On shadowing property for inverse limit spaces

We study here the G-shadowing property of the shift map σ on the inverse limit space X _f, generated by an equivariant self-map f on a metric G-space X.      

Extension of locally pseudocompact group actions

It is shown that: (1) any action of a Moscow group G on a first countable, Dieudonné complete (in particular, on a metrizable) space X can uniquely be extended to an action of the Dieudonné completion γG on X, (2) any action of a locally pseudocompact topological group G on a b _f -space (in particular, on a first countable space) X can uniquely be extended to an action of the Weil completion on the Dieudonné completion γX of X. As a consequence, we obtain that, for each locally pseudocompact topological group G, every G-space with the b _f -property admits an equivariant embedding into a compact Hausdorff G-space. Furthermore, for each pseudocompact group G, every metrizable G-space has a G-invariant metric compatible with its topology. We also give a direct construction of such an invariant metric. Received: June 22, 2000; in final form: May 22, 2001?Published online: June 11, 2002     

The roots of <Emphasis Type="Italic">σ</Emphasis>-polynomials

Let G be a connected graph. We denote by σ(G,x) and δ(G) respectively the σ-polynomial and the edge-density of G, where . If σ(G,x) has at least an unreal root, then G is said to be a σ-unreal graph. Let δ(n) be the minimum edgedensity over all n vertices graphs with σ-unreal roots. In this paper, by using the theory of adjoint polynomials, a negative answer to a problem posed by Brenti et al. is given and the following results are obtained: For any positive integer a and rational number 0≤c≤1, there exists at least a graph sequence {G _i}_1≤i≤a such that G _i is σ-unreal and δ(G _i)→c as n→∞ for all 1 ≤i≤a, and moreover, δ(n)→0 as n→∞. Supported by the National Natural Science Foundation of China (10061003) and the Science Foundation of the State Education Ministry of China.     

Dichotomies of the set of test measures of a Haar-null set

We prove that ifX is a Polish space andF a face ofP(X) with the Baire property, thenF is either a meager or a co-meager subset ofP(X). As a consequence we show that for every abelian Polish groupX and every analytic Haar-null set Λ⊆X, the set of test measuresT(Λ) of Λ is either meager or co-meager. We characterize the non-locally-compact groups as the ones for which there exists a closed Haar-null setF⊆X withT(F) meager, Moreover, we answer negatively a question of J. Mycielski by showing that for every non-locally-compact abelian Polish group and every σ-compact subgroupG ofX there exists aG-invariantF _σ subset ofX which is neither prevalent nor Haar-null. Research supported by a grant of EPEAEK program “Pythagoras”.     

Invariant functions in Denjoy–Carleman classes

Let V be a real finite dimensional representation of a compact Lie group G. It is well known that the algebra of G-invariant polynomials on V is finitely generated, say by σ ₁, . . . , σ _p . Schwarz (Topology 14:63–68, 1975) proved that each G-invariant C ^∞-function f on V has the form f = F(σ ₁, . . . , σ _p ) for a C ^∞-function F on . We investigate this representation within the framework of Denjoy–Carleman classes. One can in general not expect that f and F lie in the same Denjoy–Carleman class C ^M (with M = (M _k )). For finite groups G and (more generally) for polar representations V, we show that for each G-invariant f of class C ^M there is an F of class C ^N such that f = F(σ ₁, . . . , σ _p ), if N is strongly regular and satisfies

Banach spaces with a weak cotype 2 property

We study the Banach spacesX with the following property: there is a numberδ in ]0,1[ such that for some constantC, any finite dimensional subspaceE ⊂X contains a subspaceF ⊂E with dimF≧δ dimE which isC-isomorphic to a Euclidean space. We show that if this holds for someδ in ]0,1[ then it also holds for allδ in ]0,1[ and we estimate the functionC=C(δ). We show that this property holds iff the “volume ratio” of the finite dimensional subspaces ofX are uniformly bounded. We also show that (althoughX can have this property without being of cotype 2)L ₂(X) possesses this property iffX if of cotype 2. In the last part of the paper, we study theK-convex spaces which have a dual with the above property and we relate it to a certain extension property.     

Spectra of ergodic group actions

LetG be a locally compact second countable abelian group, (X, μ) aσ-finite Lebesgue space, and (g, x) →gx a non-singular, properly ergodic action ofG on (X, μ). Let furthermore Γ be the character group ofG and let Sp(G, X) ⊂ Γ denote theL ^∞-spectrum ofG on (X, μ). It has been shown in [5] that Sp(G, X) is a Borel subgroup of Γ and thatσ (Sp(G, X))<1 for every probability measureσ on Γ with lim sup_g→∞Re (g)<1, where is the Fourier transform ofσ. In this note we prove the following converse: ifσ is a probability measure on Γ with lim sup_g→∞Re (g)<1 (g)=1 then there exists a non-singular, properly ergodic action ofG on (X, μ) withσ(Sp(G, X))=1.     

