Energy function for 3-diffeomorphisms with one-dimensional surface attractor and repeller

The Lyapunov function was introduced by A.M.Lyapunov to study the stability of equilibrium states of differential equations. A smooth Lyapunov function in which the set of critical p oints coincides with the chain-recurrent set of the system is called the energy function. For example, such functions always exist for flows, whereas examples of cascades are known which do not have an energy function. The rst example of such diffeomorphism was built on the 3-sphere by Pixton [1] in 1977 based on the wild arc of Artin-Fox [2]. The more surprising is the fact that there exists an energy function for cascades with chaotic dynamics, discovered in 2015 by V.Z. Grines, M.K. Barinova and O.V. Pochinka [3,5]. They established the existence of the energy function for surface $\Omega$-stable diffeomorphisms with one-dimensional basis sets and for rough three-dimensional diffeomorphisms with surface basis sets and for rough 3-diffeomorphisms with surface basic sets. $\Omega$-stable 3-diffeomorphisms the non-wandering set of which consists of exactly one surface attractor and one surface rep eller are considered and the existence of the energy function of such diffeomorphisms is proved.
References
[1] D. Pixton. Wild unstable manifolds, Topology. 1977. V. 16. No. 2. P. 167–172.
[2] Artin E., Fox R. Some wild cells and spheres in three-dimensional space. Ann. Math. 1948. V. 49. 979–990.
[3] Grines V.Z., Noskova M.K., Po chinka O.V. Energy function for A-diffeomorphisms of surfaces with one-dimensional nontrivial basis sets, Dynamic Systems. 2015. V. 5. No. 1-2. P. 31–37.
[4] Grines V.Z., Noskova M.K., Pochinka O.V. Construction of the energy function for A-diffeomorphisms with a two-dimensional non-wandering set on 3-manifolds. Proceedings of the Middle Volga mathematical society. 2015. V. 17. No. 3. P. 12-17.
[5] Pochinka O. V., Grines V. Z., Noskova M. K. Construction of the energy function for three-dimensional cascades with a two-dimensional expanding attractor. Proceedings of the Moscow Mathematical Society. 2015. V. 76. No. 2. P. 271–286.