Weakly mixing property and chaos

In this paper, we define and study strong Kato chaos for a group action on a compact metric space. Let X be a compact metric space without isolated points, and let G be a topologically commutative group on X. If the dynamical system (X, G) is weakly mixing, then it is chaotic in the strong sense of Kato.     

On the extension of Hölder maps with values in spaces of continuous functions

We study the isometric extension problem for Hölder maps from subsets of any Banach space intoc ₀ or into a space of continuous functions. For a Banach spaceX, we prove that anyα-Hölder map, with 0<α ≤1, from a subset ofX intoc ₀ can be isometrically extended toX if and only ifX is finite dimensional. For a finite dimensional normed spaceX and for a compact metric spaceK, we prove that the set ofα’s for which allα-Hölder maps from a subset ofX intoC(K) can be extended isometrically is either (0, 1] or (0, 1) and we give examples of both occurrences. We also prove that for any metric spaceX, the above described set ofα’s does not depend onK, but only on finiteness ofK.     

An Analog of Titchmarsh’s Theorem for the Fourier–Walsh Transform

The space clos(X) of all nonempty closed subsets of an unbounded metric space X is considered. The space clos(X) is endowed with a metric in which a sequence of closed sets converges if and only if the distances from these sets to a fixed point θ are bounded and, for any r, the sequence of the unions of the given sets with the exterior balls of radius r centered at θ converges in the Hausdorff metric. The metric on clos(X) thus defined is not equivalent to the Hausdorff metric, whatever the initial metric space X. Conditions for a set to be closed, totally bounded, or compact in clos(X) are obtained; criteria for the bounded compactness and separability of clos(X) are given. The space of continuous maps from a compact space to clos(X) is considered; conditions for a set to be totally bounded in this space are found.     

Extending Lipschitz maps into <Emphasis Type="Italic">C</Emphasis>(<Emphasis Type="Italic">K</Emphasis>)-spaces

We show that if K is a compact metric space then C(K) is a 2-absolute Lipschitz retract. We then study the best Lipschitz extension constants for maps into C(K) from a given metric space M, extending recent results of Lancien and Randrianantoanina. They showed that a finite-dimensional normed space which is polyhedral has the isometric extension property for C(K)-spaces; here we show that the same result holds for spaces with Gateaux smooth norm or of dimension two; a three-dimensional counterexample is also given. We also show that X is polyhedral if and only if every subset E of X has the universal isometric extension property for C(K)-spaces. We also answer a question of Naor on the extension of Hölder continuous maps.     

The number of weakly compact sets which generate a Banach space

We consider the cardinal invariant CG(X) of the minimal number of weakly compact subsets which generate a Banach space X. We study the behavior of this index when passing to subspaces, its relation with the Lindelöf number in the weak topology and other related questions.     

Weakly compact weighted composition operators on spaces of Lipschitz functions

We prove that if X is a compact metric space and \({\text {lip}}(X,d)\) has the uniform separation property, then weakly compact weighted composition operators on spaces of Lipschitz functions \({\text {Lip}}(X,d)\) and \({\text {lip}}(X,d)\) are compact.     

On The Spaces of Linear Operators Acting Between Asymmetric Cone Normed Spaces

An asymmetric norm is a positive sublinear functional p on a real vector space X satisfying \(x=\theta _X\) whenever \(p(x)=p(-x)=0\). Since the space of all lower semi-continuous linear functionals of an asymmetric normed space is not a linear space, the theory is different in the asymmetric case. The main purpose of this study is to define bounded and continuous linear operators acting between asymmetric cone normed spaces. After examining the differences with symmetric case, we give some results related to Baire’s characterization of completeness in asymmetric cone normed spaces.     

