The existence connected characteristic space at the gradient-like diffeomorphisms of surfaces

In this paper we consider the class $G$ of orientation-preserving gradient-like diffeomorphisms $f$ defined on a smooth oriented closed surfaces $M^2$. Author establishes that for every such diffeomorphism there is a dual pair attractor-repeller $A_f,R_f$ that have topological dimension not greater than $1$ and the orbit space in their supplement $V_f$ is homeomorphic to the two-dimensional torus. The immediate consequence of this result is the same period of saddle separatrices of all diffeomorphisms $f\in G$. A lot of classification results for structurally stable dynamical systems with a non-wandering set consisting of a finite number of orbits (Morse-Smale systems) is based on the possibility of such representation for the system dynamics in the “source-sink” form. For example, for systems in dimension three there always exists a connected characteristic space associated with the choice of a one-dimensional dual attractor-repeller pair. In dimension two this is not true even in the gradient-like case. However, in this paper it is shown that there exists a one-dimensional dual pair with connected characteristic orbit space.