Nonexistence of energy function for surface diffeomorphisms with horseshoe basic sets

Abstract

The Lyapunov function is a powerful tool for studying dynamical systems. In particular, the existence of such a function for any dynamical system given on compact manifold, established by C. Conley, is the basis for proving the Ω-stability criterion of dynamical systems. Moreover, the Lyapunov function, whose set of critical points of aligns with the chain-recurrent set of the system, which called an energy function, qualitatively determines the dynamics of the system. For continuous-time systems, which models dissipative processes, such a function naturally arises as an energy of the system. The existence of an energy function for arbitrary flows on compact manifolds is known. Examples of Morse–Smale diffeomorphism without energy function exist on n-dimensional manifolds starting with n>2. For systems with chaotic dynamics on surfaces, the existence of energy functions has also been established in cases where the dimension of non-trivial basic sets is greater than or equal to 1. When the chain-recurrent set of a diffeomorphism contains at least one zero-dimensional non-trivial basic set, the question of the existence of the energy function is still open. To date, it is known that the answer is negative if the zero-dimensional non-trivial basic set does not contain pairs of conjugated points. The present paper is devoted to the question of existence of an energy function for systems having a basic set with pairs of conjugated points. The primary outcome of this study is that there is no energy function for a dynamical system containing a Smale horseshoe as a basic set.