Unstable Homeomorphisms

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.

Pseudo-anosov Homeomorphisms with Quadratic Expansion

We show that if f : M →M is a pseudo-Anosov homeomorphism on an orientable surface with oriented unstable manifolds and a quadratic expanding factor, then there is a hyperbolic toral automorphism on T2 and a map h : M → T2 such that h is a semi-conjugacy and (M, h) is a branched covering space of T2. We also give another characterization of pseudo-Anosov homeomorphisms with quadratic expansion ...

Striped structures of stable and unstable sets of expansive homeomorphisms and a theorem of K. Kuratowski on independent sets

We investigate striped structures of stable and unstable sets of expansive homeomorphisms and continuum-wise expansive homeomorphisms. The following theorem is proved: if f : X → X is an expansive homeomorphism of a compact metric space X with dimX > 0, then the decompositions {W (x) | x ∈ X} and {W (x) | x ∈ X} of X into stable and unstable sets of f respectively are uncountable, and moreover ...

Homeomorphisms of One-dimensional Inverse Limits with Applications to Substitution Tilings, Unstable Manifolds, and Tent Maps

Suppose that f and g are Markov surjections, each defined on a wedge of circles, each fixing the branch point and having the branch point as the only critical value. We show that if the points in the inverse limit spaces associated with f and g corresponding to the branch point are distinguished then these inverse limit spaces are homeomorphic if and only if the substitutions associated with f ...

Periodic Orbits and Homoclinic Loops for Surface Homeomorphisms

Let p be a saddle fixed point for an orientation-preserving surface diffeomorphism f , admitting a homoclinic point p. Let V be an open 2-cell bounded by a simple loop formed by two arcs joining p to p, lying respectively in the stable and unstable curves at p. It is shown that f |V has fixed point index ρ ∈ {1, 2} where ρ depends only on the geometry of V near p. More generally, for every n ≥ ...

On bD-Sets and Associated Separation Axioms