On holomorphic immersions into kähler manifolds of constant holomorphic sectional curvature

We study holomorphic immersions f:X →M from a complex manifoldX into a Kähler manifold of constant holomorphic sectional curvatureM, i.e. a complex hyperbolic space form, a complex Euclidean space form, or the complex projective space equipped with the Fubini-Study metric. ForX compact we show that the tangent sequence splits holomorphically if and only iff is a totally geodesic immersion. ForX not necessarily compact we relate an intrinsic cohomological invariantp(X) onX, viz. the invariant defined by Gunning measuring the obstruction to the existence of holomorphic projective connections, to an extrinsic cohomological invariant(f) measuring the obstruction to the holomorphic splitting of the tangent sequence. The two invariantsp(X) and?(f) are related by a linear map on cohomology groups induced by the second fundamental form. In some cases, especially whenX is a complex surface andM is of complex dimension 4, under the assumption thatX admits a holomorphic projective connection we obtain a sufficient condition for the holomorphic splitting of the tangent sequence in terms of the second fundamental form.     

On Symmetric Spaces With Convergence in Measure on Reflexive Subspaces

A closed subspace H of a symmetric space X on [0, 1] is said to be strongly embedded in X if in H the convergence in X-norm is equivalent to the convergence in measure. We study symmetric spaces X with the property that all their reflexive subspaces are strongly embedded in X. We prove that it is the case for all spaces, which satisfy an analogue of the classical Dunford–Pettis theorem on relatively weakly compact subsets in L₁. At the same time the converse assertion fails for a broad class of separableMarcinkiewicz spaces.     

On the Conservativeness of Some Markov Processes

We give a unified method to obtain the conservativeness of a class of Markov processes associated with lower bounded semi-Dirichlet forms on L ²(X;m), including symmetric diffusion processes, some non-symmetric diffusion processes and jump type Markov processes on X, where X is a locally compact separable metric space and m is a positive Radon measure on X with full topological support. Using the method, we give an example in each section, providing the conservativeness of the processes, that are given by the “increasingness of the volume of some sets(balls)” and “that of the coefficients on the sets” of the Markov processes.     

Cardinality of the Ellis semigroup on compact metric countable spaces

Let E(X, f) be the Ellis semigroup of a dynamical system (X, f) where X is a compact metric space. We analyze the cardinality of E(X, f) for a compact countable metric space X. A characterization when E(X, f) and \(E(X,f)^* = E(X,f) \setminus \{ f^n : n \in \mathbb {N}\}\) are both finite is given. We show that if the collection of all periods of the periodic points of (X, f) is infinite, then E(X, f) has size \(2^{\aleph _0}\). It is also proved that if (X, f) has a point with a dense orbit and all elements of E(X, f) are continuous, then \(|E(X,f)| \le |X|\). For dynamical systems of the form \((\omega ^2 +1,f)\), we show that if there is a point with a dense orbit, then all elements of \(E(\omega ^2+1,f)\) are continuous functions. We present several examples of dynamical systems which have a point with a dense orbit. Such systems provide examples where \(E(\omega ^2+1,f)\) and \(\omega ^2+1\) are homeomorphic but not algebraically homeomorphic, where \(\omega ^2+1\) is taken with the usual ordinal addition as semigroup operation.     

The <Emphasis Type="Italic">p</Emphasis>-Gelfand–Phillips property in spaces of operators and Dunford–Pettis like sets

The p-Gelfand–Phillips property (1 \({\leq}\) p < ∞) is studied in spaces of operators. Dunford–Pettis type like sets are studied in Banach spaces. We discuss Banach spaces X with the property that every p-convergent operator T:X \({\rightarrow}\) Y is weakly compact, for every Banach space Y.     

Bernstein–Doetsch-type criteria for the continuity and Lipschitz continuity of convex set-valued mappings

The Bernstein–Doetsch criterion (for convex and midconvex functionals) has been repeatedly generalized to convex and midconvex set-valued mappings F: X → 2 ^Y ; continuity and local Lipschitz continuity were understood in the sense of the Hausdorff distance. However, all such results imposed restrictive additional boundedness-type conditions on the images F(x). In this paper, the Bernstein–Doetsch criterion is generalized to arbitrary convex and midconvex set-valued mappings acting on normed linear spaces X,Y.     

Points with maximal Birkhoff average oscillation

Let f: X → X be a continuous map with the specification property on a compact metric space X. We introduce the notion of the maximal Birkhoff average oscillation, which is the “worst” divergence point for Birkhoff average. By constructing a kind of dynamical Moran subset, we prove that the set of points having maximal Birkhoff average oscillation is residual if it is not empty. As applications, we present the corresponding results for the Birkhoff averages for continuous functions on a repeller and locally maximal hyperbolic set.     

