8 research outputs found
Essential curves in handlebodies and topological contractions
If is a compact set, a {\it topological contraction} is a self-embedding
such that the intersection of the successive images , ,
consists of one point. In dimension 3, we prove that there are smooth
topological contractions of the handlebodies of genus whose image is
essential. Our proof is based on an easy criterion for a simple curve to be
essential in a handlebody
Dynamically ordered energy function for Morse-Smale diffeomorphisms on 3-manifolds
This note deals with arbitrary Morse-Smale diffeomorphisms in dimension 3 and
extends ideas from \cite{GrLaPo}, \cite{GrLaPo1}, where gradient-like case was
considered. We introduce a kind of Morse-Lyapunov function, called dynamically
ordered, which fits well dynamics of diffeomorphism. The paper is devoted to
finding conditions to the existence of such an energy function, that is, a
function whose set of critical points coincides with the non-wandering set of
the considered diffeomorphism. We show that the necessary and sufficient
conditions to the existence of a dynamically ordered energy function reduces to
the type of embedding of one-dimensional attractors and repellers of a given
Morse-Smale diffeomorphism on a closed 3-manifold
Quasi-energy function for diffeomorphisms with wild separatrices
According to Pixton, there are Morse-Smale diffeomorphisms of the 3-sphere
which have no energy function, that is a Lyapunov function whose critical
points are all periodic points of the diffeomorphism. We introduce the concept
of quasi-energy function for a Morse-Smale diffeomorphism as a Lyapunov
function with the least number of critical points and construct a quasi-energy
function for any diffeomorphism from some class of Morse-Smale diffeomorphisms
on the 3-sphere
Self-indexing energy function for Morse-Smale diffeomorphisms on 3-manifolds
The paper is devoted to finding conditions to the existence of a
self-indexing energy function for Morse-Smale diffeomorphisms on a 3-manifold.
These conditions involve how the stable and unstable manifolds of saddle points
are embedded in the ambient manifold. We also show that the existence of a
self-indexing energy function is equivalent to the existence of a Heegaard
splitting of a special type with respect to the considered diffeomorphism