Compact Metric Space Research Articles

Expansiveness, generators and Lyapunov exponents for random bundle transformations

We generalize Fathi's results by showing that a compact metrizable space admits an fiber expansive homeomorphism if and only if it has a compatible hyperbolic metric. Moreover, we prove that a compact metrizable space admits an fiber expansive homeomorphism if and only if it has a generator in detail. Furthermore, we show that a fiber expansive homeomorphism has finite fiber topological entropy. Finally, we show that fiber Lyapunov exponents for a fiber expansive system are different from zero, indicating that the system presents a chaotic system. Meanwhile, we also prove that negative fiber Lyapunov exponents for compact invariant sets of a dynamical system imply that the compact set is a fiber attractor.

Bauer simplices and the small boundary property

We show that, for every minimal action of a countably infinite discrete group on a compact metrizable space, if the extreme boundary of the simplex of invariant Borel probability measures is closed and has finite covering dimension then the action has the small boundary property.

Nonfree almost finite actions for locally finite‐by‐virtually Z${\mathbb {Z}}$ groups

AbstractIn this paper, we study almost finiteness and almost finiteness in measure of nonfree actions. Let be a minimal action of a locally finite‐by‐virtually group on the Cantor set . We prove that under certain assumptions, the action is almost finite in measure if and only if is essentially free. As an application, we obtain that any minimal topologically free action of a virtually group on an infinite compact metrizable space with the small boundary property is almost finite. This is the first general result, assuming only topological freeness, in this direction, and these lead to new results on uniform property and ‐stability for their crossed product ‐algebras. Some concrete examples of minimal topological free (but nonfree) subshifts are provided.

Compact spaces homeomorphic to their respective squares

We deal with topological spaces homeomorphic to their respective squares. Primarily, we investigate the existence of large families of such spaces in some subclasses of compact metrizable spaces. As our main result we show that there is a family of size continuum of pairwise non-homeomorphic compact metrizable zero-dimensional spaces homeomorphic to their respective squares. This answers a question of Włodzimierz J. Charatonik. We also discuss the situation in the classes of continua, Peano continua and absolute retracts.

Expansive actions with specification of sofic groups, strong topological Markov property, and surjunctivity

A dynamical system is a pair (X,G), where X is a compact metrizable space and G is a countable group acting by homeomorphisms of X. An endomorphism of (X,G) is a continuous selfmap of X which commutes with the action of G. One says that a dynamical system (X,G) is surjunctive provided that every injective endomorphism of (X,G) is surjective (and therefore is a homeomorphism). We show that when G is sofic, every expansive dynamical system (X,G) with nonnegative sofic topological entropy and satisfying the weak specification and the strong topological Markov properties, is surjunctive.

Uniqueness and homogeneity of non-separable Urysohn universal ultrametric spaces

We first introduce the new notion of petaloid ultrametric spaces, which is intended to be a standard class of non-separable Urysohn universal ultrametric spaces. Next we prove that all petaloid spaces are isometric to each other and homogeneous for all finite subspaces (and compact subspaces). Moreover, we show that the following spaces are petaloid, and hence they are isometric to each other and homogeneous: (1) The space of all continuous functions, whose images contain the zero, from the Cantor set into the space of non-negative real numbers equipped with the nearly discrete topology, (2) the space of all continuous ultrametrics on a zero-dimensional infinite compact metrizable space, (3) the non-Archimedean Gromov–Hausdorff space, and (4) the space of all maps from the set of non-negative real numbers into the set of natural numbers whose supports are finite or decreasing sequences convergent to the zero.

Glueing spaces without identifying points

In this paper we develop the theory of Artin-Wraith glueings for topological spaces. It means that we are glueing two topological spaces to each other without identifying points on them. These constructions are far from unique and this theory is used to deal with all of them.As an application, we show that some categories of compactifications of coarse spaces that agree with the coarse structures are invariant under coarse equivalences. As a consequence, if X and Y are some well behaved metric spaces that are coarse equivalent, then they have the same space of ends (generalizing the well known fact that works on quasi-isometric proper geodesic metric spaces). As another application, we show that for every compact metrizable space Y, there exists only one, up to homeomorphisms, compactification of the Cantor set minus one point such that the remainder is homeomorphic to Y.

Plasticity of the unit ball of some C(K) spaces

We show that if K is a compact metrizable space with finitely many accumulation points, then the closed unit ball of C(K) is a plastic metric space, which means that any non-expansive bijection from BC(K) onto itself is in fact an isometry. We also show that if K is a zero-dimensional compact Hausdorff space with a dense set of isolated points, then any non-expansive homeomorphism of BC(K) is an isometry.

