Dense Gδ Subset Research Articles

A particular minimization property implies [formula omitted]-integrability

The manifold M being compact and connected and H being a Tonelli Hamiltonian such that T⁎M is equal to the dual tiered Mañé set, we prove that there is a partition of T⁎M into invariant C0-Lagrangian graphs. Moreover, among these graphs, those that are C1 cover a dense Gδ-subset of T⁎M. The dynamic restricted to each of these sets is non-wandering.

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.

Generic measures for hyperbolic flows on non-compact spaces

We consider the geodesic flow on a complete connected negatively curved manifold. We show that the set of invariant borel probability measures contains a dense G δ -subset consisting of ergodic measures fully supported on the non-wandering set. We also treat the case of non-positively curved manifolds and provide general tools to deal with hyperbolic systems defined on non-compact spaces.

On Finite-to-One Maps

AbstractLet f : X → Y be a σ-perfect k-dimensional surjective map of metrizable spaces such that dim Y ≤ m. It is shown that for every positive integer p with p ≤ m + k + 1 there exists a dense Gδ-subset (k,m, p) of C(X, k+p) with the source limitation topology such that each fiber of f△g, g ∈ (k, m, p), contains at most max ﹛k + m – p + 2, 1﹜ points. This result provides a proof the following conjectures of S. Bogatyi, V. Fedorchuk and J. van Mill. Let f : X → Y be a k-dimensional map between compact metric spaces with dimY ≤ m. Then: (1) there exists a map h: X → m+2k such that f△h: X → Y × m+2k is 2-to-one provided k ≥ 1; (2) there exists a map h: X → m+k+1 such that f△h: X →Y × m+k+1 is (k + 1)-to-one.

On Generic Well–posedness of Restricted Chebyshev Center Problems in Banach Spaces

Let ℬ (resp. \({\fancyscript K}\) , ℬ\({\fancyscript C}\), \({\fancyscript K}\)\({\fancyscript C}\)) denote the set of all nonempty bounded (resp. compact, bounded convex, compact convex) closed subsets of the Banach space X, endowed with the Hausdorff metric, and let G be a nonempty relatively weakly compact closed subset of X. Let ℬostand for the set of all F ∈ ℬ such that the problem (F,G) is well–posed. We proved that, if X is strictly convex and Kadec, the set \({\fancyscript K}\)\({\fancyscript C}\) ∩ℬo is a dense Gδ–subset of \({\fancyscript K}\)\({\fancyscript C}\) \ G. Furthermore, if X is a uniformly convex Banach space, we will prove more, namely that the set ℬ\ℬo (resp.\({\fancyscript K}\) \ℬo, ℬ\({\fancyscript C}\) \ℬo, \({\fancyscript K}\)\({\fancyscript C}\) \ℬo) is σ–porous in ℬ (resp. \({\fancyscript K}\) , ℬ\({\fancyscript C}\), \({\fancyscript K}\)\({\fancyscript C}\)). Moreover, we prove that for most (in the sense of the Baire category) closed bounded subsets G of X, the set \({\fancyscript K}\) \ℬo is dense and uncountable in \({\fancyscript K}\).

Mixing and proximal cells along sequences

A dynamical system (X, T) is -transitive if for each pair of open and non-empty subsets U and V of X, , where is a collection of subsets of that is hereditary upward. (X, T) is -mixing if (X × X, T × T) is -transitive. For a subset S of , (x, y) ϵ X × X is S-proximal if and the S-proximal cell PS(x) is the set of points that are S-proximal to x ϵ X. We show that if (X, T) is -mixing, then for each (the dual family of ) and x ϵ X, PS(x) is a dense Gδ subset of X, and when (X, T) is minimal and is a filter the reciprocal is true. Moreover, other conditions under which the reciprocal is true are obtained. Finally the structure of proximal cells for -mixing systems is discussed, and a new and simpler proof of the Xiong–Yang theorem is presented.

Some generic results on nonattaining functionals

We prove that (a) in a reflexive space, for any linearly bounded but unbounded closed convex subset the nonsupport functionals are a dense Gδ subset of the polar set, and (b) any nonsemicoercive proper convex lsc [weak*-lsc] function in a [dual] Banach space has a generic [dense Gδ] set of L∞-perturbations which do not attain their infimum. We also characterize the proper convex functions that have inf-nonattaining L∞-perturbations. This results also in a criterion for reflexivity.

