Characteristic space of orbits of Morse-Smale diffeomorphisms on surfaces

TL;DR

This paper investigates the conditions under which the characteristic space of orbits for Morse-Smale diffeomorphisms on surfaces is connected, revealing that certain non-orientability or heteroclinic points lead to disconnected spaces.

Contribution

It constructively demonstrates that violating specific conditions results in Morse-Smale diffeomorphisms on surfaces with disconnected characteristic spaces of orbits.

Findings

Connected characteristic space exists for orientation-preserving gradient-like diffeomorphisms on orientable surfaces.

Violating conditions like non-orientability or presence of heteroclinic points causes disconnection.

The work provides a constructive approach to understanding the topology of orbit spaces in Morse-Smale systems.

Abstract

The classical approach to the study of dynamical systems consists in representing the dynamics of the system in the form of a "source-sink", that means identifying an attractor-repeller pair, which are attractor-repellent sets for all other trajectories of the system. If there is a way to choose this pair so that the space orbits in its complement (the characteristic space of orbits) is connected, this creates prerequisites for finding complete topological invariants of the dynamical system. It is known that such a pair always exists for arbitrary Morse-Smale diffeomorphisms given on any manifolds of dimension $n \geqslant 3$. Whereas for $n=2$ the existence of a connected characteristic space has been proved only for orientation-preserving gradient-like (without heteroclinic points) diffeomorphisms defined on an orientable surface. In the present work, it is constructively shown that the violation of at least one of the above conditions (absence of heteroclinic points, orientability of a surface, orientability of a diffeomorphism) leads to the existence of Morse-Smale diffeomorphisms on surfaces that do not have a connected characteristic space of orbits.