Criterion for the existence of a connected characteristic space of orbits in a gradient-like diffeomorphism of a surface

The classical approach to the study of dynamical systems consists in representing the dynamics of the system in the “source–sink” form, that is, by singling out a dual attractor–repeller pair, consisting of the attracting and repelling sets for all other trajectories of the system. A choice of an attractor-repeller dual pair so that the space of orbits in their complement (the characteristic space of orbits) is connected paves the way for finding complete topological invariants of the dynamical system. In this way, in particular, several classification results for Morse–Smale systems were obtained. Thus, a complete topological classification of Morse–Smale 3-diffeomorphisms is essentially based on the existence of a connected characteristic space of orbits associated with the choice of a one-dimensional dual attractor–repeller pair. For Morse–Smale diffeomorphisms with heteroclinic points on surfaces, there are examples in which the characteristic spaces of orbits are disconnected in all cases. In this paper, we prove a criterion for the existence of a connected characteristic space of orbits for gradient-like (without heteroclinic points) diffeomorphisms on surfaces. This result implies, in particular, that any orientation-preserving diffeomorphism admits a connected characteristic space. For an orientable surface of any kind, we also construct an orientation-changing gradient-like diffeomorphism that does not have a connected characteristic space. On any non-orientable surface of any kind, we also construct a gradient-like diffeomorphism which does not admit a connected characteristic space.