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T -neighborhood groups.

Ahsanullah, T.M.G., Al-Thukair, Fawzi (2004)

International Journal of Mathematics and Mathematical Sciences

Tests à la Hurewicz dans le plan

Dominique Lecomte (1998)

Fundamenta Mathematicae

Nous donnons, pour une certaine catégorie de boréliens d'un produit de deux espaces polonais, comprenant les boréliens à coupes dénombrables, une caractérisation du type "test d'Hurewicz" de ceux ne pouvant pas être rendus différence transfinie d'ouverts par changement des deux topologies polonaises.

The Abel equation and total solvability of linear functional equations

G. Belitskii, Yu. Lyubich (1998)

Studia Mathematica

We investigate the solvability in continuous functions of the Abel equation φ(Fx) - φ(x) = 1 where F is a given continuous mapping of a topological space X. This property depends on the dynamics generated by F. The solvability of all linear equations P(x)ψ(Fx) + Q(x)ψ(x) = γ(x) follows from solvability of the Abel equation in case F is a homeomorphism. If F is noninvertible but X is locally compact then such a total solvability is determined by the same property of the cohomological equation φ(Fx)...

The Baire property in remainders of topological groups and other results

Aleksander V. Arhangel'skii (2009)

Commentationes Mathematicae Universitatis Carolinae

It is established that a remainder of a non-locally compact topological group G has the Baire property if and only if the space G is not Čech-complete. We also show that if G is a non-locally compact topological group of countable tightness, then either G is submetrizable, or G is the Čech-Stone remainder of an arbitrary remainder Y of G . It follows that if G and H are non-submetrizable topological groups of countable tightness such that some remainders of G and H are homeomorphic, then the spaces...

The Banach contraction mapping principle and cohomology

Ludvík Janoš (2000)

Commentationes Mathematicae Universitatis Carolinae

By a dynamical system ( X , T ) we mean the action of the semigroup ( ℤ + , + ) on a metrizable topological space X induced by a continuous selfmap T : X → X . Let M ( X ) denote the set of all compatible metrics on the space X . Our main objective is to show that a selfmap T of a compact space X is a Banach contraction relative to some d 1 ∈ M ( X ) if and only if there exists some d 2 ∈ M ( X ) which, regarded as a 1 -cocycle of the system ( X , T ) × ( X , T ) , is a coboundary.

The behaviour of the nonwandering set of a piecewise monotonic interval map under small perturbations

Peter Raith (1997)

Mathematica Bohemica

In this paper piecewise monotonic maps T [ 0 , 1 ] → [ 0 , 1 ] are considered. Let Q be a finite union of open intervals, and consider the set R ( Q ) of all points whose orbits omit Q . The influence of small perturbations of the endpoints of the intervals in Q on the dynamical system ( R ( Q ) , T ) is investigated. The decomposition of the nonwandering set into maximal topologically transitive subsets behaves very unstably. Nonetheless, it is shown that a maximal topologically transitive subset cannot be completely destroyed by arbitrary...

The Bohr compactification, modulo a metrizable subgroup

W. Comfort, F. Trigos-Arrieta, S. Wu (1993)

Fundamenta Mathematicae

The authors prove the following result, which generalizes a well-known theorem of I. Glicksberg [G]: If G is a locally compact Abelian group with Bohr compactification bG, and if N is a closed metrizable subgroup of bG, then every A ⊆ G satisfies: A·(N ∩ G) is compact in G if and only if {aN:a ∈ A} is compact in bG/N. Examples are given to show: (a) the asserted equivalence can fail in the absence of the metrizability hypothesis, even when N ∩ G = {1}; (b) the asserted equivalence can hold for suitable...

The branch locus for one-dimensional Pisot tiling spaces

Marcy Barge, Beverly Diamond, Richard Swanson (2009)

Fundamenta Mathematicae

If φ is a Pisot substitution of degree d, then the inflation and substitution homeomorphism Φ on the tiling space Φ factors via geometric realization onto a d-dimensional solenoid. Under this realization, the collection of Φ-periodic asymptotic tilings corresponds to a finite set that projects onto the branch locus in a d-torus. We prove that if two such tiling spaces are homeomorphic, then the resulting branch loci are the same up to the action of certain affine maps on the torus.

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