For a Cantor set $X$, let $Homeo(X)$ denote the group of all homeomorphisms of $X$. The main result of this note is the following theorem. Let $T\in Homeo(X)$ be an aperiodic homeomorphism, let $μ_1,μ_2,...,μ_k$ be Borel probability measures on $X$, $\e> 0$, and $n \ge 2$. Then there exists a clopen set $E\subset X$ such that the sets $E,TE,..., T^{n-1}E$ are disjoint and $μ_i(E\cup TE\cup...\cup T^{n-1}E) > 1 - \e, i= 1,...,k$. Several corollaries of this result are given. In particular, it is proved that for any aperiodic $T\in Homeo(X)$ the set of all homeomorphisms conjugate to $T$ is dense in the set of aperiodic homeomorphisms.
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 $f$.
Let $Homeo(Ω)$ be the group of all homeomorphisms of a Cantor set $Ω$. We study topological properties of $Homeo(Ω)$ and its subsets with respect to the uniform $(τ)$ and weak $(τ_w)$ topologies. The classes of odometers and periodic, aperiodic, minimal, rank 1 homeomorphisms are considered and the closures of those classes in $τ$ and $τ_w$ are found.
Let f be a chain mixing continuous onto mapping from the Cantor set onto itself. Let g be a homeomorphism on the Cantor set that is topologically conjugate to a subshift. Then, homeomorphisms that are topologically conjugate to g approximate f in the topology of uniform convergence if a trivial necessary condition on the periodic points holds. In particular, if f is a chain mixing continuous onto mapping from the Cantor set onto itself with a fixed point, then homeomorphisms on the Cantor set that are topologically conjugate to a subshift approximate f in the topology of uniform convergence. I…
We develop a technique, pseudo-suspension, that applies to invariant sets of homeomorphisms of a class of annulus homeomorphisms we describe, Handel-Anosov-Katok (HAK) homeomorphisms, that generalize the homeomorphism first described by Handel. Given a HAK homeomorphism and a homeomorphism of the Cantor set, the pseudo-suspension yields a homeomorphism of a new space that admits a homeomorphism that combines features of both of the original homeomorphisms. This allows us to answer a well known open question by providing examples of hereditarily indecomposable continua that admit homeomorphisms…
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.
We continue to study topological properties of the group Homeo(X) of all homeomorphisms of a Cantor set X with respect to the uniform topology tau, which was started in the paper (S. Bezuglyi, A.H. Dooley, and J. Kwiatkowski, Topologies on the group of homeomorphisms of a Cantor set, ArXiv e-print math.DS/0410507, 2004). We prove that the set of periodic homeomorphisms is tau-dense in Homeo(X) and deduce from this result that the topological group (Homeo(X), tau) has the Rokhlin property, i.e., there exists a homeomorphism whose conjugate class is tau-dense in Homeo(X). We also show that for 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.
We study the set $M_\infty(X)$ of all infinite full non-atomic Borel measures on a Cantor space X. For a measure $μ$ from $M_\infty(X)$ we define a defective set $M_μ= \{x \in X : for any clopen set U which contains x we have μ(U) = \infty \}$. We call a measure $μ$ from $M_\infty(X)$ non-defective ($μ\in M_\infty^0(X)$) if $μ(M_μ) = 0$. The paper is devoted to the classification of measures $μ$ from $M_\infty^0(X)$ with respect to a homeomorphism. The notions of goodness and clopen values set $S(μ)$ are defined for a non-defective measure $μ$. We give a criterion when two good non-defective m…
In this paper we define a new class of metric spaces, called multi-model Cantor sets. We compute the Hausdorff dimension and show that the Hausdorff measure of a multi-model Cantor set is finite and non-zero. We then show that a bilipschitz map from one multi-model Cantor set to another has constant Radon-Nikodym derivative on some clopen. We use this to obtain an invariant up to bilipschitz homeomorphism.
Let $f$ be an $R$-closed homeomorphism on a connected orientable closed surface $M$. In this paper, we show that If $M$ has genus more than one, then each minimal set is either a periodic orbit or an extension of a Cantor set. If $M = \mathbb{T}^2$ and $f$ is neither minimal nor periodic, then either each minimal set is finite disjoint union of essential circloids or there is a minimal set which is an extension of a Cantor set. If $M = \mathbb{S}^2$ and $f$ is not periodic but orientation-preserving (resp. reversing), then the minimal sets of $f$ (resp. $f^2$) are exactly two fixed points and …
We examine topological dynamical systems on the Cantor set from the point of view of the continuous model theory of commutative C*-algebras. After some general remarks we focus our attention on the generic homeomorphism of the Cantor set, as constructed by Akin, Glasner, and Weiss. We show that this homeomorphism is the prime model of its theory. We also show that the notion of "generic" used by Akin, Glasner, and Weiss is distinct from the notion of "generic" encountered in Fraisse theory.
