Energy Function for Diffeomorphisms with Expanding Attractors and Contracting Repellers

In this paper we consider $\Omega$-stable diffeomorphisms defined on smooth closed orientable manifolds of dimension $n\ge 3$, whose all nontrivial basic sets are either expanding attractors or contracting repellers of co-dimension 1. Due to the simple topological structure of the basins of such attractors and repellers, one can make a transition from a given dynamical system with nontrivial basic sets to a regular system which is a homeomorphism with a finite hyperbolic chain-recurrent set. It is well known that not every discrete dynamical systems has energy functions, i.e. a global Lyapunov function whose set of critical points coincides with the chain-recurrent set of the system. Counterexamples were found both among regular diffeomorphisms and among diffeomorphisms with chaotic dynamics. The main result of this paper is the proof of the fact that the topological energy functions for the original diffeomorphism and for its corresponding regular homeomorphism exist or do not exist simultaneously. Thus, numerous results obtained in the field of existence of energy functions for systems with regular dynamics, e.g., for Morse–Smale diffeomorphisms, may be applied to the study of the diffeomorphisms with expanding attractors and contracting repellers of co-dimension 1.