Energy function for $\Omega$-stable flows without limit cycles on surfaces

The paper is devoted to the study of the class of $\Omega$-stable flows without limit cycles on surfaces, i.e. flows on surfaces with non-wandering set consisting of a finite number of hyperbolic fixed points. This class is a generalization of the class of gradient-like flows, differing by forbiddance of saddle points connected by separatrices. The results of the work are the proof of the existence of a Morse energy function for any flow from the considered class and the construction of such a function for an arbitrary flow of the class. Since the results are a generalization of the corresponding results of K. Meyer for Morse-Smale flows and, in particular, for gradient-like flows, the methods for constructing the energy function for the case of this article are a further development of the methods used by K. Meyer, taking in sense the specifics of $\Omega$-stable flows having a more complex structure than gradient-like flows due to the presence of the so-called “chains” of saddle points connected by their separatrices.