On Iso-dense and Scattered Spaces without textbf{AC}

A topological space is iso-dense if it has a dense set of isolated points, and it is scattered if each of its non-empty subspaces has an isolated point. In textbf{ZF} (i.e. Zermelo–Fraenkel set theory without the Axiom of Choice (textbf{AC})), basic properties of iso-dense spaces are investigated. A new permutation model is constructed, in which there exists a discrete weakly Dedekind-finite space having the Cantor set as a remainder; the result is transferable to textbf{ZF}. This settles an open problem posed by Keremedis, Tachtsis and Wajch in 2021. A metrization theorem for a class of quasi-metric spaces is deduced. The statement “Every compact scattered metrizable space is separable” and several other statements about metric iso-dense spaces are shown to be equivalent to the axiom of countable choice for families of finite sets. Results related to the open problem of the set-theoretic strength of the statement “Every non-discrete compact metrizable space contains an infinite compact scattered subspace” are also included.

Countable products and countable direct sums of compact metrizable spaces in the absence of the Axiom of Choice

The main aim of the article is to show, in the absence of the Axiom of Choice, relationships between the following, independent of ZF, statements: “Every countable product of compact metrizable spaces is separable (respectively, compact)” and “Every countable product of compact metrizable spaces is metrizable”. Statements related to the above-mentioned ones are also studied. Permutation models (among them two new ones) are shown in which a countable direct sum (also a countable product) of metrizable spaces need not be metrizable, countable unions of countable sets are countable and there is a countable family of non-empty sets of size at most 2ℵ0 which does not have a choice function. A new permutation model is constructed in which every uncountable compact metrizable space is of size at least 2ℵ0 but a denumerable family of denumerable sets need not have a multiple choice function.

Mean dimension of product spaces: a fundamental formula

Mean dimension is a topological invariant of dynamical systems, which originates with Mikhail Gromov in 1999 and which was studied with deep applications around 2000 by Elon Lindenstrauss and Benjamin Weiss within the framework of amenable group actions. Let a countable discrete amenable group G act continuously on compact metrizable spaces X and Y. Consider the product action of G on the product space $$X\times Y$$ . The product inequality for mean dimension is well known: $${{\,\mathrm{\textrm{mdim}}\,}}(X\times Y,G)\le {{\,\mathrm{\textrm{mdim}}\,}}(X,G)+{{\,\mathrm{\textrm{mdim}}\,}}(Y,G)$$ , while it was unknown for a long time if the product inequality could be an equality. In 2019, Masaki Tsukamoto constructed the first example of two different continuous actions of G on compact metrizable spaces X and Y, respectively, such that the product inequality becomes strict. However, there is still one longstanding problem which remains open in this direction, asking if there exists a continuous action of G on some compact metrizable space X such that $${{\,\mathrm{\textrm{mdim}}\,}}(X\times X,G)<2\cdot {{\,\mathrm{\textrm{mdim}}\,}}(X,G)$$ . We solve this problem. Somewhat surprisingly, we prove, in contrast to (topological) dimension theory, a rather satisfactory theorem: If an infinite (countable discrete) amenable group G acts continuously on a compact metrizable space X, then we have $${{\,\mathrm{\textrm{mdim}}\,}}(X^n,G)=n\cdot {{\,\mathrm{\textrm{mdim}}\,}}(X,G)$$ , for any positive integer n. Our product formula for mean dimension, together with the example and inequality (stated previously), eventually allows mean dimension of product actions to be fully understood.

Iterated Minkowski sums, horoballs and north-south dynamics

Given a finite generating set A for a group \Gamma , we study the map W \mapsto WA as a topological dynamical system – a continuous self-map of the compact metrizable space of subsets of \Gamma . If the set A generates \Gamma as a semigroup and contains the identity, there are precisely two fixed points, one of which is attracting. This supports the initial impression that the dynamics of this map is rather trivial. Indeed, at least when \Gamma= \mathbb{Z}^d and A \subseteq \mathbb{Z}^d is a finite positively generating set containing the identity, the natural invertible extension of the map W \mapsto W+A is always topologically conjugate to the unique “north-south” dynamics on the Cantor set. In contrast to this, we show that various natural “geometric” properties of the finitely generated group (\Gamma,A) can be recovered from the dynamics of this map, in particular, the growth type and amenability of \Gamma . When \Gamma = \mathbb{Z}^d , we show that the volume of the convex hull of the generating set A is also an invariant of topological conjugacy. Our study introduces, utilizes and develops a certain convexity structure on subsets of the group \Gamma , related to a new concept which we call the sheltered hull of a set. We also relate this study to the structure of horoballs in finitely generated groups, focusing on the abelian case.

CLASSIFICATION OF ONE DIMENSIONAL DYNAMICAL SYSTEMS BY COUNTABLE STRUCTURES

AbstractWe 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 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 homeomorphisms is Borel bireducible to the universal orbit equivalence relation.

