Hiroki Matui
H. Lin and the author introduced the notion of approximate conjugacy of dynamical systems. In this paper, we will discuss the relationship between approximate conjugacy and full groups of Cantor minimal systems. An analogue of Glasner-Weiss's theorem will be shown. Approximate conjugacy of dynamical systems on the product of the Cantor set and the circle will also be studied.
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The conjugacy and flip conjugacy problem for Cantor minimal systems
本文证明 Cantor 极小系统的拓扑共轭与翻转共轭，以及图同态可归约和图同构，均为完全解析关系而非 Borel 关系。
We survey, and extend, results on the adjoint action of the homeomorphism group $H(X)$ on the space of surjective continuous maps, $C_s(X)$, where $X$ is a Cantor set. We look also at the restriction of the action to various dynamically defined subsets of $C_s(X)$, e. g. the sets of topologically transitive maps, chain transitive maps, chain mixing maps, etc. In each case, we consider whether there exist elements with a dense conjugacy class and if so, what the generic elements look like.
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…
One residual set of Cantor graphs with dense conjugacy classes
本文研究了康托尔集上闭合图空间中残余同构类和共轭类的性质，证明了具有密集共轭类的元素集合在该空间中为稠密$G_\delta$子集，并探讨了零拓扑熵元素的稠密性。
Given a $C^{1+γ}$ hyperbolic Cantor set $C$, we study the sequence $C_{n,x}$ of Cantor subsets which nest down toward a point $x$ in $C$. We show that $C_{n,x}$ is asymptotically equal to an ergodic Cantor set valued process. The values of this process, called limit sets, are indexed by a Hölder continuous set-valued function defined on D. Sullivan's dual Cantor set. We show the limit sets are themselves $C^{k+γ}, C^\infty$ or $C^ω$ hyperbolic Cantor sets, with the highest degree of smoothness which occurs in the $C^{1+γ}$ conjugacy class of $C$. The proof of this leads to the following rigidi…
The Borel complexity of conjugacy for Cantor minimal systems
本文证明 Cantor 极小系统共轭的 Borel 复杂度与可数图同构相同，通过编码 pro-有限群和逆向极限构造回答 Foreman 问题。
In the paper, we consider the full group $[ϕ]$ and topological full group $[[ϕ]]$ of a Cantor minimal system $(X,\f)$. We prove that the commutator subgroups $D([\f])$ and $D([[\f]])$ are simple and show that the groups $D([\f])$ and $D([[\f]])$ completely determine the class of orbit equivalence and flip conjugacy of $\f$, respectively. These results improve the classification found in \cite{gps:1999}. As a corollary of the technique used, we establish the fact that $\f$ can be written as a product of three involutions from $[\f]$.
Giordano, Putnam and Skau showed that topological full groups of Cantor minimal systems are complete invariants for flip conjugacy. We will completely determine the structure of normal subgroups of the topological full group. Moreover, a necessary and sufficient condition for the topological full group to be finitely generated will be given.
A Cantor action is a minimal equicontinuous action of a countably generated group G on a Cantor space X. Such actions are also called generalized odometers in the literature. In this work, we introduce two new conjugacy invariants for Cantor actions, the stabilizer limit group and the centralizer limit group. An action is wild if the stabilizer limit group is an increasing sequence of stabilizer groups without bound, and otherwise is said to be stable if this group chain is bounded. For Cantor actions by a finitely generated group G, we prove that stable actions satisfy a rigidity principle, a…
We describe a self-homeomorphism $R$ of the Cantor set $X$ and then show that its conjugacy class in the Polish group $H(X)$ of all homeomorphisms of $X$ forms a dense $G_δ$ subset of $H(X)$. We also provide an example of a locally compact, second countable topological group which has a dense conjugacy class.
Automorphism groups of measures on the Cantor space. Part I: Good measures and Rokhlin properties
本文研究 Cantor 空间上测度自同构群中存在稠密或余瘦共轭类的判据，利用 Fraïssé 理论将好测度刻画为超齐次测度的子类，并确定了取有理值且自同构群承认余瘦共轭类的好测度。
Automorphism groups of measures on the Cantor space. Part II: Abstract homogeneous measures
本文引入了一类在 Cantor 空间 clopen 子集 Boolean 代数上超齐次的半群值测度，通过三值谱刻画了其自同构群的传递性与极小性，并证明了存在不可数个满足极小作用且具稠密共轭类的全非原子概率 Borel 测度。
Let $f$ be a chain mixing continuous onto mapping from the Cantor set onto itself.Let $g$ be an aperiodic homeomorphism on the Cantor set. We show that homeomorphisms that are topologically conjugate to g approximate $f$ in the topology of uniform convergence if a trivial necessary condition on periodic points is satisfied.In particular, let $f$ be a chain mixing continuous onto mapping from the Cantor set onto itself with a fixed point and g, an aperiodic homeomorphism on the Cantor set. Then, homeomorphisms that are topologically conjugate to $g$ approximate $f$.