Near convexity,near smoothness and approximative compactness of half spaces in Banach spaces

The authors discuss the dual relation of nearly very convexity and property WS. By two kinds of near convexities and two kinds of near smoothness, the authors prove a series of characterizations such that every half-space in Banach space X and every weak* half-space in the dual space X* are approximatively weakly compact and approximatively compact. They show a sufficient condition such that a Banach space X is a Asplund space. Using upper semi-continuity of duality mapping, the authors also give two characterizations of property WS and property S.     

On operators which attain their norm

The following problem is considered. LetX andY be Banach spaces. Are those operators fromX toY which attain their norm on the unit cell ofX, norm dense in the space of all operators fromX toY? It is proved that this is always the case ifX is reflexive. In general the answer is negative and it depends on some convexity and smoothness properties of the unit cells inX andY. As an application a refinement of the Krein-Milman theorem and Mazur’s theorem concerning the density of smooth points, in the case of weakly compact sets in a separable space, is obtained.     

Uniform continuity in {\omega_\mu}-metric spaces and uc {\omega_\mu}-metric extendability

Generally, the term uc-ness means some continuity is uniform. A metric space X is uc when any continuous function fromX to [0, 1] is uniformly continuous and a metrizable space X is a Nagata space when it can be equipped with a uc metric. We consider natural forms of uc-ness for the \({\omega_\mu}\)-metric spaces, which fill a very large and interesting class of uniform spaces containing the usual metric ones, and extend to them various different formulations of the metric uc-ness, by additionaly proving their equivalence. Furthermore, since any \({\omega_\mu}\)-compact space is uc and any uc \({\omega_\mu}\)-metric space is complete, in the line of constructing dense extensions which preserve some structure, such as uniform completions, we focus on the existence for an \({\omega_\mu}\)-metrizable space of dense topological extensions carrying a uc \({\omega_\mu}\)-metric. In this paper we show that an \({\omega_\mu}\)-metrizable space X is uc-extendable if and only if there exists a compatible \({\omega_\mu}\)-metric d on X such that the set X′ of all accumulation points in X is crowded, i.e., any \({\omega_\mu}\)-sequence in X′ has a d-Cauchy \({\omega_\mu}\)-subsequence in X′.     

A note on transitively <Emphasis Type="Italic">D</Emphasis>-spaces

In this note, we show that if for any transitive neighborhood assignment φ for X there is a point-countable refinement ? such that for any non-closed subset A of X there is some V ∈ ? such that |V ∩ A| ? ω, then X is transitively D. As a corollary, if X is a sequential space and has a point-countable wcs*-network then X is transitively D, and hence if X is a Hausdorff k-space and has a point-countable k-network, then X is transitively D. We prove that if X is a countably compact sequential space and has a pointcountable wcs*-network, then X is compact. We point out that every discretely Lindelöf space is transitively D. Let (X, τ) be a space and let (X, ?) be a butterfly space over (X, τ). If (X, τ) is Fréchet and has a point-countable wcs*-network (or is a hereditarily meta-Lindelöf space), then (X, ?) is a transitively D-space.     

On the singularity of Quillen metrics

Let π: X → S be a holomorphic map from a compact Kähler manifold (X,g _X ) to a compact Riemann surface S. Let Σ_π be the critical locus of π and let Δ  =  π(Σ_π) be the discriminant locus. Let (ξ, h _ξ) be a holomorphic Hermitian vector bundle on X. We determine the singularity of the Quillen metric on det Rπ_*ξ near Δ with respect to g _X | _TX/S and h _ξ.     

On the <Emphasis Type="Italic">σ</Emphasis>-countable compactness of spaces of continuous functions with the set-open topology

For a Tychonoff space X, we obtain a criterion of the σ-countable compactness of the space of continuous functions C(X) with the set-open topology. In particular, for the class of extremally disconnected spaces X, we prove that the space C _λ(X) is σ-countably compact if and only if X is a pseudocompact space, the set X(P) of all P-points of the space X is dense in X, and the family λ consists of finite subsets of the set X(P).