In this paper we investigate the group of homeomorphisms of topological spaces. A subgroup K of the group S(X) of all permutations of a set X is called t-representable on X if there exists a topology T on X such that the group of homeomorphisms of (X, T ) = K. We determine the t-representability of groups generated by a permutation which is a product of disjoint cycles having equal lengths. AMS...

We exploit the techniques developed in [Le] to study N -expansive homeomorphisms on surfaces. We prove that when f is a 2-expansive homeomorphism defined on a compact boundaryless surface M with nonwandering set Ω(f) being the whole of M then f is expansive. This condition on the nonwandering set cannot be relaxed: we present an example of a 2-expansive homeomorphisms on a surface with genus 2 ...

Abstract In this paper weaker and stronger forms of fuzzy b-mappings are introduced and studied using the concept of fuzzy generalized b-closed sets which are named as fagb-continuous, fagb-irresolute and fagb-closed maps. Also fgbT ∗ 1/2 spaces, fuzzy b-locally closed sets, fb-homeomorphisms, fb*-homeomorphisms, fgb-neighbourhood and fgbq-neighbourhood are introduced and studied. Several inter...

Let Homeo(S) represent the full group of homeomorphisms of the unit circle S, and let A represent the set of subgroups of Homeo(S) satisfying the two properties that if G ∈ A then 1) G contains only orientationpreserving homeomorphisms of S and 2) G contains no non-abelian free subgroups. In this article we use classical results about homeomorphisms of the circle and elementary dynamical method...

I. We consider here the groups of homeomorphisms on Euclidean w-space and the «-sphere Sn. Chiefly we will be concerned with the question of whether or not these groups reduce in an homotopy sense to the ordinary orthogonal group acting on these spaces. Such questions are intimately connected with the theory of fibre bundles in which these spaces occur as fibres. We will restrict ourselves to t...

The main result of this paper is a universal finiteness theorem for the set of all small dilatation pseudo-Anosov homeomorphisms φ : S → S, ranging over all surfaces S. More precisely, we consider pseudo-Anosov homeomorphisms φ : S → S with |χ(S)| log(λ(φ)) bounded above by some constant, and we prove that, after puncturing the surfaces at the singular points of the stable foliations, the resul...

Surface Homeomorphisms That Do Not Extend to Any Handlebody and the Johnson Filtration

Notes for the summer school Symbolic dynamics and homeomorphisms of the Cantor set,

My research primarily uses close analysis of the dynamics of a group or semigroup action on a space to discern properties of the group or semigroup. In particular, I am primarily interested in groups of automorphisms of a space and in semigroups of endomorphisms of a space. Supporting this, I have a secondary focus on topological and symbolic dynamics, which itself has lead me to an interest in...