Mazur Intersection Properties and Differentiability of Convex Functions in Banach Spaces

It is proved that the dual of a Banach space with the Mazur intersection property is almost weak* Asplund. Analogously, the predual of a dual space with the weak* Mazur intersection property is almost Asplund. Through the use of these arguments, it is found that, in particular, almost all (in the Baire sense) equivalent norms on [script l]1(Γ) and [script l][infty infinity](Γ) are Frechet differentiable on a dense Gδ subset. Necessary conditions for Mazur intersection properties in terms of convex sets satisfying a Krein–Milman type condition are also discussed. It is also shown that, if a Banach space has the Mazur intersection property, then every subspace of countable codimension can be equivalently renormed to satisfy this property.

On Mutually Nearest and Mutually Furthest Points in Reflexive Banach Spaces

Let G be a nonempty closed (resp. bounded closed) subset in a reflexive strictly convex Kadec Banach space X. Let K(X) denote the space of all nonempty compact convex subsets of X endowed with the Hausdorff distance. Moreover, let KG(X) denote the closure of the set {A∈K(X):A∩G=∅}. A minimization problem min(A, G) (resp. maximization problem max(A, G)) is said to be well posed if it has a unique solution (x0, z0) and every minimizing (resp. maximizing) sequence converges strongly to (x0, z0). We prove that the set of all A∈KG(X) (resp. A∈K(X)) such that the minimization (resp. maximization) problem min(A, G) (resp. max(A, G)) is well posed contains a dense Gδ-subset of KG(X) (resp. K(X)), extending the results in uniformly convex Banach spaces due to Blasi, Myjak and Papini.

Generic Gateaux differentiability via smooth perturbations

We prove that in a Banach space with a Lipschitz uniformly Gateaux smooth bump function, every continuous function which is directionally differentiable on a dense Gδ subset of the space, is Gateaux differentiable on a dense Gδ subset of the space. Applications of this result are given.

Generic continuity of minimal set-valued mappings

AbstractWe study classes of Banach spaces where every set-valued mapping from a complete metric space into subsets of the Banach space which satisfies certain minimal properties, is single-valued and norm upper semi-continuous at the points of a dense Gδ subset of its domain. Characterisations of these classes are developed and permanence properties are established. Sufficiency conditions for membership of these classes are defined in terms of fragmentability and σ-fragmentability of the weak topology. A characterisation of non membership is used to show that l∞ (N) is not a member of our classe of generic continuity spaces.

The Limit Points of Weyl Sums and other Continuous Cocycles

This paper examines the limit points of continuous Z-cocycles defined on a minimal dynamical system and taking values in an abelian metrisable group. This is motivated by and applied to the study of classical exponential series Σ n = 1 N e 2 π i ( n 2 θ + n x ) to show, in particular, that if θ is a transcendental number with lim infqq3/2‖qθ‖ < ∞, then for each x chosen from a dense Gδ subset of [0, 1], these partial sums are dense in C. Indeed the associated cocycle is topologically transitive. Several generalisations to higher degree polynomials and higher dimensions are presented.

Almost Periodic Ergodic Rn-Homeomorphisms

A homeomorphism of Rn is called stationary if it is the uniform limit of volume preserving homeomorphisms which are spatially periodic and have mean translation zero. We prove that ergodic homeomorphisms form a uniform topology dense Gδ subset of the stationary homeomorphisms, thus establishing examples which are uniformly continuous with bounded and almost periodic displacements. In the two-dimensional orientation preserving case, ergodic homeomorphisms have fixed points (by Brouwer′s Plane Translation Theorem). Hence so do orientation-preserving area-preserving torus homeomorphisms with mean translation zero lifts (Conley-Zehnder-Frank Theorem).

A generic factorization theorem

Let F:Z→X be a minimal usco map from the Baire space Z into the compact space X. Then a complete metric space P and a minimal usco G:P→X can be constructed so that for every dense Gδ-subset P1 of P there exist a dense Gδ Z1 of Z and a (single-valued) continuous map f: Z1→P1 such that F(Z)⊂G(f(z)) for every z∈Z1. In particular, if G is single valued on a dense Gδ-subset of P, then F is also single-valued on a dense Gδ-subset of its domain. The above theorem remains valid if Z is Čech complete space and X is an arbitrary completely regular space. These factorization theorems show that some generalizations of a theorem of Namioka concerning generic single-valuedness and generic continuity of mappings defined in more general spaces can be derived from similar results for mappings with complete metric domains. The theorems can be used also as a tool to establish that certain topological spaces contain dense completely metrizable subspaces.

