Compact Metric Space Research Articles

Minimal sets of R-closed surface homeomorphisms

Let M be an orientable connected closed surface and f be a homeomorphism on M. Suppose that the set Eˆf:={(x,y)|y∈Of(x)¯} is closed. Then the suspension of f satisfies one of the following conditions: 1) the closure of each element of it is toral; 2) there is a minimal set which is not locally connected. Moreover, we show that if the set Eˆf of a homeomorphism f on a compact metrizable space is closed, then so is Eˆfk for any natural number k.

Minimality, transitivity, mixing and topological entropy on spaces with a free interval

AbstractWe study dynamics of continuous maps on compact metrizable spaces containing a free interval (i.e. an open subset homeomorphic to an open interval). Special attention is paid to relationships between topological transitivity, weak and strong topological mixing, dense periodicity and topological entropy as well as to the topological structure of minimal sets. In particular, a trichotomy for minimal sets and a dichotomy for transitive maps are proved.

Topological pressure and the variational principle for actions of sofic groups

AbstractWe introduce topological pressure for continuous actions of countable sofic groups on compact metrizable spaces. This generalizes the classical topological pressure for continuous actions of countable amenable groups on such spaces. We also establish the variational principle for topological pressure in this sofic context.

Automorphisms on algebras of operator-valued Lipschitz maps

Let Lip(X; B(H)) and lip (X; B(H)) (0 < < 1) be the big and little Banach -algebras of B(H)-valued Lipschitz maps on X, respectively, where X is a compact metric space and B(H) is the C-algebra of all bounded linear operators on a complex in nite-dimensional Hilbert space H. We prove that every linear bijective map that preserves zero products in both directions from Lip(X; B(H)) or lip (X; B(H)) onto it- self is biseparating.We give a Banach{Stone type description for the -automorphisms on such Lipschitz -algebras, and we show that the algebraic reexivity of the - automorphism groups of Lip(X; B(H)) and lip (X; B(H)) holds for H separable.

On Borel mappings and [formula omitted]-ideals generated by closed sets

We obtain some results about Borel maps with meager fibers on Polish spaces. The results are related to a recent dichotomy by Sabok and Zapletal, concerning Borel maps and σ-ideals generated by closed sets. In particular, we give a “classical” proof of this dichotomy.We shall also show that for certain natural σ-ideals I generated by closed sets in compact metrizable spaces X, every Borel map on a Borel set in X not in I, either has a fiber not in I or else it is injective on a Borel set not in I. This is the case for the σ-ideal generated by finite-dimensional closed sets in the Hilbert cube, which provides an answer to a question asked by M. Elekes.

On the classes of remainders of locally compact noncompact spaces

We consider the classes \(\mathcal{R}(X)\) of remainders of locally compact noncompact separable metrizable spaces X in metrizable compactifications. For each integer n≧1 we describe the classes \(\mathcal{R}(X)\) of spaces X additionally having the n-complementation property in the sense of Cain, Jr. Then we show that for each α<ω1 there is a locally compact noncompact separable metrizable space Hα such that \(\{\mathcal{R}(H_{\alpha}): \alpha < \omega_{1}\}\) is a strictly increasing sequence in the family \(\mathcal{R}\) of all classes \(\mathcal{R}(X)\).

The C*-algebra of a vector bundle

We prove that the Cuntz–Pimsner algebra OE of a vector bundle E of rank ≧ 2 over a compact metrizable space X is determined up to an isomorphism of C(X)-algebras by the ideal (1 − [E])K0(X) of the K-theory ring K0(X). Moreover, if E and F are vector bundles of rank ≧ 2, then a unital embedding of C(X)-algebras OE ⊂ OF exists if and only if 1 − [E] is divisible by 1 − [F] in the ring K0(X). We introduce related but more computable K-theory and cohomology invariants for OE and study their completeness. As an application we classify the unital separable continuous fields with fibers isomorphic to the Cuntz algebra Om + 1 over a finite connected CW complex X of dimension d ≦ 2m + 3 provided that the cohomology of X has no m-torsion.

Orders of accumulation of entropy

For a continuous map $T$ of a compact metrizable space $X$ with finite topological entropy, the order of accumulation of entropy of $T$ is a countable ordinal that arises in the context of entropy structures and symbolic extensions. We show that every cou

Independent families and resolvability

Let τ and γ be infinite cardinal numbers with τ⩽γ. A subset Y of a space X is called Cτ-compact if f[Y] is compact for every continuous function f:X→Rτ. We prove that every Cτ-compact dense subspace of a product of γ non-trivial compact spaces each of them of weight ⩽τ is 2τ-resolvable. In particular, every pseudocompact dense subspace of a product of non-trivial metrizable compact spaces is c-resolvable. As a consequence of this fact we obtain that there is no σ-independent maximal independent family. Also, we present a consistent example, relative to the existence of a measurable cardinal, of a dense pseudocompact subspace of {0,1}2λ, with λ=2ω1, which is not maximally resolvable. Moreover, we improve a result by W. Hu (2006) [17] by showing that if maximal θ-independent families do not exist, then every dense subset of □θ{0,1}γ is ω-resolvable for a regular cardinal number θ with ω1⩽θ⩽γ. Finally, if there are no maximal independent families on κ of cardinality γ, then every Baire dense subset of {0,1}γ of cardinality ⩽κ and every Baire dense subset of [0,1]γ of cardinality ⩽κ are ω-resolvable.