Characterizations and Extensions of Lipschitz-α Operators

In this work, we prove that a map F from a compact metric space K into a Banach space X over F is a Lipschitz-α operator if and only if for each σ in X^＊ the map σoF is a Lipschitz-α function on K. In the case that K = [a, b], we show that a map f from [a, b] into X is a Lipschitz-1 operator if and only if it is absolutely continuous and the map σ→ （σ o f）＇ is a bounded linear operator from X^＊ into L^∞（[a, b]）. When K is a compact subset of a finite interval （a, b） and 0 〈 α ≤ 1, we show that every Lipschitz-α operator f from K into X can be extended as a Lipschitz-α operator F from [a, b] into X with Lα（f） ≤ Lα（F） ≤ 3^1-α Lα（f）. A similar extension theorem for a little Lipschitz-α operator is also obtained.     

Chaos caused by a strong-mixing measure-preserving transformation

The chaos caused by a strong-mixing preserving transformation is discussed and it is shown that for a topological spaceX satisfying the second axiom of countability and for an outer measurem onX satisfying the conditions: (i) every non-empty open set ofX ism-measurable with positivem-measure; (ii) the restriction ofm on Borel σ-algebra ℬ(X) ofX is a probability measure, and (iii) for everyY⊂X there exists a Borel setB⊂ℬ(X) such thatB⊃Y andm(B) =m(Y), iff:X→X is a strong-mixing measure-preserving transformation of the probability space (X, ℬ(X),m), and if {m}, is a strictly increasing sequence of positive integers, then there exists a subsetC⊂X withm (C) = 1, finitely chaotic with respect to the sequence {m _i}, i.e. for any finite subsetA ofC and for any mapF:A→X there is a subsequencer _i such that lim_i→∞ f ^r i(a) =F(a) for anya ∈A. There are some applications to maps of one dimension. the National Natural Science Foundation of China.     

Harmonic Functions on Homogeneous Spaces

 Given a locally compact group G acting on a locally compact space X and a probability measure σ on G, a real Borel function f on X is called σ-harmonic if it satisfies the convolution equation . We give conditions for the absence of nonconstant bounded harmonic functions. We show that, if G is a union of σ-admissible neighbourhoods of the identity, relative to X, then every bounded σ-harmonic function on X is constant. Consequently, for spread out σ, the bounded σ-harmonic functions are constant on each connected component of a [SIN]-group and, if G acts strictly transitively on a splittable metric space X, then the bounded σ-harmonic functions on X are constant which extends Furstenberg’s result for connected semisimple Lie groups. (Received 13 June 1998; in revised form 31 March 1999)     

Gromov hyperbolic cubic graphs

If X is a geodesic metric space and x ₁; x ₂; x ₃ ∈ X, a geodesic triangle T = {x ₁; x ₂; x ₃} is the union of the three geodesics [x ₁ x ₂], [x ₂ x ₃] and [x ₃ x ₁] in X. The space X is δ-hyperbolic (in the Gromov sense) if any side of T is contained in a δ-neighborhood of the union of the two other sides, for every geodesic triangle T in X. We denote by δ(X) the sharp hyperbolicity constant of X, i.e., δ(X) = inf {δ ≥ 0: X is δ-hyperbolic}. We obtain information about the hyperbolicity constant of cubic graphs (graphs with all of their vertices of degree 3), and prove that for any graph G with bounded degree there exists a cubic graph G* such that G is hyperbolic if and only if G* is hyperbolic. Moreover, we prove that for any cubic graph G with n vertices, we have δ(G) ≤ min {3n/16 + 1; n/4}. We characterize the cubic graphs G with δ(G) ≤ 1. Besides, we prove some inequalities involving the hyperbolicity constant and other parameters for cubic graphs.     

Harmonic Functions on Homogeneous Spaces

 Given a locally compact group G acting on a locally compact space X and a probability measure σ on G, a real Borel function f on X is called σ-harmonic if it satisfies the convolution equation . We give conditions for the absence of nonconstant bounded harmonic functions. We show that, if G is a union of σ-admissible neighbourhoods of the identity, relative to X, then every bounded σ-harmonic function on X is constant. Consequently, for spread out σ, the bounded σ-harmonic functions are constant on each connected component of a [SIN]-group and, if G acts strictly transitively on a splittable metric space X, then the bounded σ-harmonic functions on X are constant which extends Furstenberg’s result for connected semisimple Lie groups.     

