Characteristic space of orbits of Morse–Smale diffeomorphisms on surfaces

The classical approach to the study of dynamical systems consists in representing the dynamics of the system in the form of a “source-sink”, which means identifying attractor-repeller pairs 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.