Критерий существования энергетической функции у регулярного гомеоморфизма 3-сферы

Труды Математического института им. В.А. Стеклова РАН. 2023. Т. 321. С. 45–61.

Barinova M., Grines V., Pochinka O.

C. Conley’s fundamental theorem of the theory of dynamical systems states that every dynamical system, even a nonsmooth one (i.e., a continuous flow or a discrete dynamical system generated by a homeomorphism), admits a continuous Lyapunov function. A Lyapunov function is strictly decreasing along the trajectories of the dynamical system outside the chain recurrent set and is constant on the chain component. A Lyapunov function whose set of critical points coincides with the chain recurrent set of the dynamical system is called an energy function; it has the closest relationship with the dynamics. However, not every dynamical system has an energy function. In particular, according to D. Pixton, even a structurally stable diffeomorphism with nonwandering set consisting of four fixed points may not have a smooth energy function. Our main result in this paper is a criterion for the existence of a continuous
Morse energy function for regular homeomorphisms of the 3-sphere, according to which the existence of such a function is equivalent to the asymptotic triviality of one-dimensional saddle manifolds. The criterion generalizes the results of V. Z. Grines, F. Laudenbach, and O. V. Pochinka for Morse–Smale 3-diffeomorphisms in the case when the ambient manifold is the three-dimensional sphere. In particular, our criterion implies that Pixton’s examples do not
admit even a continuous energy function.

Keywords: энергетическая функция, energy function, дикое вложение, wild embedding, regular homeomorphism, регулярный гомеоморфизм

Publication based on the results of: Теория динамических систем и ее приложения (2023)