This paper is a survey devoted to the study of probability and infinite ergodic invariant measures for aperiodic homeomorphisms of a Cantor set. We focus mostly on the cases when a homeomorphism has either a unique ergodic invariant measure or finitely many such measures (finitely ergodic homeomorphisms). Since every Cantor dynamical system $(X,T)$ can be realized as a Vershik map acting on the path space of a Bratteli diagram, we use combinatorial methods developed in symbolic dynamics and Bratteli diagrams during the last decade to study the simplex of invariant measures.
We provide a classification of minimal sets of homeomorphisms of the two-torus, in terms of the structure of their complement. We show that this structure is exactly one of the following types: (1) a disjoint union of topological disks, or (2) a disjoint union of essential annuli and topological disks, or (3) a disjoint union of one doubly essential component and bounded topological disks. Periodic bounded disks can only occur in type 3. This result provides a framework for more detailed investigations, and additional information on the torus homeomorphism allows to draw further conclusions. I…
We define a notion of tiling of the full infinite $p$-ary tree, establishing a series of equivalent criteria for a subtree to be a tile, each of a different nature; namely, geometric, algebraic, graph-theoretic, order-theoretic, and topological. We show how these results can be applied in a straightforward and constructive manner to define homeomorphisms between two given spaces of $p$-adic integers, $\mathbb{Z}_{p}$ and $\mathbb{Z}_{q}$, endowed with their corresponding standard non-archimedean metric topologies.
As was known to H. Poincare, an orientation preserving circle homeomorphism without periodic points is either minimal or has no dense orbits, and every orbit accumulates on the unique minimal set. In the first case the minimal set is the circle, in the latter case a Cantor set. In this paper we study a two-dimensional analogue of this classical result: we classify the minimal sets of non-resonant torus homeomorphisms; that is, torus homeomorphisms isotopic to the identity for which the rotation set is a point with rationally independent irrational coordinates.
University of Maribor（马里博尔大学）；Abdus Salam International Centre for Theoretical Physics (ICTP)（阿卜杜斯·萨拉姆国际理论物理中心（ICTP））；University of Louisville（路易斯维尔大学）；Charles University Prague（布拉格查理大学）
Let X be a h-homogeneous zero-dimensional compact Hausdorff space, i.e. X is a Stone dual of a homogeneous Boolean algebra. Using the dual Ramsey theorem and a detailed combinatorial analysis of what we call stable collections of subsets of a finite set, we obtain a complete list of the minimal sub-systems of the compact dynamical system (Exp(Exp(X)),Homeo(X)), where Exp(X) stands for the hyperspace comprising the closed subsets of X equipped with the Vietoris topology. The importance of this dynamical system stems from Uspenskij's characterization of the universal ambit of G = Homeo(X). The r…
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.
We study diagonal actions $φ:\mathbb{F}_2\curvearrowright\partial\mathbb{F}_2\times K$ on the Cantor set which are given by $φ_a=\partial_a\timesα,φ_b=\partial_b\timesβ$. Under some restrictions on $α,β$ we compute $K_*(C(\partial\mathbb{F}_2\times K)\rtimes_r\mathbb{F}_2)$. As an application in the case of $α$ is Denjoy homeomorphism of the Cantor set and $β=id$ we will show that $C(\partial\mathbb{F}_2\times K)\rtimes_r\mathbb{F}_2$ is Kirchberg algebra with $K_*(C(\partial\mathbb{F}_2\times K)\rtimes_r\mathbb{F}_2)=(\mathbb{Z}^\infty,\mathbb{Z}^\infty)$. Also we will check that $C^*$-crosse…
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 full groups of minimal actions of countable groups by homeomorphisms on a Cantor space $X$, showing that these groups do not admit a compatible Polish group topology and, in the case of $\Z$-actions, are coanalytic non-Borel inside $\Homeo(X)$. We point out that the full group of a minimal homeomorphism is topologically simple. We also study some properties of the closure of the full group of a minimal homeomorphism inside $\Homeo(X)$.
