Homeomorphisms of the Plane

The Automorphism Group on the Riemann Sphere

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...

A stable/unstable manifold theorem for local homeomorphisms of the plane

We use a notion (introduced in Topology 41 (2002), 1119–1212), which is stronger than the concept of filtration pair, to prove a stable/unstable manifold general theorem for local homeomorphisms of the plane in a neighborhood of an isolated fixed point.

Mappings of the Sierpinski Curve onto Itself

Given two points p and q of the Sierpinski universal plane curve S, necessary and/ or sufficient conditions are discussed in the paper under which there is a mapping I of S onto itself such that I(P) = q and I belongs to one of the following: homeomorphisms, local homeomorphisms, local homeomorphisms in the large sense, open, simple or monotone mappings.

Homeomorphisms of the Mandelbrot Set

On subsets of the Mandelbrot set, EM ⊂M, homeomorphisms are constructed by quasi-conformal surgery. When the dynamics of quadratic polynomials is changed piecewise by a combinatorial construction, a general theorem yields the corresponding homeomorphism h : EM → EM in the parameter plane. Each h has two fixed points in EM , and a countable family of mutually homeomorphic fundamental domains. Po...

On bD-Sets and Associated Separation Axioms

Smooth Projective Planes

Using symplectic topology and the Radon transform, we prove that smooth 4-dimensional projective planes are diffeomorphic to CP. We define the notion of a plane curve in a smooth projective plane, show that plane curves in high dimensional regular planes are lines, prove that homeomorphisms preserving plane curves are smooth collineations, and prove a variety of results analogous to the theory ...