Residually finite actions and crossed products

AbstractWe study a notion of residual finiteness for continuous actions of discrete groups on compact Hausdorff spaces and how it relates to the existence of norm microstates for the reduced crossed product. Our main result asserts that an action of a free group on a zero-dimensional compact metrizable space is residually finite if and only if its reduced crossed product admits norm microstates, i.e., is an MF algebra.

Orders of accumulation of entropy on manifolds

For a continuous self-map T of a compact metrizable space with finite topological entropy, the order of accumulation of entropy of T is a countable ordinal that arises in the theory of entropy structure and symbolic extensions. Given any compact manifold M and any countable ordinal α, we construct a continuous, surjective self-map ofM having order of accumulation of entropy α. If the dimension of M is at least 2, then the map can be chosen to be a homeomorphism.

Analytic equivalence relations and bi-embeddability

AbstractLouveau and Rosendal [5] have shown that the relation of bi-embeddability for countable graphs as well as for many other natural classes of countable structures is complete under Borel reducibility for analytic equivalence relations. This is in strong contrast to the case of the isomorphism relation, which as an equivalence relation on graphs (or on any class of countable structures consisting of the models of a sentence of ) is far from complete (see [5, 2]).In this article we strengthen the results of [5] by showing that not only does bi-embeddability give rise to analytic equivalence relations which are complete under Borel reducibility, but in fact any analytic equivalence relation is Borel equivalent to such a relation. This result and the techniques introduced answer questions raised in [5] about the comparison between isomorphism and bi-embeddability. Finally, as in [5] our results apply not only to classes of countable structures defined by sentences of , but also to discrete metric or ultrametric Polish spaces, compact metrizable topological spaces and separable Banach spaces, with various notions of embeddability appropriate for these classes, as well as to actions of Polish monoids.

FIXED POINT THEOREMS FOR GENERALIZED NONEXPANSIVE SET-VALUED MAPPINGS IN CONE METRIC SPACES

Abstract. In 2007, Huang and Zhang [1] introduced a cone metric spacewith a cone metric generalizing the usual metric space by replacing thereal numbers with Banach space ordered by the cone. They consideredsome xed point theorems for contractive mappings in cone metric spaces.Since then, the xed point theory for mappings in cone metric spaces hasbecome a subject of interest in [1-6] and references therein. In this paper,we consider some xed point theorems for generalized nonexpansive set-valued mappings under suitable conditions in sequentially compact conemetric spaces and complete cone metric spaces. 1. Introduction and PreliminariesIn 2007, Huang and Zhang [1] introduced a cone metric space with a conemetric generalizing the usual metric space by replacing the real numbers withBanach space ordered by the cone. They considered some xed point theoremsfor contractive mappings in cone metric spaces. Since then, the xed pointtheory for mappings in cone metric spaces has become a subject of interest in[1-6] and references therein. Especially, for single-valued mappings, Choudhuryand Metiya [6] considered some xed point theorems for weak contraction incone metric spaces in 2010. In 2011, Wardowski [2] introduced H-cone metricin the collection of subsets of a given cone metric space. And he consideredthe concept of set-valued contraction of Nadler-type and proved a xed pointtheorem for contractive set-valued mappings in H-cone metric spaces. Inspiredand encouraged by the previous works, in this paper we consider some xedpoint theorems for generalized nonexpansive set-valued mappings under suit-able conditions in sequentially compact cone metric spaces and complete conemetric spaces.Let Ebe a real Banach space and Pbe a subset of E.Pis called a cone if and only if

Some cohomological aspects of the Banach fixed point principle

Let $T\colon X\to X$ be a continuous selfmap of a compact metrizable space $X$. We prove the equivalence of the following two statements: (1) The mapping $T$ is a Banach contraction relative to some compatible metric on $X$. (2) There is a countable point separating family $\mathcal {F}\subset \mathcal {C}(X)$ of non-negative functions $f\in \mathcal {C}(X)$ such that for every $f\in \mathcal {F}$ there is $g\in \mathcal {C}(X)$ with $f=g-g\circ T$.

Bishop’s theorem and differentiability of a subspace of C b (K)

Let K be a Hausdorff space and Cb(K) be the Banach algebra of all complex bounded continuous functions on K. We study the Gâteaux and Frechet differentiability of subspaces of Cb(K). Using this, we show that the set of all strong peak functions in a nontrivial separating separable subspace H of Cb(K) is a dense Gδ subset of H, if K is compact. This gives a generalized Bishop’s theorem, which says that the closure of the set of all strong peak points for H is the smallest closed norming subset of H. The classical Bishop’s theorem was proved for a separating subalgebra H and a metrizable compact space K.

