Generic Existence of Solutions of Symmetric Optimization Problems

In this paper we study a class of symmetric optimization problems which is identified with a space of objective functions, equipped with an appropriate complete metric. Using the Baire category approach, we show the existence of a subset of the space of functions, which is a countable intersection of open and everywhere dense sets, such that for every objective function from this intersection the corresponding symmetric optimization problem possesses a solution.

The set of badly approximable vectors is strongly C1 incompressible

We prove that the countable intersection of C1-diffeomorphic images of certain Diophantine sets has full Hausdorff dimension. For example, we show this for the set of badly approximable vectors in ℝd, improving earlier results of Schmidt and Dani. To prove this, inspired by ideas of McMullen, we define a new variant of Schmidt's (α,β)-game and show that our sets are hyperplane absolute winning (HAW), which in particular implies winning in the original game. The HAW property passes automatically to games played on certain fractals, thus our sets intersect a large class of fractals in a set of positive dimension. This extends earlier results of Fishman to a more general set-up, with simpler proofs.

Transitivity of Euclidean-type extensions of hyperbolic systems

Letf:X→Xbe the restriction to a hyperbolic basic set of a smooth diffeomorphism. We show that in the class ofCr(r>0) cocycles with fiber the special Euclidean group SE(n), those that are transitive form a residual set (countable intersection of open dense sets). This result is new for odd values ofn≥3. More generally, we consider Euclidean-type groupsG⋉ℝnwhereGis a compact connected Lie group acting linearly on ℝn. When Fix G={0}, it is again the case that the transitive cocycles are residual. When Fix G≠{0}, the same result holds upon restriction to the subset of cocycles that avoid an obvious and explicit obstruction to transitivity.

Ubiquity and large intersections properties under digit frequencies constraints

We are interested in two properties of real numbers: the first one is the property of being well-approximated by some dense family of real numbers {xn}n≥1, such as rational numbers and more generally algebraic numbers, and the second one is the property of having given digit frequencies in some b-adic expansion.We combine these two ways of classifying the real numbers, in order to provide a finer classification. We exhibit sets S of points x which are approximated at a given rate by some of the {xn}n, those xn being selected according to their digit frequencies. We compute the Hausdorff dimension of any countable intersection of such sets S, and prove that these sets enjoy the so-called large intersection property.

Searching for Absolute ᘓℛ–Epic Spaces

In previous papers, Barr and Raphael investigated the situation of a topological space Y and a subspace X such that the induced map C(Y ) → C(X) is an epimorphism in the category ᘓℛ of commutative rings (with units). We call such an embedding a ᘓℛ-epic embedding and we say that X is absolute ᘓℛ-epic if every embedding of X is ᘓℛ-epic. We continue this investigation. Our most notable result shows that a Lindelöf space X is absolute ᘓℛ-epic if a countable intersection of βX-neighbourhoods of X is a βX-neighbourhood of X. This condition is stable under countable sums, the formation of closed subspaces, cozero-subspaces, and being the domain or codomain of a perfect map. A strengthening of the Lindelöf property leads to a new class with the same closure properties that is also closed under finite products. Moreover, all σ-compact spaces and all Lindelöf P-spaces satisfy this stronger condition. We get some results in the non-Lindelöf case that are sufficient to show that the Dieudonné plank and some closely related spaces are absolute ᘓℛ-epic.

A porosity result in convex minimization

We study the minimization problemf(x)→min,x∈C, wherefbelongs to a complete metric spaceℳof convex functions and the setCis a countable intersection of a decreasing sequence of closed convex setsCiin a reflexive Banach space. Letℱbe the set of allf∈ℳfor which the solutions of the minimization problem over the setCiconverge strongly asi→∞to the solution over the setC. In our recent work we show that the setℱcontains an everywhere denseGδsubset ofℳ. In this paper, we show that the complementℳ\ℱis not only of the first Baire category but also aσ-porous set.

Generic uniqueness of minimal configurations with rational rotation numbers in Aubry-Mather theory

We study(h)-minimal configurations in Aubry-Mather theory, wherehbelongs to a complete metric space of functions. Such minimal configurations have definite rotation number. We establish the existence of a set of functions, which is a countable intersection of open everywhere dense subsets of the space and such that for each elementhof this set and each rational numberα, the following properties hold: (i) there exist three different(h)-minimal configurations with rotation numberα; (ii) any(h)-minimal configuration with rotation numberαis a translation of one of these configurations.

Monotone but not positive subsets of the Cantor space

A subset of the Cantor space ω2 is called monotone iff it is closed upward under the partial ordering ≤ defined by x ≤ y iff x(n) ≤ y(n) for all n ∈ ω. A set is -positive (-positive) iff it is monotone and -positive set is a countable union of -positive sets; a -positive set is a countable intersection of -positive sets. (See Cenzer [2] for background information on these concepts.) It is clear that any -positive set is and monotone; the converse holds for n ≤ 2 [2] and was conjectured by Dyck to hold for greater n. In this note, we will disprove this conjecture by giving examples of monotone sets (for n ≥ 3) which are not even -positive.First we note a few isomorphisms. The space (ω2, ≤) is isomorphic to the space (ω2 ≥), so instead of monotone and positive sets we may construct hereditary and negative sets (the analogous notions with “closed upward” replaced by “closed downward”). Also, (ω2, ≤) is isomorphic to ((ω), ⊆), where denotes the power set operator, or to ((S), ⊆) for any countably infinite set S.In order to remove extraneous notation from the proofs, we state the results in an abstract form (whose generality is deceptive).

Countable-codimensional subspaces of locally convex spaces

A barrel in a locally convex Hausdorff space E[τ] is a closed absolutely convex absorbent set. A σ-barrel is a barrel which is expressible as a countable intersection of closed absolutely convex neighbourhoods. A space is said to be barrelled (countably barrelled) if every barrel (σ-barrel) is a neighbourhood, and quasi-barrelled (countably quasi-barrelled) if every bornivorous barrel (σ-barrel) is a neighbourhood. The study of countably barrelled and countably quasi-barrelled spaces was initiated by Husain (2).

Minimal first countable spaces

A topological space is E0 (resp. E1) provided every point is the countable intersection of neighborhoods (resp. closed neighborhoods). For i = 0 and i = 1, characterizations of minimal Ei. spaces (Ei. spaces with no strictly coarser Ei. topology) and Ei-closed spaces (Ei. spaces which are closed in every Ei. space containing them) are given; for example, the properties of minimal Ei. and minimal first countable Ti+1 are shown to be equivalent. Minimal E0 spaces are characterized as countable spaces with the cofinite topology, and minimal E1 spaces are characterized as E1-closed and semiregular spaces. E0-closed spaces are shown to be precisely the finite discrete spaces.