on $\mathbb T^N$

In this paper, we study a natural class of groups that act as affine transformations of $\mathbb T^N$. We investigate whether these solvable, "abelian-by-cyclic," groups can act smoothly and nonaffinely on $\mathbb T^N$ while remaining homotopic to the affine actions. In the affine actions, elliptic and hyperbolic dynamics coexist, forcing a priori complicated dynamics in nonaffine perturbations. We first show, using the KAM method, that any small and sufficiently smooth perturbation of such an affine action can be conjugated smoothly to an affine action, provided certain Diophantine condition…

On Virtual Conjugacy of Generalized Odometers

The paper is focused on the study of continuous orbit equivalence for generalized odometers (profinite actions). We show that two generalized odometers are continuously orbit equivalent if and only if the acting groups have finite index subgroups (having the same index) whose actions are piecewise conjugate. This result extends M.~Boyle's flip-conjugacy theorem originally established for $\mathbb Z$-actions. As a corollary we obtain a dynamical classification of the restricted isomorphism between generalized Bunce-Deddens $C^*$-algebras. We also show that the full group associated with a gener…

Classification of one dimensional dynamical systems by countable structures

We study the complexity of the classification problem of conjugacy on dynamical systems on some compact metrizable spaces. Especially we prove that the conjugacy equivalence relation of interval dynamical systems is Borel bireducible to isomorphism equivalence relation of countable graphs. This solves a special case of the Hjorth's conjecture which states that every orbit equivalence relation induced by a continuous action of the group of all homeomorphisms of the closed unit interval is classifiable by countable structures. We also prove that conjugacy equivalence relation of Hilbert cube hom…

仿射扩张的局部-全局共轭问题

Local-global conjugacy questions for affine extensions

研究Galois群$G_n(f)$在二元根树自同构群$\operatorname{Aut}(T_n)$中的局部-全局共轭性，通过上同调论证将Goksel的结果推广到置换表示的仿射扩张，并给出Goksel主要猜想的一个反例。

Hénon 映射玩具模型的揉搓序列

Kneading sequences for toy models of Hénon maps

本文研究 Hénon 映射玩具模型中组合等价与拓扑共轭的关系，定义临界线的揉搓序列作为完全不变量，并推广 Singer 定理。

环面自同构的Z^d-作用的可测结构的刚性

Rigidity of measurable structure for Z^d-actions by automorphisms of a torus

我们证明了对于Z^d，d >= 2的某些作用类，通过环面的自同构，任何可测共轭都必须是仿射的，因此可测共轭蕴涵代数共轭;同样，任何可测因子都是代数的，代数和仿射中心化子提供了可测共轭的不变量。利用对偶模的代数机制和关于代数数域类数的信息，我们通过Bernoulli自同构构造了Z^d-作用的各种例子，其可测轨道结构是刚性的，包括弱同构但不同构的作用。我们表明，这些行动的中心化的结构可能会或可能不会作为一个区别测量理论不变。

Orbit equivalences of pseudo-Anosov flows

We prove a classification theorem for transitive Anosov and pseudo-Anosov flows on closed 3-manifolds, up to orbit equivalence. In many cases, flows on a 3-manifold $M$ are completely determined by the set of free homotopy classes of their (unoriented) periodic orbits. The exceptional cases are flows with a special structure in their orbit space called a ``tree of scalloped regions"; in these cases the set of free homotopy classes of unoriented periodic orbits together with the additional data of a choice of sign for each $π_1(M)$-orbit of tree gives a complete invariant of orbit equivalence c…

Dynamical Equivalence and Linear Conjugacy of Chemical Reaction Networks: New Results and Methods

In the first part of this paper, we propose new optimization-based methods for the computation of preferred (dense, sparse, reversible, detailed and complex balanced) linearly conjugate reaction network structures with mass action dynamics. The developed methods are extensions of previously published results on dynamically equivalent reaction networks and are based on mixed-integer linear programming. As related theoretical contributions we show that (i) dense linearly conjugate networks define a unique super-structure for any positive diagonal state transformation if the set of chemical compl…

Non-homogeneous locally free actions of the affine group

We classify smooth locally free actions of the real affine group on closed orientable three-dimensional manifolds up to smooth conjugacy. As a corollary, there exists a non-homogeneous action when the manifold is the unit tangent bundle of a closed surface with a hyperbolic metric.

Topological Conjugacy of Non-hyperbolic Linear Flows

The topological equivalence classification for linear flows on $\mathbb{R}^n$ had been completely solved by Kuiper and independently Ladis in 1973. However, Ladis' proof was published in a Russian journal which isn't easily available, Kuiper's proof is more topological and a little bit subtle. Aiming at topological conjugacy classification, mainly based on the ideas of Kuiper, we introduce other techniques and try to present an elementary and self-contained proof just using linear algebra and elementary topology.

伪装环轨迹与反应网络的仿射等价性

The disguised toric locus and affine equivalence of reaction networks

该研究针对质量作用动力学下的伪装环动力学系统，证明其对应参数在网络可逆仿射变换下保持不变，同时发现任意质量作用系统经仿射变换后正稳态集合存在双射，但定性动力学可能差异显著。

The complexity of topological conjugacy of pointed Cantor minimal systems

In this paper, we analyze the complexity of topological conjugacy of pointed Cantor minimal systems from the point of view of descriptive set theory. We prove that the topological conjugacy relation on pointed Cantor minimal systems is Borel bireducible with the Borel equivalence relation $Δ_{\mathbb{R}}^+$ on $\mathbb{R}^{\mathbb{N}}$ defined by $x Δ_{\mathbb{R}}^+ y \Leftrightarrow \{x_i:i \in \mathbb{N}\}=\{y_i:i \in \mathbb{N}\}$. Moreover, we show that $Δ_{\mathbb{R}}^+$ is a lower bound for the Borel complexity of topological conjugacy of Cantor minimal systems. Finally, we interpret our…

Topological classification of affine operators on unitary and Euclidean spaces

We classify affine operators on a unitary or Euclidean space U up to topological conjugacy. An affine operator is a map f: U-->U of the form f(x)=Ax+b, in which A: U-->U is a linear operator and b in U. Two affine operators f and g are said to be topologically conjugate if hg=fh for some homeomorphism h: U-->U.