Boundaries and generation of convex sets

We introduce a notion which is intermediate between that of taking thew*-closed convex hull of a set and taking the norm closed convex hull of this set. This notion helps to streamline the proof (given in [FLP]) of the famous result of James in the separable case. More importantly, it leads to stronger results in the same direction. For example:

Hyperspaces of nowhere topologically complete spaces

It is proved that ifX is a connected locally continuumwise connected coanalytic nowhere topologically complete space, then the hyperspace 2 ^X of all nonempty compact subsets ofX is strongly universal in the class of all coanalytic spaces. Moreover, 2 ^X is homeomorphic to Π₂ ifX is a Baire space, and toQ∖Π₁ ifX contains a dense absoluteG _δ-setG ⊂X such that the intersectionG ∩U is connected for any open connectedU ⊂X. (Here Π₁, Π₁⊂X are the standard subsets of the Hilbert cubeQ absorbing for the classes of analytic and coanalytic spaces, respectively.) Similar results are obtained for higher projective classes. Translated fromMatematicheskie Zametki, Vol. 62, No. 1, pp. 35–51, July, 1997. Translated by O. V. Sipacheva     

Teichmüller sequences on trajectories invariant under a Kleinian group

We extend the results of [T2] to the situation where there is a compatibility with the action of a Kleinian group. A classical Techmüller sequence is a sequence of quasiconformal mapsf _i with complex dilatations of the form , where ϕ is a quadratic differential and 0<-k _i<1 are numbers such thatk _i→1 asi→∞. We proved in [T2] that if τ is a vertical trajectory associated to ϕ, then there is often, for instance if the sequence is normalized so thatf _i fix 3 points, a subsequence such thatf _i tend either toward a constant or an injective map of τ. If there is compatibility with the action of a non-elementary finitely generated Kleinian groupG, we can given a precise characterization which of these cases occurs. Suppose thatf _i induce isomorphisms ϕ_i ofG onto another Kleinian group and that ϕ_i have algebraic limit ϕ. If the quadratic differential is defined on a component of the ordinary set ofG, if there are no parabolic elements, and if τ is extended maximally so that all branches coming together at a singular point are included, then we can state the main result as follows. The limit is a constantc if the stabilizerG _τ of τ is elementary; and, if it is non-elementary, then the limit is injective. In the first case, ϕ(g) is parabolic with fixpointc wheneverg∈G _τ is of infinite order; and in the latter case, the limitf is an embedding of τ in a natural topology of τ, andf embeds τ into a component of the limit set of ϕG whose stabilizer is ϕG _τ. Various extensions and generalizations are presented. The research for this paper has been supported by the project 51749 of the Academy of Finland.     

On thin-complete ideals of subsets of groups

Let F ì P_G \mathcal{F} \subset {\mathcal{P}_G} be a left-invariant lower family of subsets of a group G. A subset A ⊂ G is called F \mathcal{F} -thin if xA ?yA ? F xA \cap yA \in \mathcal{F} for any distinct elements x, y ∈ G. The family of all F \mathcal{F} -thin subsets of G is denoted by t( F ) \tau \left( \mathcal{F} \right) . If t( F ) = F \tau \left( \mathcal{F} \right) = \mathcal{F} , then F \mathcal{F} is called thin-complete. The thin-completion t^*( F ) {\tau^*}\left( \mathcal{F} \right) of F \mathcal{F} is the smallest thin-complete subfamily of P_G {\mathcal{P}_G} that contains F \mathcal{F} . Answering questions of Lutsenko and Protasov, we prove that a set A ⊂ G belongs to τ*(G) if and only if, for any sequence (g _n ) _n∈ω of nonzero elements of G, there is n ∈ ω such that

Subcontinuity

We give interesting characterizations using subcontinuity. Let X, Y be topological spaces. We study subcontinuity of multifunctions from X to Y and its relations to local compactness, local total boundedness and upper semicontinuity. If Y is regular, then F is subcontinuous iff [`(F)]\bar F is USCO. A uniform space Y is complete iff for every topological space X and for every net {F _a }, F _a ⊂ X × Y, of multifunctions subcontinuous at x ∈ X, uniformly convergent to F, F is subcontinuous at x. A Tychonoff space Y is Čech-complete (resp. G _m-space) iff for every topological space X and every multifunction F ⊂ X × Y the set of points of subcontinuity of F is a G _δ -subset (resp. G _m-subset) of X.     

A note on reflexivity in Banach spaces

For any normed spaceX, the unit ball ofX is weak *-dense in the unit ball ofX ^**. This says that for any ε>0, for anyF in the unit ball ofX ^**, and for anyf ₁,…,f _n inX ^*, the system of inequalities |f _i(x)?F(f _i)|≤ε can be solved for somex in the unit ball ofX. The author shows that the requirement that ε be strictly positive can be dropped only ifX is reflexive.     

Sugli insiemi di Baire e la compattificazione

Sunto Sia ê un insieme compatto e E un sottoinsieme denso in E. Si tratta di caratterizzare E come insieme diBaire in ê. La caratterizzazione è data per tre casi: E è aperto; E è un F_σ; E è un G_δ, G_n, dove ogni G_n aperto è un F_σ. A Enrico Bompiani in occasione del suo Giubileo scientifico. Questo lavoro è stato preparato con la sovvenzione della National Science Foundation, Grant 14357.