Measure-theoretical sensitivity and equicontinuity

For an invariant measure μ in a topological dynamics, notions of μ-sensitivity, μ-complexity and μ-equicontinuity are introduced and investigated. It turns out that μ-sensitivity defined here is equivalent to pairwise sensitivity defined by Cadre and Jacob. For an ergodic μ, μ-equicontinuity, no μ-complexity pair and non-μ-sensitivity are equivalent, which implies minimality and equicontinuity when restricted to the support. Moreover, the notion of μ-sensitive set is introduced, it is shown that a transitive system with an ergodic measure of full support has zero topological entropy if there is no uncountable μ-sensitive set, and a non-minimal transitive system with dense minimal points has infinite sequence entropy for some sequence.     

Topological continuous convergence

Let X be a limit space, Y a topological space. We show that c(X,Y), the limitierung of continuous convergence on LIM(X,Y), is topological whenever X is basic locally compact. For regular Y, local compactness of X is sufficient. In both cases, c(X,Y) coincides with the compact-open topology. If X satisfies a certain regularity condition, the fact that c(X,Y) is topological implies, conversely, that X is (basic) locally compact.The author would like to thank S. Weck for some inspiring discussions.     

On fuzzy almost continuous convergence in fuzzy function spaces

In this paper, we study the fuzzy almost continuous convergence of fuzzy nets on the set FAC(X, Y) of all fuzzy almost continuous functions of a fuzzy topological space X into another Y. Also, we introduce the notions of fuzzy splitting and fuzzy jointly continuous topologies on the set FAC(X, Y) and study some of its basic properties.     

On the almost monotone convergence of sequences of continuous functions

A sequence (f _n ) _n of functions f _n : X → ℝ almost decreases (increases) to a function f: X → ℝ if it pointwise converges to f and for each point x ∈ X there is a positive integer n(x) such that f _n+1(x) ≤ f _n (x) (f _n+1(x) ≥ f _n (x)) for n ≥ n(x). In this article I investigate this convergence in some families of continuous functions.     

Residual properties and almost equicontinuity

A propertyP of a compact dynamical system (X,f) is called a residual property if it is inherited by factors, almost one-to-one lifts and surjective inverse limits. Many transitivity properties are residual. Weak disjointness from all propertyP systems is a residual property providedP is a residual property stronger than transitivity. Here two systems are weakly disjoint when their product is transitive. Our main result says that for an almost equicontinuous system (X, f) with associated monothetic group Λ, (X, f) is weakly disjoint from allP systems iff the onlyP systems upon which Λ acts are trivial. We use this to prove that every monothetic group has an action which is weak mixing and topologically ergodic.     

Graphical convergence of continuous functions

Let X and Y be metrizable spaces. We show that convergence of a net of continuous functions 〈f _λ 〉 to a continuous function f in the graph topology for C(X,Y) is equivalent to the uniform convergence of the net of associated distance functionals for the graphs with respect to each compatible metric on X×Y. Remarkably, no weaker convergence results if uniform convergence is replaced by pointwise convergence in the last statement.     

Constructive notions of equicontinuity

In the informal setting of Bishop-style constructive reverse mathematics we discuss the connection between the antithesis of Specker’s theorem, Ishihara’s principle BD-N, and various types of equicontinuity. In particular, we prove that the implication from pointwise equicontinuity to uniform sequential equicontinuity is equivalent to the antithesis of Specker’s theorem; and that, for a family of functions on a separable metric space, the implication from uniform sequential equicontinuity to uniform equicontinuity is equivalent to BD-N.      

Ideal convergence of continuous functions

For a given ideal I⊆P(ω), IC(I) denotes the class of separable metric spaces X such that whenever is a sequence of continuous functions convergent to zero with respect to the ideal I then there exists a set of integers {m₀<m₁<?} from the dual filter F(I) such that limi→∞fmi(x)=0 for all x∈X. We prove that for the most interesting ideals I, IC(I) contains only singular spaces. For example, if I=Id is the asymptotic density zero ideal, all IC(Id) spaces are perfectly meager while if I=Ib is the bounded ideal then IC(Ib) spaces are σ-sets.     

Observations about equicontinuity and related concepts

The concepts of equicontinuity, even continuity, topological equicontinuity and the newly defined concepts of compact equicontinuity and compact topological equicontinuity are compared. It is shown that for a set of group homomorphisms from a semitopological group to a topological group all these notions of equicontinuities coincide. It is shown also that an infinite-dimensional normed space endowed with its weak topology is an example of a space which does not satisfy the Ascoli theorem in Noble's sense.     

On existence,uniqueness and convergence of multi-valued stochastic diferential equations driven by continuous semimartingales

In this paper we study the existence and uniqueness of solutions of multi-valued stochastic diferential equations driven by continuous semimartingales when the coefcients are stochastically Lipschitz continuous.We also show the convergence results when the random coefcients or the diferentials converge.     

On existence,uniqueness and convergence of multi-valued stochastic differential equations driven by continuous semimartingales

In this paper we study the existence and uniqueness of solutions of multi-valued stochastic differential equations driven by continuous semimartingales when the coefficients are stochastically Lipschitz continuous. We also show the convergence results when the random coefficients or the differentials converge.     

Even continuity and topological equicontinuity in topologized semigroups

A topologized semigroup X having an evenly continuous resp., topologically equicontinuous, family RX of right translations is investigated. It is shown that: (1) every left semitopological semigroup X with an evenly continuous family RX is a topological semigroup, (2) a semitopological group X is a paratopological group if and only if the family RX is evenly continuous and (3) a semitopological group X is a topological group if and only if the family RX is topologically equicontinuous. In particular, we get that for any paratopological group X which is not a topological group, the family RX provides an example of a transitive group of homeomorphisms of X that is evenly continuous and not topologically equicontinuous. The last conclusion answers negatively a question posed by H.L. Royden.     

Weak stability and weak discrete convergence of continuous mappings

Summary The present paper establishes new equivalence theorems concerning weak discrete convergence, weak stability and consistency for sequences of continuous mappings as well as associated estimates. The method of increasing precisions yields an algorithm for the numerical solution of weakly discretely convergent problems. The general results are illustrated by an application to double sequences and difference quotients.