In this article, we study countable sofic shifts of Cantor-Bendixson rank at most 2. We prove that their conjugacy problem is complete for GI, the complexity class of graph isomorphism, and that the existence problems of block maps, factor maps and embeddings are NP-complete.
Minimal Cantor systems of finite topological rank (that can be represented by a Bratteli-Vershik diagram with a uniformly bounded number of vertices per level) are known to have dynamical rigidity properties. We establish that such systems, when they are expansive, define the same class of systems, up to topological conjugacy, as primitive and recognizable ${\mathcal S}$-adic subshifts. This is done establishing necessary and sufficient conditions for a minimal subshift to be of finite topological rank. As an application, we show that minimal subshifts with non-superlinear complexity (like all…
We show that the group of homeomorphisms of the Cantor set $H(K)$ has ample generics, that is, for every $m$ the diagonal conjugacy action $g\cdot(h_1,h_2,..., h_m)=(gh_1g^{-1},gh_2g^{-1},..., gh_mg^{-1})$ of $H(K)$ on $H(K)^m$ has a comeager orbit. This answers a question of Kechris and Rosendal. We show that the generic tuple in $H(K)^m$ can be taken to be the limit of a certain projective Fraisse family. We also present a proof of the existence of the generic homeomorphism of the Cantor set in the context of the projective Fraisse theory.
It will be shown that the smooth conjugacy class of an $S-$unimodal map which does not have a periodic attractor neither a Cantor attractor is determined by the multipliers of the periodic orbits. This generalizes a result by M.Shub and D.Sullivan for smooth expanding maps of the circle.
Given a dynamical simplex $K$ on a Cantor space $X$, we consider the set $G_K^*$ of all homeomorphisms of $X$ which preserve all elements of $K$ and have no nontrivial clopen invariant subset. Generalising a theorem of Yingst, we prove that for a generic element $g$ of $G_K^*$ the set of invariant measures of $g$ is equal to $K$. We also investigate when there exists a generic conjugacy class in $G_K^*$ and prove that this happens exactly when $K$ has only one element, which is the unique invariant measure associated to some odometer; and that in that case the conjugacy class of this odometer …
We show that conjugacy of reversible cellular automata is undecidable, whether the conjugacy is to be performed by another reversible cellular automaton or by a general homeomorphism. This gives rise to a new family of finitely-generated groups with undecidable conjugacy problems, whose descriptions arguably do not involve any type of computation. For many automorphism groups of subshifts, as well as the group of asynchronous transducers and the homeomorphism group of the Cantor set, our result implies the existence of two elements such that every finitely-generated subgroup containing both ha…
Conjugacy problems in higher-dimensional Thompson groups
该研究证明了n≥2时Brin-Thompson群$nV$中元素的拓扑共轭性不可判定，还得到了Cantor空间同胚群有限生成超群中Whitehead问题及群论共轭的不可判定性。
Let X be an infinite compact metric space with finite covering dimension. Let $\afhpa,\bt: X\to X$ be two minimal homeomorphisms. Suppose that the range of $K_0$-groups of both crossed product C*-algebras s are dense in the space of real affine continuous functions. We show that $\af$ and $\bt$ are approximately conjugate uniformly in measure if and only if they have affine homeomorphic invariant probability measure spaces.
Limits of Quadratic Rational Maps: The Cantor Locus
本文研究二次有理映射 Cantor 轨迹边界的抛物动力学，提出动力学标记与局部参数化，证明固定标记序列在亚循环极限下收敛于抛物参数或经重标度后趋于边界。
We study topological properties of conjugacy classes in Polish groups, with emphasis on automorphism groups of homogeneous countable structures. We first consider the existence of dense conjugacy classes (the topological Rokhlin property). We then characterize when an automorphism group admits a comeager conjugacy class (answering a question of Truss) and apply this to show that the homeomorphism group of the Cantor space has a comeager conjugacy class (answering a question of Akin-Hurley-Kennedy). Finally, we study Polish groups that admit comeager conjugacy classes in any dimension (in which…
Let $α$ and $β$ be two Furstenberg transformations on 2-torus associated with irrational numbers $θ_1,$ $θ_2,$ integers $d_1, d_2$ and Lipschitz functions $f_1$ and $f_2.$ We show that $α$ and $β$ are approximately conjugate in a measure theoretical sense if (and only if) $\bar{θ_1\pm θ_2}=0$ in $\R/\Z.$ Closely related to the classification of simple amenable $C^*$-algebras, we show that $α$ and $β$ are approximately $K$-conjugate if (and only if) $\bar{θ_1\pm θ_2}=0$ in $\R/\Z$ and $|d_1|=|d_2|.$ This is also shown to be equivalent to that the associated crossed product $C^*$-algebras are is…
On the Rokhlin lemma for infinite measure-preserving bijections
本文研究无限测度保持双射中的罗克林引理，完全分类此类双射到λ近共轭，其中λ是保持的无限测度，改进了经典版本仅到μ近共轭的分类。
In this article, we extend H. Matui and H. Lin's notions of approximate $K$-conjugacies and $C^*$-strongly approximate conjugacies to general minimal dynamical systems. In particular, upon modifying a result of the existence of minimal skew products, we answer a question of H. Lin and show that, associated with any Cantor minimal system $(K,\tildeα)$, there is a class $R_0(\tildeα)$ of minimal skew products on $K\timesΩ$, such that for any two rigid homeomorphisms $α\in R_0(\tildeα)$ and $β\in R_0(\tildeβ)$, the notions of approximate $K$-conjugacy and $C^*$-strongly approximate conjugacy coin…
It is introduced an analogue of the orbit-breaking subalgebra for the case of free flows on locally compact metric spaces, which has a natural approximate structure in terms of a fixed point and any nested sequence of central slices around this point. It is shown that in the case of minimal flows admitting a compact Cantor central slice, the resulting $C^*$-algebra is the stabilization of the Putnam orbit-breaking subalgebra associated to the induced homeomorphism on the central slice.