Flow equivalence of topological Markov shifts and Ruelle algebras

We study discrete flow equivalence of two-sided topological Markov shifts by using extended Ruelle algebras. We characterize flow equivalence of two-sided topological Markov shifts in terms of conjugacy of certain actions weighted by ceiling functions of two-dimensional torus on the stabilized extended Ruelle algebras for the Markov shifts.

Cellini下降代数与李型有限群的半单共轭类

Cellini's descent algebra and semisimple conjugacy classes of finite groups of Lie type

利用代数群理论，给出了从有限群的半单共轭类到外尔群的共轭类的映射.随机一致选取一个半单类，得到外尔群共轭类的一个概率测度。我们猜想，这一措施同意的第二个措施的共轭类的外尔群诱导的建设Cellini使用仿射外尔群。在C型奇特征和A型单位共轭类等特殊情况下，证明了这一猜想。洗牌的模式，无论新旧，都是自然产生的。A型洗牌导致有趣的数论涉及拉马努金和。证明了我们的猜想在C型偶特征中的存在性，并给出了动力系统中一个问题的另一种解法。一个想法是如何提供的，至少在类型A，关联到一个半单共轭类的元素的外尔群，细化映射共轭类。这是最简单的非平凡的例子证实。

Coded equivalence of one-sided topological Markov shifts

We introduce a notion of coded equivalence in one-sided topological Markov shifts. The notion is inspired by coding theory. One-sided topological conjugacy implies coded equivalence. We will show that coded equivalence implies continuous orbit equivalence of one-sided topological Markov shifts.

可解群与直线上的仿射作用

Solvable Groups and Affine Actions on the Line

本文研究有限生成可解群在实区间上的保向作用，证明C^1微分同胚作用必半共轭于仿射作用，并给出同胚情形下的局部半共轭结果及仿射作用族的鲁棒性条件。

Topological conjugacy of topological Markov shifts and Cuntz-Krieger algebras

For an irreducible non-permutation matrix $A$, the triplet $({\mathcal{O}_A},{\mathcal{D}_A},ρ^A)$ for the Cuntz-Krieger algebra ${\mathcal{O}_A}$, its canonical maximal abelian $C^*$-subalgebra ${\mathcal{D}_A}$, and its gauge action $ρ^A$ is called the Cuntz-Krieger triplet. We introduce a notion of strong Morita equivalence in the Cuntz-Krieger triplets, and prove that two Cuntz-Krieger triplets $({\mathcal{O}_A},{\mathcal{D}_A},ρ^A)$ and $({\mathcal{O}_B},{\mathcal{D}_B},ρ^B)$ are strong Morita equivalent if and only if $A$ and $B$ are strong shift equivalent. We also show that the general…

A groupoid approach to $C^*$-algebras associated with $λ$-graph systems and continuous orbit equivalence of subshifts

A $λ$-graph system $\frak L$ is a labeled Bratteli diagram with shift operation. It is a generalized notion of finite labeled graph and presents a subshifts. We will study continuous orbit equivalence of one-sided subshifts and topological conjugacy of two-sided subshifts from the view points of groupoids and $C^*$-algebras constructed from $λ$-graph systems.

Twisted conjugacy in residually finite groups of finite Prüfer rank

Suppose, $G$ is a residually finite group of finite upper rank admitting an automorphism $φ$ with finite Reidemeister number $R(φ)$ (the number of $φ$-twisted conjugacy classes). We prove that such $G$ is soluble-by-finite (in other words, any residually finite group of finite upper rank, which is not soluble-by-finite, has the $R_\infty$ property). This reduction is the first step in the proof of the second main theorem of the paper: suppose, $G$ is a residually finite group of finite Prüfer rank and $φ$ is its automorphism with $R(φ)<\infty$; then $R(φ)$ is equal to the number of equivalence…

近似同态与轨道等价关系的sofic逼近

Approximate homomorphisms and sofic approximations of orbit equivalence relations

本文证明可数群的近似同态序列可实现为轨道等价关系的sofic逼近的限制，并由不变随机子群唯一确定，进而恢复了顺从群近似同态的已知稳定性与共轭刻画。

微分方程的等价性

Equivalence for Differential Equations

我们将研究常微分方程关于仿射变换群的等价问题。

The Borel complexity of von Neumann equivalence

We prove that for a countable discrete group $Γ$ containing a copy of the free group $\F_n$, for some $2\leq n\leq\infty$, as a normal subgroup, the equivalence relations of conjugacy, orbit equivalence and von Neumann equivalence of the ergodic a.e. free actions of $Γ$ are analytic non-Borel equivalence relations in the Polish space of probability measure preserving $Γ$ actions. As a consequence we obtain that the isomorphism relation in the spaces of separably acting factors of type $\II_1$, $\II_\infty$ and $\III_λ$, $0\leqλ\leq 1$, are analytic and not Borel when these spaces are given the…