This paper is concerned with the problem of approximating a homeomorphism by piecewise affine homeomorphisms. The main result is as follows: every homeomorphism from a planar domain with a polygonal boundary to R that is globally Hölder continuous of exponent α ∈ (0, 1], and whose inverse is also globally Hölder continuous of exponent α can be approximated in the Hölder norm of exponent β by pi...

We obtain a homotopy splitting of the forget control map for controlled homeomorphisms over closed manifolds of nonpositive curvature.

We introduce the notions of weakly and strongly positively expansive (wPE and sPE, respectively) discrete dynamical systems. Both are topological generalizations of the well-known metric notion of positive expansiveness (PE). We prove that the three notions are identical on compact metrizable spaces, but not on noncompact spaces. We investigate properties of PE, wPE, and sPE dynamical systems a...

In order to study the geometries of a hyperbolic plane, it is necessary to understand the set of transformations that map from the space to itself. This paper shows that in the Poincaré half-plane model, this group of transformations is the general Möbius group. This is done by showing that the general Möbius group is a group of homeomorphisms that map circles to circles on the extended complex...

Guba and Sapir asked, in their joint paper [3], if the simultaneous conjugacy problem was solvable in Diagram Groups or, at least, for Thompson’s group F . We give an elementary proof for the solution of the latter question. This relies purely on the description of F as the group of piecewise linear orientationpreserving homeomorphisms of the unit interval with dyadic breakpoints and slopes tha...