This paper introduces fundamental ideas of bitopological dynamical systems. Here, notions transitivity, point pairwise iterated compactness, weakly etc. are introduced. Later, it is shown that under homeomorphism, transitivity implies transitivity. Moreover, homeomorphism; compactness and found to be equivalent. we apply our results in the development process a human embryo from zygote until bi...

The paper initially studies both the s-T3-separation and semi-T3-separation axiom of Khalimsky (K-for brevity) topological spaces. To do this work, first we investigate some properties semi-open semi-closed sets with respect to operations union or intersection further, a homeomorphism, semi-homeomorphism. Next, study various semi-topological K-topological spaces such as simple K-paths. Finally,...

We study those Artin groups which, modulo their centers, are finite index subgroups of the mapping class group of a sphere with at least 5 punctures. In particular, we show that any injective homomorphism between these groups is parameterized by a homeomorphism of a punctured sphere together with a map to the integers. We also give a generating set for the automorphism group of the pure braid g...

A homeomorphism f : X → X of a compactum X with metric d is expansive if there is c > 0 such that if x, y ∈ X and x 6= y, then there is an integer n ∈ Z such that d(f(x), f(y)) > c. In this paper, we prove that if a homeomorphism f : X → X of a continuum X can be lifted to an onto map h : P → P of the pseudoarc P , then f is not expansive. As a corollary, we prove that there are no expansive ho...

This paper is a contribution to the sensitivity analysis of piecewise smooth equations. A piecewise smooth function is a Lipschitzian homeomorphism near a given point if and only if it is coherently oriented and has an invertible B-derivative at this point. We emphasise the role of functions of the type f = g h where g is piecewise smooth and h is smooth and present veriiable conditions which e...

We present a method for estimating the minimal periodic orbit structure, the topological entropy and a fat representative of the homeomorphism associated to the existence of a finite collection of periodic orbits of an orientation preserving homeomorphism of the disk D. The method focuses on the concept of fold and recurrent bogus transition and is more direct than existing techniques. In parti...

We present two examples of nice normal spaces X having the property that for some xed-point free homeomorphism on X its Cech-Stone extension has a xed point. One of the spaces presented here is locally countable, locally compact, separable , normal, countably paracompact and weakly zero-dimensional. The other one is hereditarily normal and strongly zero-dimensional. Our construction of this exa...

An ideal on a set X is a nonempty collection of subsets of X with heredity property which is also closed finite unions. The concept of Γδ : P(X)→ τ defined as follows for every A ∈ X, Γδ(A) = {x ∈ X : there exists a U ∈ τ δ(x) such that U − A ∈ I}, was introduced by Al-Omari and Hatir [1]. In this paper, we introduce and study δ∗-homeomorphism and Γδ-homeomorphism. Also we give some application...

In this work, we show that a strongly continuous semigroup generated by a normal operator N is conjugate to the semigroup generated by the real part of N , provided zero is not an eigenvalue of the real part of N . We also show that in case N satisfies a certain sectorial property, the homeomorphism establishing the conjugacy, as well as its inverse, is locally Hölder continuous. Moreover, in c...

This answers a question apparently first raised by Tietze [T, p. 83]. It was previously known that there were at most two knots with a given complement [CGLS, Corollary 3]. The notion of equivalence of knots can be strengthened by saying that K and K' are isotopic if the above homeomorphism h is isotopic to the identity, or, equivalently, orientation-preserving. The analog of Theorem I holds in...