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Noninvertible minimal maps

Sergiĭ Kolyada, L'ubomír Snoha, Sergeĭ Trofimchuk (2001)

Fundamenta Mathematicae

For a discrete dynamical system given by a compact Hausdorff space X and a continuous selfmap f of X the connection between minimality, invertibility and openness of f is investigated. It is shown that any minimal map is feebly open, i.e., sends open sets to sets with nonempty interiors (and if it is open then it is a homeomorphism). Further, it is shown that if f is minimal and A ⊆ X then both f(A) and f - 1 ( A ) share with A those topological properties which describe how large a set is. Using these results...

Nonlinear contractive conditions: A comparison and related problems

Jacek Jachymski, Izabela Jóźwik (2007)

Banach Center Publications

We establish five theorems giving lists of nonlinear contractive conditions which turn out to be mutually equivalent. We derive them from some general lemmas concerning subsets of the plane which may be applied both in the single- or set-valued case as well as for a family of mappings. A separation theorem for concave functions is proved as an auxiliary result. Also, we discuss briefly the following problems for several classes of contractions: stability of procedure of successive approximations,...

Non-locally compact Polish groups and two-sided translates of open sets

Maciej Malicki (2008)

Fundamenta Mathematicae

This paper is devoted to the following question. Suppose that a Polish group G has the property that some non-empty open subset U is covered by finitely many two-sided translates of every other non-empty open subset of G. Is then G necessarily locally compact? Polish groups which do not have the above property are called strongly non-locally compact. We characterize strongly non-locally compact Polish subgroups of S ∞ in terms of group actions, and prove that certain natural classes of non-locally...

Nonmeasurable algebraic sums of sets of reals

Marcin Kysiak (2005)

Colloquium Mathematicae

We present a theorem which generalizes some known theorems on the existence of nonmeasurable (in various senses) sets of the form X+Y. Some additional related questions concerning measure, category and the algebra of Borel sets are also studied.

Nonmetrizable topological dynamical characterization of central sets

Hong-Ting Shi, Hong-Wei Yang (1996)

Fundamenta Mathematicae

Without the restriction of metrizability, topological dynamical systems ( X , ⟨ T s ⟩ s ∈ G ) are defined and uniform recurrence and proximality are studied. Some well known results are generalized and some new results are obtained. In particular, a topological dynamical characterization of central sets in an arbitrary semigroup (G,+) is given and shown to be equivalent to the usual algebraic characterization.

Non-self mappings in modular spaces and common fixed point theorems

Abdolrahman Razani, Valdimir Rakočević, Zahraa Goodarzi (2010)

Open Mathematics

The aim of this paper, is to introduce the convex structure (specially, Takahashi convex structure) on modular spaces. Moreover, we are interested in proving some common fixed point theorems for non-self mappings in modular space.

Non-transitive points and porosity

T. K. Subrahmonian Moothathu (2013)

Colloquium Mathematicae

We establish that for a fairly general class of topologically transitive dynamical systems, the set of non-transitive points is very small when the rate of transitivity is very high. The notion of smallness that we consider here is that of σ-porosity, and in particular we show that the set of non-transitive points is σ-porous for any subshift that is a factor of a transitive subshift of finite type, and for the tent map of [0,1]. The result extends to some finite-to-one factor systems. We also show...

Non-universal families of separable Banach spaces

Ondřej Kurka (2016)

Studia Mathematica

We prove that if 𝓒 is a family of separable Banach spaces which is analytic with respect to the Effros Borel structure and no X ∈ 𝓒 is isometrically universal for all separable Banach spaces, then there exists a separable Banach space with a monotone Schauder basis which is isometrically universal for 𝓒 but not for all separable Banach spaces. We also establish an analogous result for the class of strictly convex spaces.

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