Typical Transitivity for Lifts of Rotationless Annulus or Torus Homeomorphisms

We say that a homeomorphism h of the base space X (which may be either the annulus or n-torus, n⩾2) is rotationless if it is area-preserving and has a lift h∼ to the covering space X∼([0,1] × R or Rn) with mean translation zero (∫Ω(h∼(x)–x)dx=0, where Ω is [0,1] × [0,1]). We prove (Theorem 1) that in the space of rotationless homeomorphisms of X with the uniform topology, the subspace consisting of homeo-morphisms with transitive lifts to X ∼ contains a dense Gδ subset. This extends our earlier result, valid only when the base space is the annulus, that typical rotationless homeomorphisms have recurrent lifts. Our result also extends that of Besicovitch, who in 1937 exhibited the first transitive homeomorphism of the plane. In this context we establish such a homeomorphism which is additionally spatially periodic.

A continuity property related to an index of non-separability and its applications

For a set E in a metric space X the index of non-separability is β(E) = inf{r &gt; 0: E is covered by a countable-family of balls of radius less than r}.Now, for a set-valued mapping Φ from a topological space A into subsets of a metric space X we say that Φ is β upper semi-continuous at t ∈ A if given ε &gt; 0 there exists a neighbourhood U of t such that β(Φ(U)) &lt; ε. In this paper we show that if the subdifferential mapping of a continuous convex function Φ is β upper semi-continuous on a dense subset of its domain then Φ is Fréchet differentiable on a dense Gδ subset of its domain. We also show that a Banach space is Asplund if and only if every weak* compact subset has weak* slices whose index of non-separability is arbitrarily small.

The smooth variational principle and generic differentiability

A modified version of the smooth variational principle of Borwein and Preiss is proved. By its help it is shown that in a Banach space with uniformly Gâteaux differentiable norm every continuous function, which is directionally differentiable on a dense Gδ subset of the space, is Gâteaux differentiable on a dense Gδ subset of the space.

Smooth Banach spaces, weak asplund spaces and monotone or usco mappings

It is shown that if a real Banach spaceE admits an equivalent Gateaux differentiable norm, then for every continuous convex functionf onE there exists a denseGδ subset ofE at every point of whichf is Gateaux differentiable. More generally, for any maximal monotone operatorT on such a space, there exists a denseGδ subset (in the interior of its essential domain) at every point of whichT is single-valued. The same techniques yield results about stronger forms of differentiability and about generically continuous selections for certain upper-semicontinuous compact-set-valued maps.

Topological spaces with dense subspaces that are homeomorphic to complete metric spaces and the classification of C(K) Banach spaces

In [S1] we introduced and in [S2, S3, S4] developed a class of topological spaces that is useful in the study of the classification of Banach spaces and Gateaux differentiation of functions defined in Banach spaces. The class C may be most succinctly defined in the following way: a Hausdorff space T is in C if any upper semicontinuous compact valued map (usco) that is minimal and defined on a Baire space B with values in T must be point valued on a dense Gδ subset of B. This definition conceals many interesting properties of the family C. See [S2] for a discussion of the various definitions. Our main result here is that if X is a Banach space such that the dual space X* in the weak* topology is in C and K is any weak* compact subset of X* then the extreme points of K contain a dense, necessarily Gδ, subset homeomorphic to a complete metric space. In [S4] we studied the class K of κ-analytic spaces in C. Here we shall show that many elements of K contain dense subsets homeomorphic to complete metric spaces. It is easy to see that C contains all metric spaces and it is proved in [S4] that analytic spaces are in K. We obtain a number of topological results that may be of independent interest. We close with a discussion of various examples that show the interaction of these ideas between functional analysis and topology

Generic Fr�chet differentiability of the pressure in certain lattice systems

This note has two goals. The first is to give an explicit description (Theorem 1) of the duals of certain weighted products ℛ h of a countable family of Banach spaces. These products include the usual spaces of interactions which arise in statistical mechanics. The second goal is to use this description to prove that if the factor spaces are finite dimensional and the weight functionh satisfies a certain growth condition, then the pressure is Frechet differentiable wherever it is Gateaux differentiable (hence is Frechet differentiable in a denseG δ subset).