On Existence of an Energy Function for Ω-Stable Surface Diffeomorphisms

If the chain recurrent set of a diffeomorphism f given on a closed n-manifold M-n is hyperbolic (equivalently, f is an Omega-stable) then it coincides with the closure of the periodic points set Per(f) and its chain recurrent components coincide with the basic sets. Due to C. Conley for such a diffeomorphism there is a Lyapunov function which is a continuous function phi : M-n -> R increasing out of the chain recurrent set and a constant on the chain components. But in general a Lyapunov function has critical points out of the chain recurrent set, that is it is not an energy function. In this paper we investigate the problem of the existence of an energy function for diffeomorphisms of a surface. D. Pixton constructed a Morse energy function for Morse-S male 2-diffeomorphisms (all basic sets are trivial). It was proved by M. Barinova, V. Grines and O. Pochinka that every Omega-stable diffeomorphism f : M-2 -> M-2, whose all non-trivial basic sets are attractors or repellers, possesses a smooth energy function which is a Morse function outside non-trivial basic sets. The question about an existence of an energy function for 2-diffeomorphisms with zero-dimensional basic sets was open until now. The main result of this paper is that every Omega-stable 2-diffeomorphism with a zero-dimensional non-trivial basic set without pairs of conjugated points does not possess an energy function.