Abstract
In this paper, the problem of the existence of an energy function for discrete dynamical systems defined by structurally stable diffeomorphisms is studied. For the considered class of dynamical systems, such a function is defined as a continuous function that decreases along the orbits of the system outside the non-wandering set and remains constant on the non-wandering orbits. It is currently known that if the non-wandering set of a structurally stable surface diffeomorphism does not contain nontrivial saddle basic sets, then it possesses an energy function. However, if such a set exists and does not contain a pair of conjugated points, then the diffeomorphism does not admit such a function. The so-called serpentine geometric type of Markov partitioning is introduced, which generalizes the geometric type for the classical Smale horseshoe, and diffeomorphisms with nontrivial saddle basic sets admitting Markov partitions of this type are considered. Such sets contain pairs of conjugated points, accordingly, their study extends the class of diffeomorphisms for which the question of the existence of an energy function has been resolved. The main result of this paper is the proof that an orientation-preserving structurally stable 2-diffeomorphism with a basic set admitting a Markov partition of serpentine geometric type does not possess an energy function.