Unique ergodicity of horospheric foliations revisited

With each rational function on the Riemann sphere, Lyubich–Minsky construction (1997) associates an abstract topological space called the quotient hyperbolic lamination. The latter space carries the so-called vertical geodesic flow with Anosov property. Its unstable foliation is what we call the quotient horospheric lamination. We consider the case of hyperbolic rational function, and more generally, functions postcritically finite on the Julia set without parabolics, that do not belong to the following list of exceptions: powers, Chebyshev polynomials and Latt‘es examples. In this case the quotient horospheric lamination is known to be minimal, while restricted to the union of nonisolated hyperbolic leaves (Glutsyuk, 2007). In the present paper we prove its unique ergodicity. To this end, we introduce the so-called transversely contracting flows and homeomorphisms (on abstract compact metrizable topological spaces), which include the vertical geodesic flows under consideration and the usual Anosov flows and diffeomorphisms. We prove a version of our unique ergodicity result for the transversely contracting flows and homeomorphisms. Particular cases for Anosov flows and diffeomorphisms are given by classical results by Bowen, Marcus, Furstenberg, Margulis, et al. We give a new and purely geometric proof, which seems to be simpler than the classical ones (which use either Markov partitions, K-property, or harmonic analysis).

Representations of Currents Taking Values in a Totally Disconnected Group

Let $G={\mathcal{A}ut(\mathcal{T})}$ be the group of all automorphisms of a homogeneous tree $\mathcal{T}$ of degree q + 1 ≥ 3 and (X, m) a compact metrizable measure space with a probability measure m. We assume that μ has no atoms. The group $\mathcal{G}={\mathcal{A}ut(\mathcal{T})}^X=G^X$ of bounded measurable currents is the completion of the group of step functions $f:X\to{\mathcal{A}ut(\mathcal{T})}$ with respect to a suitable metric. Continuos functions form a dense subgroup of $\mathcal{G}$ . Following the ideas of I.M. Gelfand, M.I. Graev and A.M. Vershik we shall construct an irreducible family of representations of $\mathcal{G}$ . The existence of such representations depends deeply from the nonvanisching of the first cohomology group $H^1({\mathcal{A}ut(\mathcal{T})},\pi)$ for a suitable infinite dimensional π.

Extension theory and finite products of copies of [formula omitted

We study products of the first uncountable ordinal space [ 0 , Ω ) with itself. We show that any product of copies of [ 0 , Ω ) is pseudo-compact and note the classical result that any countable product of copies of [ 0 , Ω ) is normal. Our Main Result yields that if X is a finite product of copies of [ 0 , Ω ) , Z is a compact metrizable space, and K is a CW-complex with K an absolute extensor for Z, then K is an absolute extensor for Y = Z × X . It will also show that K is an absolute extensor for the Stone–Čech compactification, β ( Y ) , of Y.

Bockstein basis and resolution theorems in extension theory

We prove a generalization of the Edwards–Walsh Resolution Theorem: Theorem Let G be an abelian group with P G = P , where P G = { p ∈ P : Z ( p ) ∈ Bockstein basis σ ( G ) } . Let n ∈ N and let K be a connected CW -complex with π n ( K ) ≅ G , π k ( K ) ≅ 0 for 0 ⩽ k < n . Then for every compact metrizable space X with XτK ( i.e., with K an absolute extensor for X) , there exists a compact metrizable space Z and a surjective map π : Z → X such that (a) π is cell-like, (b) dim Z ⩽ n , and (c) ZτK.

On metric spaces with the properties of de Groot and Nagata in dimension one

A metric space ( X , d ) has the de Groot property GP n if for any points x 0 , x 1 , … , x n + 2 ∈ X there are positive indices i , j , k ⩽ n + 2 such that i ≠ j and d ( x i , x j ) ⩽ d ( x 0 , x k ) . If, in addition, k ∈ { i , j } then X is said to have the Nagata property NP n . It is known that a compact metrizable space X has dimension dim ( X ) ⩽ n iff X has an admissible GP n -metric iff X has an admissible NP n -metric. We prove that an embedding f : ( 0 , 1 ) → X of the interval ( 0 , 1 ) ⊂ R into a locally connected metric space X with property GP 1 (resp. NP 1 ) is open, provided f is an isometric embedding (resp. f has distortion Dist ( f ) = ‖ f ‖ Lip ⋅ ‖ f − 1 ‖ Lip < 2 ). This implies that the Euclidean metric cannot be extended from the interval [ − 1 , 1 ] to an admissible GP 1 -metric on the triode T = [ − 1 , 1 ] ∪ [ 0 , i ] . Another corollary says that a topologically homogeneous GP 1 -space cannot contain an isometric copy of the interval ( 0 , 1 ) and a topological copy of the triode T simultaneously. Also we prove that a GP 1 -metric space X containing an isometric copy of each compact NP 1 -metric space has density ⩾ c .