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In this paper we give a lower estimate for the number of critical points ofthe Lyapunov function for Pixton diffeomorphisms (i.e. Morse-Smalediffeomorphisms in dimension 3 whose chain recurrent set consists of fourpoints: one source, one saddle and two sinks). Ch. Bonatti and V. Grines provedthat the class of topological equivalence of such diffeomorphism $f$ iscompletely defined by the equivalency class of the Hopf knot $L_{f}$ that isthe knot in the generating class of the fundamental group of the manifold$\mathbb S^2\times\mathbb S^1$. They also proved that there are infinitely manysuch classes and that any Hopf knot can be realized by a Pixton diffeomorphism.D.~Pixton proved that diffeomorphisms defined by the standard Hopf knot$L_0=\{s\}\times \mathbb S^1$ have an energy function (Lyapunov function) whoseset of critical points coincide with the chain recurrent set whereas the set ofcritical points of any Lyapunov function for Pixton diffeomorphism withnontrivial (i.e. non equivalent to the standard) Hopf knot is strictly largerthan the chain recurrent set of the diffeomorphism. The Lyapunov function forPixton diffeomorphism with minimal number of critical points is called thequasi-energy function. In this paper we construct a quasi-energy function forPixton diffeomorphisms defined by a generalized Mazur knot.

A quasi-energy function for Pixton diffeomorphisms defined by generalized Mazur knots