Baire property of some function spaces

A compact space X is called \(\pi\)-monolithic if for any surjective continuous mapping \(f \colon X \rightarrow K\) where K is a metrizable compact space there exists a metrizable compact space \(T \subseteq X\) such that \(f(T)=K\). A topological space X is Baire if the intersection of any sequence of open dense subsets of X is dense in X. Let \(C_p(X, Y)\) denote the space of all continuous Y-valued functions C(X,Y) on a Tychonoff space X with the topology of pointwise convergence. In this paper we have proved that for a totally disconnected space X the space \(C_p(X,\{0,1\})\) is Baire if, and only if, \(C_p(X,K)\) is Baire for every \(\pi\)-monolithic compact space K.For a Tychonoff space X the space \(C_p(X,\mathbb{R})\) is Baire if, and only if, \(C_p(X,L)\) is Baire for each Fréchet space L.We construct a totally disconnected Tychonoff space T such that \(C_p(T,M)\) is Baire for a separable metric space M if, and only if, M is a Peano continuum. Moreover, \(C_p(T,[0,1])\) is Baire but \(C_p(T,\{0,1\})\) is not.

Random walks on convergence groups

We extend some properties of random walks on hyperbolic groups to random walks on convergence groups. In particular, we prove that if a convergence group G acts on a compact metrizable space M with the convergence property, then we can provide G\cup M with a compact topology such that random walks on G converge almost surely to points in M . Furthermore, we prove that if G is finitely generated and the random walk has finite entropy and finite logarithmic moment with respect to the word metric, then M , with the corresponding hitting measure, can be seen as a model for the Poisson boundary of G .

Weak Mean Equicontinuity for a Countable Discrete Amenable Group Action

The weak mean equicontinuity for a countable discrete amenable group G acting continuously on a compact metrizable space X is studied. It is shown that weak mean equicontinuity, continuously pointwise ergodicity and uniformity are coincided. Moreover, we prove that (X, G) is mean equicontinuous if and only if the product system \((X \times X,G)\) is weak mean equicontinuous.

Directional mean dimension and continuum-wise expansive ℤ^{𝕜}-actions

We study directional mean dimension of Z k \mathbb {Z}^k -actions (where k k is a positive integer). On the one hand, we show that there is a Z 2 \mathbb {Z}^2 -action whose directional mean dimension (considered as a [ 0 , + ∞ ] [0,+\infty ] -valued function on the torus) is not continuous. On the other hand, we prove that if a Z k \mathbb {Z}^k -action is continuum-wise expansive, then the values of its ( k − 1 ) (k-1) -dimensional directional mean dimension are bounded. This is a generalization (with a view towards Meyerovitch and Tsukamoto’s theorem on mean dimension and expansive multiparameter actions) of a classical result due to Mañé: Any compact metrizable space admitting an expansive homeomorphism (with respect to a compatible metric) is finite-dimensional.

Radius of comparison and mean cohomological independence dimension

We introduce a notion of mean cohomological independence dimension for actions of discrete amenable groups on compact metrizable spaces, as a variant of mean dimension, and use it to obtain lower bounds for the radius of comparison of the associated crossed product C⁎-algebras. Our general theory, gives the following for the minimal subshifts constructed by Dou in 2017. For any countable amenable group G and any polyhedron Z, Dou's subshift T of ZG with density parameter ρ satisfiesrc(C(X)⋊TG)>12mdim(T)(1−1−ρρ)−2.If k=dim⁡(Z) is even and Hˇk(Z;Q)≠0, thenrc(C(X)⋊TG)>12mdim(T)−1,regardless of what ρ is.

Minimal direct products

A space is called minimal if it admits a minimal continuous selfmap. We give examples of metrizable continua $X$ admitting both minimal homeomorphisms and minimal noninvertible maps, whose squares $X\times X$ are not minimal, i.e., they admit neither minimal homeomorphisms nor minimal noninvertible maps, thus providing a definitive answer to a question posed by Bruin, Kolyada and the second author in 2003. (In 2018, Boronski, Clark and Oprocha provided an answer in the case when only homeomorphisms were considered.) Then we introduce and study the notion of product-minimality. We call a compact metrizable space $Y$ product-minimal if, for every minimal system $(X,T)$ given by a metrizable space $X$ and a continuous selfmap $T$, there is a continuous map $S\colon Y\to Y$ such that the product $(X\times Y,T\times S)$ is minimal. If such a map $S$ always exists in the class of homeomorphisms, we say that $Y$ is a homeo-product-minimal space. We show that many classical examples of minimal spaces, including compact connected metrizable abelian groups, compact connected manifolds without boundary admitting a free action of a nontrivial compact connected Lie group, and many others, are in fact homeo-product-minimal.

Polynomial growth, comparison, and the small boundary property

We show that a minimal action of a finitely generated group of polynomial growth on a compact metrizable space has comparison. It follows that if such an action is free and has the small boundary property then it is almost finite and its C⁎-crossed product is Z-stable, and consequently that such crossed products are classified by their Elliott invariant.