1.1. The main results. Let T denote the unit circle and D the unit disc. Suppose that f : T → T is a homeomorphism. Let f̂ : D → D be a homeomorphism too. We say that f̂ extends f if f̂ and f agree on T. All mappings in this paper are sense preserving (see the remark at the end of introduction). Definition 1.1. We say that a homeomorphism f : T → T is K-quasisymmetric if there exists a K-quasiconf...

Let f : M → N be a homotopy equivalence between closed negatively curved manifolds. The fundamental existence results of Eells and Sampson [5] and uniqueness of Hartmann [15] and Al’ber [1] grant the existence of a unique harmonic map h homotopic to f . Based on the enormous success of the harmonic map technique Lawson and Yau conjectured that the harmonic map h should be a diffeomorphism. This...

Given uniformly homeomorphic metric spaces $$X$$ and $$Y$$ , it is proven that hyperspaces $$C(X)$$ $$C(Y)$$ are homeomorphic, where denotes the collection of all nonempty closed subsets endowed with Hausdorff distance. Gerald Beer has proved hyperspace an Atsuji space when either compact or discrete. An a generalization as well discrete spaces. In this paper, we investigate space, class subspa...

Abstract. According to Thurston’s classification, the set of homotopy classes orientation-preserving homeomorphisms orientable surfaces is split into four disjoint subsets. A class from each subset characterized by existence a homeomorphism called canonical form, namely: periodic homeomorphism, reducible nonperiodic algebraically finite order, that not an and pseudo-Anosov homeomorphism. forms ...

Let S : E → R where TwE = R for all w ∈ E, be a C-differential system on an n-dimensional Euclidean w-space E, which naturally gives rise to a flow φ : (t, w) 7→ t·w on E, and let Λ be a φ-invariant closed subset containing no any singularities of S. If Λ is compact and hyperbolic, then Anosov’s theorem asserts that S is structurally stable on Λ in the sense of topological equivalence; that is,...

This paper is concerned with homeomorphisms of Euclidean spaces onto themselves, with bounded orbits. The following results are obtained. (1) A homeomorphism of E onto itself has both bounded orbits and an equicontinuous family of iterates iff it is a conjugate of either a rotation or a reflection; (2) An example of Bing is modified to produce a fixed point free, orientation preserving homeomor...