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 $[\f]$.
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.
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.
The conjugacy and flip conjugacy problem for Cantor minimal systems
Instituto de Matemáticas, Universidad Nacional Autónoma de México (UNAM)（墨西哥国立自治大学数学研究所）；Faculty of Mathematics and Computer Science, Jagiellonian University in Kraków（雅盖隆大学数学与计算机科学系）；Department of Mathematics, Oregon State University（俄勒冈州立大学数学系）
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。
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…
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.
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…
We study asymptotic continuous orbit equivalence of Smale spaces. We prove that two irreducible Smale spaces are flip conjugate if and only if there exists a periodic point preserving homeomorphism giving an asymptotic continuous orbit equivalence between them. We introduce a notion of asymptotic topological conjugacy and asymptotic flip conjugacy in Smale spaces and characterize them in terms of the associated Ruelle algebras with dual actioons. We finally characterize the flip conjugacy classes of irreducible two-sided topological Markov shifts in terms of the associated Ruelle algebras with…
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 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.
A $D_{\infty}$-topological Markov chain can be represented by a pair of zero-one square matrices, which is called a flip pair. We introduce the concepts of $D_{\infty}$-strong shift equivalence and $D_{\infty}$-shift equivalence, which are equivalence relations between flip pairs. We investigate the relationships between the existence of a $D_{\infty}$-conjugacy, the existence of a $D_{\infty}$-strong shift equivalence, the existence of a $D_{\infty}$-shift equivalence and the coincidence of the Lind zeta functions.
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.
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…
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…
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…
Bounded orbit injection equivalence is an equivalence relation defined on minimal free Cantor systems which is a candidate to generalize flip Kakutani equivalence to actions of the Abelian free groups on more than one generator. This paper characterizes bounded orbit injection equivalence in terms of a mild strengthening of Rieffel-Morita equivalence of the associated C*-crossed-product algebras. Moreover, we construct an ordered group which is an invariant for bounded orbit injection equivalence, and does not agrees with the K\_0 group of the associated C*-crossed-product in general. This new…
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 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…
A $D_{\infty}$-topological Markov chain is a topological Markov chain provided with an action of the infinite dihedral group $D_{\infty}$. It is defined by two zero-one square matrices $A$ and $J$ satisfying $AJ=JA^{\textsf{T}}$ and $J^2=I$. Flip signature is obtained from symmetric bilinear forms with respect to $J$ on the eventual kernel of $A$. We modify Williams' decomposition theorem to prove flip signature is a $D_{\infty}$-conjugacy invariant. We introduce natural $D_{\infty}$-actions on Ashley's eight-by-eight and the full two-shift. The Flip signatures show that Ashley's eight-by-eigh…
Cohomology for actions of free abelian groups on the Cantor set has (when endowed with an order structure) provided a complete invariance for orbit equivalence. In this paper, we study a particular class of actions of such groups called odometers (or profinite actions) and investigate their cohomology. We show that for a free, minimal $\Z^{d}$-odometer, the first cohomology group provides a complete invariant for the action up to conjugacy. This is in contrast with the situation for orbit equivalence where it is the cohomology in dimension $d$ which provides the invariant. We also consider cla…
Switched server systems are mathematical models of manufacturing, traffic and queueing systems that have being studied since the early 1990s. In particular, it is known that typically the dynamics of such systems is asymptotically periodic: each orbit of the system converges to one of its finitely many limit cycles. In this article, we provide an explicit example of a switched server system with exotic behavior: each orbit of the system converges to the same Cantor attractor. To accomplish this goal, we bring together recent advances in the understanding of the topological dynamics of piecewis…
We consider Moebius and conformal homeomorphisms $f : \partial X \to \partial Y$ between boundaries of CAT(-1) spaces $X,Y$ equipped with visual metrics. A conformal map $f$ induces a topological conjugacy of the geodesic flows of $X$ and $Y$, which is flip-equivariant if $f$ is Moebius. We define a function $S(f) : \partial ^2 X \to \mathbb{R}$, the {\it integrated Schwarzian} of $f$, which measures the deviation of the topological conjugacy from being flip-equivariant, in particular vanishing if $f$ is Moebius. Conversely if $X,Y$ are simply connected complete manifolds with pinched negative…
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…