Dense and comeager conjugacy classes in zero-dimensional dynamics
研究了可数群在Cantor空间上的最小和拓扑transitive作用的波兰空间，探讨了共轭类的存在性，发现自由群的最小作用和概率测度保持作用存在稠密且来ager共轭类，而可数非有限生成阿贝尔群则不存在。
This paper considers a nonlinear dynamical system on a complex, finite dimensional Banach space which has an asymptotically stable, hyperbolic fixed point. We investigate the connection between the so-called principle eigenfunctions of the Koopman operator and the existence of a topological conjugacy between the nonlinear dynamics and its linearization in the neighborhood of the fixed point. The principle eigenfunctions generate an algebra of observables for the linear dynamics --- called the principle algebra --- which can be used to generate a sequence of approximate conjugacy maps in the sa…
Isogenies of minimal Cantor systems: from Sturmian to Denjoy and interval exchanges
本文以实数连分数展开为背景，引入零维系统同源映射概念，完成斯特姆、德若瓦等系统的同源映射分类，刻画区间交换变换的流等价性，证明相关猜想。
Let β: S^n \to S^n, for n = 2k + 1, k \geq 1, be one of the known examples of a non-uniquely ergodic minimal diffeomorphism of an odd dimensional sphere. For every such minimal dynamical system (S^n, β) there is a Cantor minimal system (X, α) such that the corresponding product system (X x S^n, αx β) is minimal and the resulting crossed product C*-algebra C(X x S^n) \rtimes_{αx β} \mathbb{Z} is tracially approximately an interval algebra (TAI). This entails classification for such C*-algebras. Moreover, the minimal Cantor system (X, α) is such that each tracial state on C(X x S^n) \rtimes_β \m…
M. Gromov introduced the mean dimension for a continuous map in the late 1990's, which is an invariant under topological conjugacy. On the other hand, the notion of metric mean dimension for a dynamical system was introduced by Lindenstrauss and Weiss in 2000 and this refines the topological entropy for dynamical systems with infinite topological entropy. In this paper we will show if $N$ is a $n$ dimensional compact riemannian manifold then, for any $a\in [0,n]$, the set consisting of continuous maps with metric mean dimension equal to $a$ is dense in $C^{0}(N)$ and for $a=n$ this set is resi…
Effective dynamical systems beyond dimension zero and factors of SFTs
该研究借助可计算分析定义一般动力系统的有效性，证明有限生成递归表现群的有效动力系统是康托空间有效闭集上可计算作用的拓扑因子，并将模拟结果推广至零维空间外，指出一大类群的诸多自然作用是有限型子移位的拓扑因子。
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。
The main aim of this paper is to detect dynamical properties of the Györgyi-Field model of the Belousov-Zhabotinsky chemical reaction. The corresponding three-variable model given as a set of nonlinear ordinary differential equations depends on one parameter, the flow rate. As certain values of this parameter can give rise to chaos, the analysis was performed in order to identify different dynamics regimes. Dynamical properties were qualified and quantified using classical and also new techniques. Namely, phase portraits, bifurcation diagrams, the Fourier spectra analysis, the 0-1 test for cha…
We initiate the study of a measurable analogue of small topological full groups that we call $\mathrm L^1$ full groups. These groups are endowed with a Polish group topology which admits a natural complete right invariant metric. We mostly focus on $\mathrm L^1$ full groups of measure-preserving $\mathbb Z$-actions which are actually a complete invariant of flip conjugacy. We prove that for ergodic actions the closure of the derived group is topologically simple although it can fail to be simple. We also show that the closure of the derived group is connected, and that for measure-preserving f…
Dynamical Perturbing and $C^*$-algebra Lifting Problems
本文研究了近似群作用的稳定性问题，并引入了新的$C^*$-代数提升问题，探讨了有限群作用的稳定性及条件半射影性。
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