Let X be a h-homogeneous zero-dimensional compact Hausdorff space, i.e. X is a Stone dual of a homogeneous Boolean algebra. It is shown that the universal minimal space M(G) of the topological group G=Homeo(X), is the space of maximal chains on X introduced by Uspenskij. If X is metrizable then clearly X is homeomorphic to the Cantor set and the result was already known (Glasner-Weiss 2003). However many new examples arise for non-metrizable spaces. These include, among others, the generalized Cantor sets X={0,1}^{kappa} for non-countable cardinals kappa, and the corona or remainder of omega, …
We give a characterization of sets K of probability measures on a Cantor space X with the property that there exists a minimal homeomorphism g of X such that the set of g-invariant probability measures on X coincides with K. This extends theorems of Akin (corresponding to the case when K is a singleton) and Dahl (when K is finite-dimensional). Our argument is elementary and different from both Akin's and Dahl's.
Let $K$ be the Cantor set. We prove that arbitrarily close to a homeomorphism $T:K\rightarrow K$ there exists a homeomorphism $\widetilde T:K\rightarrow K$ such that the $α$-limit and the $ω$-limit of every orbit is a periodic orbit. We also prove that arbitrarily close to an endomorphism $T:K\rightarrow K$ there exists an endomorphism $\widetilde T:K\rightarrow K$ close to $T$ such that every orbit is finally periodic.
Three dimensional Hńon-like map $$ F(x,y,z) = (f(x) - ε(x,y,z),\ x,\ δ(x,y,z)) $$ is defined on the cubic box $ B $. An invariant space under renormalization would appear only in higher dimension. Consider renormalizable maps each of which satisfies the condition $$ \partial_y δ\circ F(x,y,z) + \partial_z δ\circ F(x,y,z) \cdot \partial_x δ(x,y,z) \equiv 0 $$ for $ (x,y,z) \in B $. Denote the set of maps satisfying the above condition be $ \mathcal N $. Then the set $ \mathcal N \cap \mathcal I(\bar ε) $ is invariant under the renormalization operator where $ \mathcal I(\bar ε) $ is the set of …
We study the set M(X) of full non-atomic Borel (finite or infinite) measures on a non-compact locally compact Cantor set X. For an infinite measure $μ$ in M(X), the set $\mathfrak{M}_μ= \{x \in X : {for any compact open set} U \ni x {we have} μ(U) = \infty \}$ is called defective. We call $μ$ non-defective if $μ(\mathfrak{M}_μ) = 0$. The class $M^0(X) \subset M(X)$ consists of probability measures and infinite non-defective measures. We classify measures $μ$ from $M^0(X)$ with respect to a homeomorphism. The notions of goodness and compact open values set $S(μ)$ are defined. A criterion when t…
This paper studies the behavior under iteration of the maps T_{ab}(x,y) = (F_{ab}(x)- y, x) of the plane R^2, in which F_{ab}(x)= ax if x>0 and bx if x<0. These maps are area-preserving homeomorphisms of the plane that map rays from the origin into rays from the origin. Orbits of the map correspond to solutions of the nonlinear difference equation x_{n+2}= 1/2(a-b)|x_{n+1}| + 1/2(a+b)x_{n+1} - x_n. This difference equation can be written in an eigenvalue form for a nonlinear difference operator of Schrodinger type, in which μ= 1/2(a-b) is viewed as fixed and the energy E=2- 1/2(a+b). The paper…
This article presents a generalization of the notion of \emph{Poincaré rotation set} to homeomorphisms of the adèle class group $\mathbb{A}/\mathbb{Q}$ of the rational numbers $\mathbb{Q}$, which is a connected compact abelian group which can be identified with the one-dimensional universal solenoid $\mathbf{S}$, \ie the algebraic universal covering of the circle. The definition is first introduced in general for homeomorphisms of $\mathbf{S}$ which are isotopic to a translation, and then specializing in homeomorphisms of $\mathbf{S}$ isotopic to the identity, in which case the rotation set is…
Let $U$ and $V$ be open subsets of the Cantor set with finite disjoint complements, and let $h:U\to V$ be a homeomorphism with dense orbits. Building from the ideas of Herman, Putnam, and Skau, we show that the partial action induced by $h$ can be realized as the Vershik map on a Bratteli diagram, and that any two such diagrams are equivalent.
We show that the (topological) full group of a minimal pseudogroup over the Cantor set satisfies various rigidity phenomena of topological dynamical and combinatorial nature. Our main result applies to its possible homomorphisms into other groups of homeomorphisms, and implies that arbitrary homomorphisms between the full groups of a vast class of pseudogroups must extend to continuous morphisms between pseudogroups (in particular giving rise to equivariant maps at the level of spaces). As applications, we obtain explicit obstructions to the existence of embeddings between full groups in terms…
